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ANL-78-XX-95

Energy Level Structure and Transition Probabilities

in the Spectra of the Trivalent Lanthanides in LaF3

M. T. Carnall

Chemistry Division

jonne National Laboratory

Hannah Crosswhite and H. M. Crosswhite

Department of Physics

The Johns Hopkins University

.. -. -wd" "e -t" A . 0mwd OM

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a ntuuTBua U snM wuVa NT wiajm

CONTENTS

Page

I. Introduction 2

II. Physical and Crystallographic Properties of LaF3 5

Fig. 1. LaF3 Structure Viewed Down the C-axis 6

III. Treatment of Experimental Data 8

Fig. 2. Crystal-field Parameters for Ln3+:LaCl3 and Ln3+:LaF3 13

IV. Energy Level Correlations - Survey of Experimental Data 14

1. f2(f12 ) 14

2. f3(f11) 14

3. f4(f10 ) 16

4. f5(f9) 16

5. f6(f8) 17

6. f7 17

V. Th' retical Interpretation of Excited State Relaxation 19

A. Theoretical Treatment of Absorption Spectra 19

1. General Concepts 19

2. Induced Electric-Dipole Transitions 21

3. Magnetic Dipole Transitions 23

4. Comparison of Calculated and Observed TransitionProbabilities 24

B. Relaxation of Excited States 24

1. General Considerations 24

2. Radiative Relaxation 25

Fig. 3. Energy-level diagrams of Ln3+:LaCl3 27

3. Non-radiative Relaxation 28

4. Comparison of Computed Excited State Lifetimes withThose Observed Experimentally 30

5. Connents on the Use of the Tables 31

Acknowledgements 33

References 34

APPENDIX I

Fig. 1. Variation of the Slater Parameter Fk with Lanthanide Atomic Number.

Fig. 2. Variation of the Spin-Orbit Constant with Lanthanide Atomic Number.

Table 1. Parameters for Ln3+:LaCl3.

Table 2. Relativistic Hartree-Fock Integrals for 4fN(IV).

APPENDIX II

Table 1. Atomic Parameters for Ln3+:LaF3.

Table 2. Crystal-Field Parameters for Ln3+:LaF3.

APPENDIX III

Table 1. Experimental and Calculated Energy Levels for Pr3+:LaF3.

Table 2. Matrix Elements of [U ]2 for Selected 4f2-transitions.

APPENDIX IV

Table 1. Experimental and Calculated Energy Levels for Nd3+:LaF3.

Table 2. Matrix Elements of [U(X)]2 for Selected 4f3-transitions.

Figs. 1-21. The Absorption Spectrum of Nd3+:LaF3 in the Region ".4000-50000 cm1

recorded at '4 K.

APPENDIX V

Tale 1. Computed Energy Levels for Pm3+:LaF3.

Table 2. Matrix Elements of [U 12 for Selected 4f4-transitions.

APPENDIX VI

Table 1. Experimental and Calculated Energy Levels for Sm3+:LaF3.

Table 2. Matrix Elements of [u)]2 for Selected 4f5-transitions.

APPENDIX JII

Table 1. Computed Energy Levels for Eu3+:LaF3.

Table 2. Matrix Elements of [U ]2 for Selected 4f6-transitions.

APPENDIX VIII

Table 1. Experimental and Calculated Energy Levels for Gd3+:LaF3'

Table 2. Matrix Elements of [U ]2 for Selected 4f7-transitions.APPENDIX IX

Table 1. Experimental and Calculated Energy Levels for Tb3+:LaF 3.

Table 2. Matrix Elements of [U']2 for Selected 4f8-transitions.

APPENDIX X

Table 1. Experimental and Calculated Energy Levels for Dy3+:LaF3.

Table 2. Matrix Elements of [U(A) 12 for Selected 4f9-transitions.APPENDIX XI

Table 1. Experimental and Calculated Energy Levels for Ho3+:LaF3.

Table 2. Matrix Elements of [U']2 for Selected 4f10-transitionsAPPENDIX XII

Table 1. Experimental and Calculated Energy Levels for Er3+:LaF3.

Table 2. Matrix Elements of [U ]2 for Selected 4f11-transitions

APPENDIX XIII

Table 1. Experimental and Calculated Energy Levels for Tm3+:LaF3

Table 2. Matrix Elements of [U 12 for Selected 4f12-transitions

APPENDIX XIV

Table 1. Calculated Values of A for Ln3+:LaF 3.

Table 2. Observed and Calculated Excited State Lifetimes for Ln 3+

Table 3. Partial Lifetimes and Branching Ratios in the RelaxationExcited States in Tb3+:LaF 3.

:LaF3

of

i.

1.

."

FOREWORD

The spectroscopic investigations of lanthanide doped LaF3 crystals

and their theoretical interpretation reported here are the result of an

extensive joint effort of the Argonne National Laboratory and The Johns

Hopkins University over the last decade. It had its roots in similar

studies at both laboratories of other crystal hosts, particularly LaCl 3,

for which a summary is also given in an appendix to this report. This

work was supported at Argonne by the United States Atomic Energy

Commission and the United States Energy Research and Development

Administration, and at Hopkins by the United States Army Research Office

and the National Science Foundation. An identical report is being

issued under separate cover by each institution.

-2-

I. Introduction

The low-temperature absorption spectra of trivalent lanthanide compounds

reveal the sharp-line structure characteristic of transitions between states

within the 4fN-configuration, induced by the ionic environment. These transi-

tions can be directly represented as upper-state energy levels. In some cases,

notably ions in LaCl3 host crystals, the empirical levels have been interpreted

in terms of a model which exhibits in detail the structure of the 4fN elec-

tronic configuration itself. The atomic parameters derived from these studies

vary slowly and systematically across the lanthanide series and in fact are

depressed from corresponding values derived from atomic and ionic vapor

spectra.

G. H. Dieke has given a definitive review of the spectra of lanthanides

in crystals including a comprehensive summary of the data in LaCl3 host crys-

tals, Dieke (1968). The early interpretive papers concentrated on analysis of

the crystal-field parameters. More recently, the emphasis has been on devel-

oping models which include structural details of the 4fN configurations them-

selves with simultaneous diagonalization of the complete Hamiltonian rather

than a perturbation-theory treatment. The most recent analyses have been

given in a series of papers, Crosswhite et al. (1976), Carnall et al. (1976),

Crosswhite et al. (1977). These include effective two-body and three-body

operators approximating the electrostatic effects of configuration-mixing

interactions, plus two-body double-vector effective operators which allow for

the variation of spin-orbit interactions with spectroscopic term and for the

spin-other-orbit corrections for relativistic effects. The present analysis

of spectroscopic data for the lanthanides in single-crystal LaF3 employs the

most recent model.

-3-

As a crystal matrix in which to study the absorption spectra of lantha-

nide ions, LaF3 has several advantageous features. In addition to being a

transparent host over a wide region of the spectrum, it is chemically inert

so that the crystals can be handled in air. Thus even though the site sym-

metry of the metal ion is low, the commercial availability of the crystals in

a wide range of dopings has encouraged extensive experimental studies.

We report here two types of correlation with experimental results. For

even-f-electron systems we have computed a center of gravity based on the en-

ergies of the observed states, and calculated optimized sets of atomic energy

level parameters. For odd-electron systems we have performed complete crystal-

field calculations in which the parameters of both the atomic and crystal-

field parts of the interaction have been adjusted to experimental data. Trends

in the free-ion parameters over the series have been examined and in some cases

further restrictions on their values are set, based on observed parametric reg-

ularities across the periodic series. The crystal field parameters for the

odd-f-electron ions have also been compared and regularities over the series

are noted. The end result of this effort is a set of eigenvectors for all of

the ionic states in each configuration. We then turn to the interpretation of

intensities of absorption bands and develop the basis for computing transition

probabilities using a semi-empirical theory of induced electric dipole transi-

tions. Finally, we apply the results of the analysis of absorption spectra at

room temperature to the related process of excited state relaxation, and pro-

vide the basis for computing the radiative lifetime for any given state in any

member of the series.

Spectroscopic results for all lanthanides doped into LaF3 (Ln 3+:LaF 3)

except Pm3+ and Eu3+ are reported. Since the crystals were obtained from

commercial sources, the fact that the only isotope of Pm available in macro

-4-

amounts is 147Pm which is S--unstable with a relatively short half-life, ex-

cluded it from consideration. The tendency of EuF3 to reduce to EuF2 at high

temperatures even in the presence of F2 vapors, and the very strong broad band

structure associated with Eu2+ in the visible and ultraviolet range due to

4f7 + 4f65d transitions, made it impossible to obtain the requisite experi-

mental data for the Eu-system. The energy level schemes for Pm3+:LaF3 and

Eu3+:LaF3 were therefore calculated using parameters obtained by interpolation.

Some of the experimental results utilized in the analyses represent as

yet unpublished work done over a period of years in the Chemistry Division of

Argonne National Laboratory with the help of senior thesis students and guest

scientists. These results were obtained using several different instruments.

Spectra in the range 4000-15000 cm 1 were recorded using a Cary Model 14R

(crystal-grating- 0.5 meter) recording spectrophotometer. In the region

15000-50000 cm-1, both a 1-meter Hilger-Engis Model 1000 spectrograph equipped

with an EMI 9558 Q photomultiplier, and the Argonne 30' Paschen-Runge spectro-

graph (in second order) were used. Observations were made at %298, 77, and

4 K. Crystals of LaF3 doped with selected lanthanides in various concentra-

tions were obtained from Optovac Inc., North Brookfield, Mass. 01535.

There continues to be a wide interest in lanthanide-doped crystals and

glasses as laser materials. In some cases quantitative intensity calculations

have been carried out and related to the fluorescing properties in a given

host. The most extensive calculations have been limited to the f2(f12) and

f3(f11) members of the series since the matrix elements of the tensor opera-

tors connecting the states of interest required in the intensity analysis for

more complex systems have not been published. We report here a consistent set

of the matrix elements for all 4fN-configurations based on the systematic atomic

parameters generated in this work.

-5-

II. Physical and Crystallographic Properties of LaF3

There have been conflicting reports suggesting both C2V, Oftedal (1929,

1931) and D3h, Schlyter (1953), site symmetries of the La3+ ions in LaF3 '

More recent studies, Mansmann (1964, 1965), Zalkin et al. (1966), Lowndes

et al. (1969), indicate that the nine nearest-neighbor F~ ions present a suf-

ficiently distorted environment so that the symmetry is Dd (Pcl) with a C2

site symmetry, Fig. 1. A recent powder neutron-diffraction study of LaF3 and

CeF3, Cheetham et al. (1976), provided additional confirmation of the latter

structure. Isostructural members of the series are LaF3, CeF3, PrF3, and

NdF3; SmF3 and the heavier trifluorides are dimorphic. They also crystallize0

in the orthorhombic YF 3 lattice where each Y3+ has 8-F at 2.3 A and one at

2.6 A, Zalkin and Templeton (1953).

The crystallographic evidence for a low site symmetry in LaF3 is con-

sistent with the results of an early spectroscopic study of PrF3 in which

Sayre and Freed (1955) pointed out that the number of lines observed at low

temperature for electronic transitions associated with several excited states

excluded a site symmetry higher than C2 V. The Raman spectrum of LaF3 has been

interpreted in terms of a C2 site symmetry of the La3+, but these results also

emphasize that the deviation of the symmetry from more symmetric models is

very small indeed, Bauman and Porto (1967). Spectroscopic evidence for hidden

selection rules in the polarized spectrum of Nd3+:LaF3 is also suggestive of

an approach to a symmetry higher than C2, Wong et al. (1963b), Kumar et al.

(1976).

A recent investigation of the normal and vacuum ultraviolet absorption

bands of trivalent lanthanides doped into LaF3 has provided evidence that LaF30

is transparent down to the normal ultraviolet limit of '2000 A, Heaps et al.

(1976).

Fig. 1

LoF3 STRUCTURE VIEWED DOWN THE c-AXIS

to"N F(3)

F(F) .oa

F(.58)

FM( )2.46 L o

F(2) .31 2.42 .58 2.49 4

2)400.I 2\ .(I).42''

F()F(I) F(2)

0

0 I2A

0.001 A0.001 A

Space Group

Dp -P3CI

Site symmetry C 2

A. Zolkin, D. H. Templeton, J. E. HopkinsInorg. Chem. 5, 1466 (1966)

.08)

= 7.185

7.351Co

-7-

Refractive index (n) measurements using films of LaF 3, CeF 3 , and NdF3 ,

Haas et al. (1959), provided the following values in the 0.25-2.0 p range:

2.0

1.2

0.8

0.6

n

1.57

1.575

1.58

1.585

0.45

0.40

0.30

0.27

n

1.60

1.61

1.625

1.65

For single-crystal LaF3 ,

fractive index (ordinary

Winch (1966), was

n - 1.57376 +

____ n0(ob

2 5 3 6 .5 1 .6 55

3131.5 -

3663.3 -

4046.5 1.617

4358.3 1.616

5460.7 1.605

The M.P. of LaF3

the expression obtained for the variation of the re-

ray) with wavelength in the visible-near ultraviolet,

153.137

) - 686.2

s)

87

97

64

97

n0(calc)

1.65652

1.63639

1.62520

1.61933

1.61546

1.60583

is 14930C and its density is 554 p/cm3. Brown (1958).

-8-

III. Treatment of Experimental Data

In recent years there have been numerous reports of the spectroscopic

properties of different lanthanides in host LaF 3 , and many of these investiga-

tions have included low-temperature spectra. Consequently, the energies of

the crystal-field components associated with many of the free-ion states have

been recorded. The extensive data available led us to begin a systematic

theoretical analysis of the 4fN energy level structures over the series.

Part of the motivation to undertake a systematic analysis of the data

stumned from the results described in two papers by Onopko (1968a,b) in which

he pointed out that the energies of crystal-field components of several of the

lowest-lying free-ion states in Nd3+:LaF 3 and Er3+:LaF3 could be computed in

reasonably good agreement with experiment by assuming that the site symetry

approaches D3h. The approximation also appeared to be justified in treating

the crystal-field levels in Gd3+:LaF3, Schwiesw and Crosswhite (1969). We

had already developed computer programs that allow a complete diagonalization

of the atomic and crystal-field Hamiltonian for the 4f3(4f 11) case in 03h

symmetry. We had also determined the atomic parameters for both Nd3+:NdF3

and Er3+:LaF3. We therefore extended the analysis using Onopko's crystal-

field parameters. The initial results confirmed the validity of Onopko's

analysis by providing a coputed set of crystal-field levels that were in

reasonably good agreement with experiment throughout the spectral range to

'000 of l'

The analysis was extended by assigning the experimentally observed en-

ergy to those crystal-field states identified by the initial computation.

This permitted a least-squares adjustment of the original crystal-field param-

eters. Additional assignments were made based entirely on the correlation

-9-

between the optimized crystal-field parameter calculations and observed levels.

We then proceeded to perform a similar crystal-field calculation for other lan-

thanides using the optimized values for Nd3+ and Er3+ as the basis for beginning

the analysis.

For odd-electron systems the crystal field will split a level into J +

1/2 separate components in all site symetries except cubic and octahedral.

The number of possible components therefore is the same for C2 and D3h, and

because D3h appears to be a good approximation to the actual one, it is not

difficult to correlate the experimental and computed levels.

On the contrary, in even-electron systems such as Pr3+(4f 2 ), crystal-

field calculations based on D3h symetry do not remove the degeneracy of the

M :1, 2 states. It has been pointed out that the naber of lines observed

in the spectrum of Pr3+:LaF3 implies a low site syimetry consistent with the

complete removal of symmetry-related degeneracy in the indicated levels. Sayre

and Freed (1955), Carnall et al. (1969). This leads to ambiguities in making

the necessary crystal-field correlations. We have therefore attempted complete

crystal-field analyses only for odd-electron systems. For the even-electron

cases we have made the approximation of fitting the centers of the crystal

groups to the free-ion Hamiltonian only.

The basic theory used to interpret the structure observed experimentally

in lanthanide crystal spectra has been considerably refined in recent years.

A semi-emirical approach has been employed in which the attempt is made to

identify those effective interactions operating within the fN-configuration

that reproduce the observed structure. Judd (1963). ybourne (1965). Judd et

al. (1968). Crosswhite et al. (1968). Based on this method of interpretation,

the total Hamiltonian of the system can be written:

-10-

E -=EF + ECF

where EF is the atomic part of the interaction:

EF Fk(nf,nf)fk +C4fA0 + aL(L+1) + SG(G2) + yF(R7)k=0,2,4,6

+ T t + Mmk + Pkpki=2,3,4,6,7,8 k0,2,4 kk2,4,6

The Fk, 4g, a, B, y, Ti, Mk and Pk are parameters and their associated terms

are the corresponding operators. ECF represents the crystal-field interaction.

Following Onopko (1968a,b) we assumed that the symetry of the lanthanide site

in LaF3 was approximately hexagonal (D3h), that is, that the hexagonal terms

in the crystal-field expansion are dominant.

ECF 9kC( k) = 82C(2)4 C(4)+ 66) + [C(6) + C(6)ECF(. qq. 50C0 0 0 0B C 6 : C6 6C~

The above interactions which constitute the model used in this investigation

are discussed in detail in Appendix I with reference to the similar treatment

of the more extensive data for Ln3+:LaC13.

For odd-f-electron systems the total Hamiltonian can be separated into

three submatrices and all three of them diagonalized simultaneously. The

methods of truncating the very large matrices that occur for ions in the mid-

dle of the series have been discussed previously, Carnall et al. (1976). The

results of the diagonalizations--which eventually included variation of most

of the parameters in the systems, are given in Appendix II.

Onopko (1968a,b) quoted his original results in the Stevens operator

(6kq) normalization and subsequently extended the analysis to Er3+:LaF3,

-11-

Onopko (1969). The corresponding crystal-field parameters in the tensor opera-

tor normalization (Bk) used in the present study, Wybourne (1965), are given

below:

Nd3+: LaF3

cm_ 1 B = 276 cm4

B40 = 14086B0 = 1600

6 = 6796

820 =

040 -

060 -

066 =

The crystal-field parameters obtained from the

gation are compared with those for Nd3+:LaCl3 ,

the symmetry is hexagonal, below:

Nd 3+:LaF Nd 3+:LaCl33(D 3h Approximation) (C3h Symneti

2=216 cm- B0 = 163 cm

4= 1225 B4 = -3360 0B6 * 1506 6 = -71300

6 *770 B6 = 4626 6a c 17 m'I a * 8.1 cm

139 levels fitted 101 levels I

141 c

145

48.3

430

Er : LaF3

:m- B2= 282an

0BE = 1160

B6 = 7730

B 6 = 4536

Nd3+:LaF 3 data in this investi-

Crosswhite et al. (1976), where

3

ry)

fitted

It has been demonstrated that ab initio calculations of crystal-field

parameters based on an ionic model are not able to reproduce the values ob-

tained semi-empirically. However, serious efforts are being made to develop

suitable models for the LaF3 crystal, Newman and Curtis (1969), Stedman and

Newman (1971), and attempts have also been made to treat the low-synunetry LaF3

lattice case by actually using the appropriate number of terms in the poten-

tial, Matthies and Welsch (1975).

2 = 13820840 = 176

860 = 100

866 = 645

w w

-12-

The experimental results of the present efforts should provide a useful

testing ground for further theoretical treatments. Systematic trends in the

values of the crystal field parameters for Ln3+:LaF3 are compared with those

for Ln3+:LaCl 3 which are much better established experimentally, in Fig. 2.

200

100

1000

500

0

1000

500

0

Pr Nd Pm Sm Eu Gd Tb Dy Ho

Fig. 2. Crysta1-field Parameters forLn +:LaC1 3 and Ln3+:LaF 3.

I I I I T T T I I I I-2

C oo La C13

" La F3

40 "I I

- -I

I I I I I I I I I I I

0

I -

- -

N 2 3 4 5 6 7 8 9 10 II 12Er Tm

-14-

IV. Energy Level Correlations - Survey of Experimental Data

1. 4f2 (f12 )

The energy level structure in Pr3+:LaF3 has been examined experimentally

in moderate to high resolution by several groups. Wong et al. (1963a), Yen

et al. (1964), Caspers et al. (1965a), Carnall et al. (1969). Free-ion level

energies are recorded in Appendix III together with the corresponding computed

levels using parameters given in Appendix II. The weak point in the theoret-

ical analysis is the assumption of a center of gravity for the 1I6 and for the

3P1 transitions.

Most of the possible crystal-field components in the spectrum of Tm3+:LaF3

have been observed, Carnall et al. (1970), and the centers of gravity of these

groups are compared to the calculated free-ion levels in Appendix XIII.

2. 4f3fll

There are extensive published reports, Wong et al. (1963b), Caspers et

al. (1965b), of the structure observed in the low-temperature absorption and

fluorescence spectra of Nd3+:LaF3. These data have been extended by previously

unpublished work at ANL to provide as complete a set of crystal-field compo-

nents as possible. Of the 182 levels in the f3 configurations, 139 have been

assigned. The results are included in Appendix IV, and can be compared to

those obtained in the recent extensive investigation of the spectrum of

Nd3+:LaCl3, Crosswhite et al. (1976). Kumar et al. (1976) have examined the

absorption and fluorescence spectra of Nd3+:LaF3 at 77 K, the latter excited

using the 3371 A line of a N2-laser. They reported several transitions not

observed in previous investigations. Some of these are consistent with band

energies observed in the present study but there are discrepancies. In gen-

eral, there is a small shift in energy between observations at 77 K and those

-15-

reported here which refer to 4 K. The assignments made to the fluorescence

spectrum serve to further establish the energies of the lower-lying states

which had been reported earlier. The components of the 4I15/2 state reported

by Voron'ko et al. (1973) from observations made at 77 K are in good agreement

with the unpublished results recorded in Table 1, Appendix IV.

The present theory of intensities of f + f transitions treats the compos-

ite free-ion states rather than transitions between individual crystal-field

components (see Section V). A number of investigators are presently concerned

with extending the theory to the crystal-field case. In order to provide a

basis for testing the results of such calculations we include in Figs. 3-22,

Appendix IV, a set of absorption spectra of Nd3+:LaF3 taken at t4 K covering

most of the levels observed in the f3-configuration. No attempt was made to

preserve a constant resolution since two different instruments were used, and

several different crystals of varying concentrations and path lengths were em-

ployed. However, in any one group, the relative intensities of the components

are clearly evident.

The absorption and fluorescence spectra of Er3+:LaF3 measured at 77 K and

including levels up to %39500 cm- were reported by Krupke and Gruber (1963,

1964, 1965). Several higher-energy transitions were also tentatively identi-

fied. A subsequent investigation, Carnall et al. (1972), included measurements

at "4 K in the range 6000-50000 cm-1 . Additional spectroscopic measurements at

low temperature have been made, so that the levels recorded in Appendix XII

represent a composite and in a number of cases a reevaluation of results ap-

pearing in the literature. In addition to a discrepancy in the calibration

standards applied to a number of groups originally reported by Carnall et al.

(1972), the comparison of experimental energies with those computed based on

-16-

the crystal-field interpretation used here suggested u possible vibronic origin

for some of the states previously identified as crystal-field components.

3. 4f4(f10 )

The absorption spectrum of Pm3+:LaF 3 has not been reported, but an exten-

sive investigation of the absorption and fluorescence spectra of Pm3+:LaCl 3 has

been published, Carnall et al. (1976). We have therefore used the regularities

ii the energy level parameters for Ln3+:LaF3 as the basis for interpolation and

assignment of approximate parameters for Pm3+:LaF3. The :orresponding computed

free-ion levels are given in Appendix V. The chemical shifts observed on com-

paring free-ion levels in Nd3+:LaF3 and Nd3+:LaCl3 are similar to those found

in comparing the computed results for Pm3+:LaF3 with the experimental states of

Pm3 +: LaCl3.

An extensive investigation of the absorption and fluorescence spectra of

Ho3+:LaF 3 has been reported by Caspers et al. (1970). Additional experiments

have been conducted at ANL, but only minor additions or modifications of the

published data were indicated. In many cases, the number of observed components

of free-ion groups is less than allowed theoretically based on C2 site symmetry,

but the centers of gravity of these levels appear to provide the basis for cal-

culation of a consistent set of energy level parameters, as recorded in Appen-

dix XI.

4. 4f5 (f9 )

The observation and analysis of the absorption and fluorescence spectra

of Sm3+:LaF3 in the range 0-11000 cm 1 was reported by Rast et al. (1967), and

the line list was extended to "32000 cm 1 in a tabulation given in Dieke (1968).

The number of observed lines was further extended in the present investigation,

and a composite tabulation based primarily on recent work at ANL is given in

Appendix VI.

-17-

Absorption and fluorescence spectra of Dy3+:LaF 3 including levels up to

'32000 cm-1 have been reported in the literature by Fry et al. (1968). A num-

ber of new levels including groups at higher energies were recorded in the

course of the present investigation. The results are presented in Appendix X.

5. 4f6(f8

Crystals of LaF3 doped with EuF3 are found to contain some Eu2+ which

makes it difficult to observe the Eu3+ transitions in absorption in the near-

ultraviolet. Weber (1967a) observed fluorescence in Eu3+:LaF3 from the excited

states 5D0, 5D1, 5D2, and 5D3 using pulsed selective excitation. The energy

level scheme for the low-lying 5D and 7F states that can be deduced from these

measurements shows the expected red shift with respect to the corresponding

levels observed in Eu3+:LaCl3, Dieke (1968), and is consistent with the results

given in Appendix VII. The latter were computed based on energy level param-

eters deduced from systematic trends over the series.

The energy levels of Tb3+ in single-crystal TbF3 have been studied in

absorption and fluorescence by Krupka and Guggenheim (1960). From this data

the centers of gravity of the 5D4 and the ground term 7F multiplet components

could be determined. The results are in agreement with the moderately exten-

sive study of the low-temperature absorption spectrum of Tb3+:LaF3 which was

part of the present investigation. The free-ion levels of Tb3+:LaF3 are given

in Appendix IX.

6. 4f7

The energy levels of the 6P and 6I groups in Gd3+:LaF3 were reported by

Schwiesow and Crosswhite (1969) who also performed a crystal-field analysis

assuming an approximate hexagonal site symmetry. The experimental results were

subsequently extended to include the 6D states in the 40000-50,000 cm-l range,

-18-

Carnall et al. (1971). The data recorded in Appendix VIII are a composite of

the indicated published results.

-19-

V. Theoretical Interpretation of Excited State Relaxation

A. Theoretical Treatment of Absorption Spectra

1. General Concepts

The quantitative treatment of the intensities of trivalent lanthanide ab-

sorption bands relates an experimentally determined quantity, a normalized band

envelope, PEXPT' to a theoretical model based on the mechanisms by which radia-

tion can be absorbed. The terms oscillator strength or transition probability

are applied interchangeably to the symbol P. There is some magnetic dipole

character in a few transitions (PM.D.), but an induced electric-dipole mecha-

nism (PE.D.) must be invoked to account for the intensities of most lanthanide

absorption bands. The term induced or forced electric dipole is used to empha-

size that true electric dipole transitions require the initial and final states

to be of different parity, whereas no parity change is involved in transitions

within a configuration. In contrast, magnetic dipole transitions within a con-

figuration are (parity) allowed. The weak intra-fN transitions are accounted

for by assuming that a small amount of the character of higher-lying opposite-

parity configurations is mixed into the fN states via the odd terms in the

potential due to the ligand field, Wybourne (1965). We neglect higher multi-

pole mechanisms, (electric quadrupole, etc.), and write:

EXPT E.D. + PM.D.

In expressing PE.D. in terms of a theoretical model, Judd (1962) and Ofelt

(1962) summed over the intensities of the individual crystal-field components

of a given state. As a consequence, the model applies to spectra observed at

room temperature or above since it is assumed that all of the crystal-field

-20-

components of the ground state are equally populated. Some attempts have been

made to avoid this sumation and thus treat the intensities of transitions be-

tween the lowest crystal-field level of the ground state and excited crystal-

field states, Axe (1963). However, the intensity calculations reported in the

present sumary deal only with composite levels. Thus the appropriate expres-

sion for PEXPT' which represents the number of classical oscillators in one ion,

more commonly referred to as the probability for absorption of radiant energy,

Hoogschagen (1946), is:

PEXPT 2303m fc1(o)da = 4.32 x 10- Ic1(o)daN-e

where ci is the molar absorptivity of a band at the energy o1 (cm i), and the

other symbols have their usual meaning. P is here a dimensionless quantity.

The molar absorptivity at a given energy is computed from the Beer-Lambert law:

c = ' log Is/I (1)

where c is the concentration of the lanthanide ion in moles/1000 cm3, t is the

light path in the crystal (cm), and log I,/I is the absorptivity or optical

density. The expression for PEXPT is identical to that for f defined by Krupke

(1966) in his treatment of lanthanide spectra in LaF3.

In view of our interest in both absorption and fluorescence processes,

there is an advantage in pointing out the basic role of the Einstein coefficient

in expressing the transition probability due to dipole radiation:

A64w a 32A(if) - 1|i|D0f>| (2)

where I and f signify the initial and final states, A is the (spontaneous)

transition probability per unit time, o(cm 1) is the energy difference between

-21-

the states, and D is the dipole operator, Condon and Shortley (1957).

In addressing the problem of the absorption of energy, Broer et al. (1945)

expressed eq. (2) in terns of oscillator strength using the relationship P

Amc/8w2a2e2. The factor 2J + 1 was added since the matrix elements of 0 are

sunned over all components of the initial state I. A refractive index correc-

tion X was also included giving:

P 8w 2mca (XF + n (3)3he (2J+1)

where F2 and R2 represent the matrix elements of the electric dipole and mag-

netic dipole operators, respectively, joining an initial state J to the final

state J', X , and n is the refractive index of the medium.

2. Induced Electric-Dipole Transitions

Judd (1962) and Ofelt (1962) independently derived expressions for the

oscillator strength of induced electric dipole transitions within the fN con-

figuration. Since their results are similar, and were published simultaneously,

the basic. theory has become known as the Judd-Ofel t theory. However, Judd's

expression, eq. (4), was cast in a form that could be directly related to oscil-

lator strengths derived from lanthanide absorption spectra taken at 25C or

above:

P E.O. T 7,61Tv(ial||U ||$'J')2 (4)

where v(sec 1 ) is the ean frequency of the transition J * *'J', ( is a

unit tensor operator of rank A, the sue running over the three values A " 2,4,6,

and the TA are three parameters which can be evaluated from experimental data.

These parameters involve the radial parts of the 4N wave functions, the wave

functions of perturbing configurations such as 4fN-l5d, and the interaction

-22-

between the central ion and the immediate environment.

Typically, several excited states are encompassed by a single complex

absorption envelope, and the matrix elements of U are sunned over these

states. The energy in this case becomes that of the center of gravity of

the envelope. Those investigators who have studied lanthanide intensities

in crystals have followed an alternate parametrization, Axe (1963), Krupke

(1966), which has clear advantages in describing both the absorption and

fluorescence processes in terms of a single set of adjustable parameters.

The expression for TA given by Judd in eq. (4) was:

Tx Nw2 n2+n21(2A+1) (2t+1)Bt 2(t,A) (5)9n t

Substituting v * co and eq. (5) into eq. (4) gives

PE.D. ,w2m ny ,,U(A)IJ)2 (6)

where " *(2a+l)I(2t+1)Bt 2(t,A), Axe (1963), and in terms of T, eq. (3),t

S2as, ,6,2

The matrix elements of eq. (6) are calculated in the SL basis using the

relation:

(fNaSLJ|| U(A) I f~a'S'L'J' ) s o(SS' )(-1) S+L'+J+X ((2J+1)(2J'+1)l1/2

(AS I(fNSL|U A'SL') (8)

Selection rules imposed by the nature of the mechanism assumed are discussed

by Ofelt (1962). The reduced matrix elements on the right side of eq. (8)

have been tabulated by Nielson and Koster (1963). The matrix elements as

-23-

computed must be transformed from the SL basis to intermediate coupling before

being squared and substituted into eq. (6).

The intermediate-coupling eigenvectors, IfNl$J>, are expressed in terms of

SL basis states, |fNSLJ>, by:

IfN yJ> " c(a,S,L)IfNaSLJ>

where c(a,S,L) are the numerical coefficients resulting from the simultaneous

diagonalization of the atomic parts of the Hamiltonian. The matrix elements

of U , eq. (8), have been calculated for transitions between various excited

states as well as between the ground and excited states of the whole series of

lanthanide ions using the energy level parameters given in Appendix II. The

rest s are tabulated in Appendices III-XII.

3. Magnetic Dipole Transitions

Following the results of Condon and Shortly (1957), the magnetic dipole

operator is given as 14 * -e/2mc (Li+2Si). The matrix elements of the opera-

tor i in eq. (3) can then be written,

H2 * e2/4m2c2(4JIIL+2SIIq'J')2 (9)

The non-zero matrix elements will be those diagonal in the quantum numbers a,

S, and L. The selection rule on J, ftJ 0,tl, restricts consideration to

three cases:

1) J'"J (aSLJIIL+2SIIaSLJ') * gt(J(J+1)(2J+1) 11/ 2 (10)

where g " 1 + J+i.L +}-LL+1

2) J'"J-1

(aSLJI|L+2S||aSLJ-1) * "((S+L+J+1)(S+L+1-J)(J+SL)(J+L-S)J1/2 (11)4J

-24-

3) J'=J+1

(aSLJIIL+2SIIaSLJ+1) =[(-S+L+J+2)(S+J+1-L)(L+J+1-S) (S+L-J) 1/2 (12)4(J+1)

The matrix elements calculated in eqs. (10)-(12) must be transformed into

the intermediate coupling scheme before computation of the magnetic dipole con-

tribution represented by eq. (9).

Values of the quantity P' > 0.015 x 10-8 where PM.D. P'n and n is the

refractive index of the medium, have been tabulated for transitions of the tri-

valent lanthanide ions between the ground states and all excited free-ion

states, Carnall et al. (1968a).

4. Comparison of Calculated and Observed Transition Probabilities

A number of authors have determined the n intensity parameters from a

least squares fitting procedure using band envelopes for Ln 3+:LaF 3 observed at

'25 C and available sets of the matrix elements of U Krupke (1966) was the

first to show that transition probabilities in good agreement with experiment

could be computed for Pr3+:LaF3 and Nd3+:LaF3. A summary of parameter values,

n,, for Ln3+:LaF3 is presented in Table 1, Appendix XIV. The parameters for

Sm3+:LaF 3, Dy3+:LaF3 , and Tm3+:LaF3 were checked against results obtained in

the present investigation, and those for Dy3+ and Tm3+ were modified from the

values originally reported. While the values of the matrix elements of U()2

given in Appendices III-XIII differ slightly from those in the literature, these

differences are not sufficient to affect the reported values of S.

B. Relaxation of Excited States

1. General Considerations

A great deal of progress has been made in analyzing the mechanisms of

excited state relaxation of lanthanides in crystal hosts. Two modes of

relaxation can be recognized: radiative and non-radiative processes. Axe (1963)

addressed the problem of expressing the radiative process in quantitative terms

using the Juei-Ofelt theory. Non-radiative relaxation was already being formu-

lated in terms of multiphonon processes in the early 1960's. Barasch and Dieke

(1965), Riseberg and Moos (1967,1968). Such processes become less probable as

the energy gap between an excited state and the next lower energy state increases.

2. Radiative Relaxation

In treating the fluorescence process, the Einstein coefficient, eq. (2)

is used directly to express the rate of relaxation of an excited state (pJ) to a

particular final state (,'J'). Following Axe (1963), the counterpart of eq. (3)

becomes

A(J,$'J') = )644 a3 [x' F2 + n3 2] (13)

where a(cm~l) represents the energy gap between states (pJ) and ('J'), X' =

n n2+2i2, and n is the refractive index of the medium. As in the absorption

process, there is an implicit assumption that all crystal-field components of

the initial state are equally populated. In principal, if fluorescence can be

detected, the lifetime of the state is long compared to the rate at which it is

populated in the excitation process, so thermal equilibrium at the temperature

of the system can be achieved prior to emission.

The matrix elements of the electric and magnetic dipole operators, F

and M2, are identical to those in eq. (7) and eq. (9), respectively. However,

the form of the refractive index correction in eq. (13) is not the same as for

the absorption process, eq. (3). Equation (13) can be evaluated using param-

eters % established from measurement of the absorption spectrum of the lan-

thanide ion in a crystal lattice identical to that studied in fluorescence.

-26-

Since excited state relaxation generally involves transitions to several

lower-lying- states, we define a total radiative relaxation rate, AT(1J)

AT(tJ) = E A(pJ,p'J') (14)

where the sum runs over all states lower in energy than the fluorescing state.

It is useful to define in addition the radiative branching ratio, BR'

from the relaxing state (pJ) to a particular final state (i'J')

8R(PJ,' J') = (15)

and the radiative lifetime of a state

TR(1JJ) = [AT('PJ)]1 (16)

The principal fluorescing states of the lanthanides in crystal hosts are

indicated in Fig. 3. However, fluorescence from many of these levels is only

observed at low temperatures since rapid relaxation of an excited state by non-

radiative processes competes strongly with the radiative mode unless the energy

gap to the next lower level is large, as discussed in the next section.

Since the parameters S have been determined for a number of the lantha-nides in LaF3 host, Table 1, Appendix XIV, we can compute the radiative life-

time of any excited state using eqs. 13, 14 and 16. For those states with

large energy gaps to the next lower level, the observed and computed radiative

lifetimes would be expected to be in approximate agreement. Weber (1967b) has

pointed out that such agreement is observed for excited states in Er3+:LaF3

where the energy gaps are in excess of 3000 cm1:

-- Jeo-O13cm-' "

L

K -

SI

0

38

36

34

32

0-

P---

0-

Um

5-

R_ST

3 p

0

'p--

---v-I

-re M r-- i OF

H TII-

Em4

--4-

- --s u

--- W-7

-I-

x - em6

m2kyhfmm

'H 4 I1 , 4Pr Nd Pm

L m

P

Z-

I=-K -

N-

"-

[ -

0m5

g-

C-C-,94

A-T-S

_ T ---- GP

T T-

X- _4syk

-- '- it

Rm':

0

-5

!am N--

Ls *

K m LLB LEEE2

.

4

-w-

KT

401V2 jT

14-

EF 1 ED %

4F

F -4 45

East ET %

4 FD GG

-4"

'He FHm TF'0

Sm Eu

Gd

Gd

SIC-4

r-

4 u--C-.

-==a

Tb yHis Io

Tb Dy Ho

4amm % -

1-I

r-

Er

Fig. 3. Energy Level Diagrams for Ln 3+:LaCl 3.

401- 4% 74

U-

0-

30

28

26

241

22

20

8

16

14

12

a-

-te

8

6

4'

-_ ,

01

CeaH

Tm

aFT

Yb

i

NEW"

TR Observed LifetimeTransition jmsecj (msec)

4113/2 i4115/2 10.9 13

411/2 + 11.6 11

2P3/2 + 0.43 0.29

3. Non-radiative Relaxation

Following tie excitation of a lanthanide ion in a crystal lattice, the

relaxation of excited states may occur by a purely radiative process, or more

generally, may occur by the transfer of some energy to lattice vibrations. It

was known experimentally for a number of years before any quantitative theory

of non-radiative processes was developed that fluorescence was not observed at

25*C from Ln3+:LaCl 3 if the energy gap between the excited state and that next

lower in energy was < 1000 cm~1, Barasch and Dieke (1965). In a systematic

investigation of multiphonon orbit-lattice relaxation of lanthanide excited

states, Riseberg and Moos (1968) have given an explicit expression for the

temperature dependent transition rate in LaF3, LaCl3 and LaBr3. The energy-gap

dependence of the multiphonon process is treated phenomenologically by assuming

that the appropriate phonon energy is that corresponding to the cut-off in the

phonon states. For LaF3 this is 350 cml, while for LaCl 3 it is ,260 cm 1 .

In terms of the ,1000 cm~1 energy gap cited earlier, it is apparent that a 3-4

phonon emission process is a relatively efficient mode of relaxation at 25C.

However, as the gap increases, demanding the simultaneous emission of a large

number of phonons, the process rapidly decreases in probability such that radia-

tive decay can efficiently compete as a relaxation mechanism.

Since in the usual case both radiative and non-radiative processes operate

to relax an excited state, we can express the total fluorescence lifetime of the

-29-

state as

(TT) = A.-- pJ) + WT(IJ) (17)

where AT is the radiative rate and WT(4J) is the sum of the rates of the various

non-radiative processes.

The dependence of the relaxation rate on energy gap alone, W(o) can be

expressed as a simple exponential, Moos (1970),

W(o) = CeaAE

where C and a are parameters characteristic of the host material, not of the

lanthanide ion. For LaF3, C = 6.6 x 108 (sec-1) and a = -5.6 x 10-3, Riseberg

and Weber (1976).

Adding the temperature dependence, Riseberg and Moos (1967,1968), results

in the expression

W(1pJ) = W(o) {[e i /kT _ 1-1 + 1}AE/fiwi

which can be rewritten

W(pJ) = CeaAEl _--f w /kT]-DE/kiwi (18)

where kiwi is the maximum phonon energy, taken as 350 cm-1 for LaF3, Riseberg

and Weber (1976). Since k = 0.695, kT = 207 cm 1 at room temperature and the

corresponding expression is:

W(*J) = CeaAE(.8155)-AE/350

For example, the radiative lifetime of the 4I9/2 state of Er3+:LaF 3 at

tl2000 cm"1 has been computed to be 20.7 msec but observed to be "..15 msec,

Weber (1967b). Since AE between 419/2 and the next lower 4I11/2 state is

-30-

2000 cm-1 , W(,J) = ol04 sec-1 . Thus the non-radiative lifetime of '40.1 msec

is rate determining.

4. Comparison of Computed Excited-State Lifetimes with Those Observed

Experimentally

Experimental measurements of excited-state lifetimes for a number of

lanthanides in LaF3 have been reported, and in many instances these values have

been compared to those computed using eq. (13), (17) and (18). Although the

results of the computations were reported, the availability of the relevant

matrix elements of U interconnecting the excited states in the various con-

figurations is very limited. The only tabulation that includes all members of

the series is restricted to matrix elements that join excited states to the

ground state, Carnall et al. (1968b). As a consequence, the results presented

in Appendices III-XIII represent the first systematic effort to make intercon-

necting matrix elements of U available for the whole series and thus enable

calculation of a wide range of lifetimes. Some minor corrections to values re-

ported in the earlier tables have also been made. Calculated radiative life-

times and observed lifetimes for some of the prominent fluorescing states in

Ln3+:LaF3 are given in Table 2, Appendix XIV. An example of the complete cal-

culation of the lifetimes associated with the radiative relaxation of two

states in Tb3+, showing branching ratios to all lower-lying states, is given

in Table 3, Appendix XIV. The strong radiative coupling of the 5D4 to the 5D3

state is clearly indicated. Recently Page et al. (1976) reported lifetimes in

the 100 ysec range for the 5D4 state in Tb3+:LaF3 at 300 K, with little change,

as expected, on cooling to 77 K. However, these values are a factor of ten

shorter than would have been predicted, see Table 2, Appendix XIV, and must be

regarded as questionable. In Tb3+:LaCl3 at 300 K the lifetimes of the 5D3

states were 570 and 1220 )sec, respectively, in good agreement with the calcu-

lated values for the LaF3 host, Barasch and Dieke (1965).

-31-

5. Comments on the Use of the Tables

(A) Atomic (free-ion) states

As discussed in the introduction, detailed crystal-field calculations

were only carried out for odd-f-electron systems. For the even-f-electron

systems a center of gravity was computed from the experimentally observed

crystal-field components of each state. These "free-ion" states are recorded

in the appendices and were used as the basis for computing the energy level

parameters. Similar tabulations of the free-ion state energies have been

included for the odd-f-electron systems in order to facilitate use of the

tables of U(K)*2. In these latter cases the computation was made using the

atomic parameter values given in Appendix I with the crystal-field parameters

set equal to zero. In all the tabulations of free-ion levels the state

designation corresponds to the largest component of the eigenvector.

(B) Tables of (U(K))2

The entries in these tables are arranged in order of increasing J-value

for the initial state. Entries are not repeated. For example, the matrix

element between an initial J = 9/2 and a final J = 3/2 state is identical

to the J = 3/2 + J = 9/2 entry. Only the latter is given. If the entry is

missing from the table, the matrix elements are zero. The following partial

tabulation of matrix elements of U(K)*2 for Nd3+:LaF3 joining the ground

(4I9/2) state with several excited states taken from Appendix IV serves as

an illustration and may be compared with results given for Nd3+(aquo) in

Carnall et al. (1968b).

State

41

41

41

41

4F

4F

2H

Ecm )

235

2114

4098

6148

11621

12660

12768

U(2)*2

.0194

0

0

0

.0006

.0095

U(4)*2

.1072

.0135

0

.2283

.2337

.0082

U(6)*2

1.1639

.4549

.0452

.0554

.3983

.1195

-32-

2J

9

11

13

15

3

5

9

-33-

Acknowledgements

Several individuals contributed substantially to the experimental inves-

tigation and/or interpretation of important segments of the data presented in

this report, and their assistance is gratefully acknowledged:

Dr. Katheryn Rajnak, Kalamazoo College, Kalamazoo, Michigan

Prof. R. Sarup, College of the Holy Cross, '4orcester, Massachusetts

Dr. H. Kramer (nee Larmermann)

and the following senior thesis students:

E. A. Flom, St. Olaf College, Northfield, Minnesota

D. W. Mehaffy, Coe College, Cedar Rapids, Iowa

J. F. Pophanken, Houston Baptist College, Houston, Texas

G. A. Robbins, Stamford University, Birmingham, Alabama

34-

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Rajnak, K. and B. G. Wybourne (1964), Phys. Rev. 134, A596.

Rajnak, K. (1965), J. Chem. Phys. 43, 847.

Rast, H. E., J. L. Fry and H. H. Caspers (1967), J. Chem. Phys. 46, 1460.

Riseberg, L. A. and H. W. Moos (1967), Phys. Rev. Ltrs. 19, 1423.

Riseberg, L. A. and H. W. Moos (1968), Phys. Rev. 174, 429.

Riseberg, L. A. and M. J. Weber (1976), Prog. in Optics 14, in press.

Sayre, E. V. and S. Freed (1955), J. Chem. Phys. 23, 2066.

Schwiesow, R. L. and H. M. Crosswhite (1969), J. Opt. Soc. Am. 59, 602.

Schlyter, K. (1953), Arkiv Kemi 5, 73.

Stedman, G. E. and D. J. Newman (1971), J. Phys. Chem. Solids 32, 109.

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Trees, R. E. (1952), Phys. Rev. 85, 382.

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Weber, M. J. (1968), J. Cnem. Phys. 48, 4774.

-38-

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APPENDIX I

-1-

Appendix I

As a result of the correspondence between the energy levels of lantha-

nides in LaCl3 and LaF3, it is convenient to summarize the present status of

the theory applied to the host LaCl3 in order to facilitate the LaF3 discus-

sion. The basic experimental work on trivalent lanthanide spectra in LaCl3

crystals has been described by G. H. Dieke, Dieke (1968), who also gave a

comprehensive historical review and a brief discussion of early efforts at

theoretical analysis. Most of the subsequent extensions of the latter have

been based on the same data, and no attempt will be made to reproduce them

here.

The effect of the crystalline neighborhood on the electronic orbitals of

the rare earth ion is appreciable, but is nevertheless small compared with the

atomic electron interactions, and to a large extent can be treated in terms of

a model whose basis states are the free-ion orbitals themselves, without need

for specific structural detail of the electronic involvement with ligand ions.

Because of the dominance of the atomic forces it is important to have an atomic

Hamiltonian which is detailed enough to accurately describe the observed crystal

level groupings. In the process we have incidently learned much about the

structure of the atomic energy levels themselves.

In a pure 4fN configuration the only interactions to be evaluated would

be the four Slater integrals F0'2'4'6 and the spin-orbit integral, r, plus rela-

tivistic correction terms representing spin-other-orbit and spin-spin inter-

actions. Ab initio evaluations for each of these can be carried out; however,

the latter values are not quite in agreement with experimental ones. This is

not because they are inaccurate in themselves but because there are additional

contributions due to ignored inter-configuration interactions which have the

-2-

same angular dependence and are therefore lumped with the former in any empirical

fitting. The experimental values are therefore only effective ones, although

we conventionally continue to give them the original names. These same inter-

actions will in addition produce completely new effective-Hamiltonian operators,

namely: aL(L+1), Trees (1951, 1952, 1964); SG(G2) and yG(R7), Racah (1949) and

Rajnak and Wybourne (1963); Tit., the three-body electrostatic effective opera-

tors discussed by Rajnak and Wybourne (1963), Rajnak (1965), Judd (1966,1972)

and Crosswhite et al. (1968); and A z , the generalized two-body double-vector

operators, Judd et al. (1968) and Crosswhite and Judd (1970).

Fortunately, not all of the theoretically possible operators are needed

in the analysis, and those that are (including the Fk and r corrections) are

all either constant or slowly varying functions of the atomic number Z. For

instance, of the fourteen possible T t., only those six which have non-zero

matrix elements in second-order perturbation theory are retained. Furthermore,

these results are in good agreement with ab initio calculations, Newman and

Taylor (1971,1972), Balasubramanian et al. (1975). A review of the whole ques-

tion of crystal energy level parametrization was given by Newman (1971).

There are thirteen generalized two-body double-vector operators, eight

having rank one in each of the spin and orbital angular momentum spaces. (An

additional one has matrix elements exactly proportional to those of the spin-

orbit interaction and can be ignored). Five have rank two in each of the

spaces. The principal contribution to the latter comes from the spin-spin

interaction and can be estimated by ab initio calculations of the Marvin inte-

grals M0'2'4, Marvin (1947). The former appears to be dominated by spin-other-

orbit effects, parametrized by the same Marvin integrals, and another effect

arising from the fact that there are spin-orbit matrix elements connecting 4fN

-3-

with configurations of the type 4fN-ln'f. Rajnak and Wybourne (1964) have

called these "electrostatically-correlated spin-orbit" effects. Matrix ele-

ments have been given by Judd et al. (1968). Their essential role is to allow

for effective spin-orbit variations with spectroscopic term and are parametrized

by the quantities P2'4'6. Ab initio calculations have been given by Newman and

Taylor (1972).

A summary of the parameters derived from LaCl3 crystal studies is given

in Table 1, Crosswhite (1977). As a rule of thumb for estimating atomic param-

eters appropriate fQr the LaF3 case, the LaCl3 FK and C values reported in

Table 1 should be increased slightly: 1.8% For F2, 1.1% for F4, 0.8% for F6

and 0.5% for C. More extensive details are given elsewhere; Crosswhite et al.

(1976), Carnall et al. (1976), Crosswhite et al. (1977), Crosswhite and

Crosswhite (1977). For the major parameters we have found on comparison with

ab initio calculations that the required corrections are remarkably uniform.

Computations with a Hartree-Fock program containing an approximate relativistic

correction, Cowan and Griffin (1976), are given in Table 2, and differences of

these and experimental Fk and C values are shown in Figs. 1 and 2. The two-

body electrostatic correction parameters a, 3 and y show similar slow varia-

tions across the series; the Ti are essentially constant; and the Pk can be

taken proportional to . The Mk(spin-other-orbit) are shown in Fiq. 2.

The crystalline environment can in principle make contributions to each

of these terms in addition to new ones specific to the particular point-group

symmetry. However, experimentally we find that there is a great similarly be-

tween the parameters for the LaC1 3-doped spectra and those of the few free ion

cases for which experimental data are complete enough to permit full param-

etrization (La II, Ce III and Pr IV 4f2 and Pr III 4f3). Systematic values can

APPENDIX I - TABLE 1

Parameters for Ln

Eave

F2

F4

F6

Alpha

Beta

Gamma

T2

T3

T4

T6

TB

Zeta

MO bP2 c

BO

B

B6

Pr

9928

68368

50008

32743

22.9

-674

[1520]

744

1.76275

10-

-342

-677

466

3+:LaCl3

aValues in brackets werebOnly M0 was varied; the

C~ny y 2 was varied; the

not freely varied.

ratios M2/M0 = 0.56,ratios P4/P2 = 0.75,

= 0.38

= 0.50

were maintained.

were maintained.

.............. 00- . - -Er

35490Nd

24186

7186652132

35473

22.1-650

1586

377

40

63

-292358

354

880

1.97

255

163

-336

-713

462

Pm

36805

7580854348

38824

21.0-645

1425

302

45

34

-315554

[400]

1022

2.1

319

143-395

-666

448

Sm

47190

78125

56809

40091

21.6-724

[1700]

291

13

34

-193

288

330

1168

2.4

341

186

-270

-623

470

Eu

64542

84399

60343

41600

16.8[-640]

[1750]

[370]

[40]

[40]

[-330]

[380]

[370]

1331

[2.38]

245

189

-287

-801

525

Gd

87538

85200

60399

44847

[19][-643]

1644

[315]

[44]

[40]

[-300]

[325]

[360]

[1513]

[2.82]

495

216

-272

-688

474

Tb

68200

90012

64327

42951

17.5

[-630]

[1880]

[340]

[40]

[40]

[-330]

[330]

[380]

1707

[3.00]

590

185

-291

-457

302

Dy55894

92750

65699

45549

17.2-622

1881

311

116

12

-474413

315

1920

2.8

591

193

-328-470

287

Ho

48193

95466

67238

46724

17.2

-621

2092

300

37

98

-316

440

372

2137

3.0

523

216

-284

-448

294

98203

69647

49087

15.9-632

[2017]

300

48

18

-342

214

449

2370

4.5

667

216

-271

-411

272

I

1

a

-5-

APPENDIX I - TABLE 2

Relativistic Hartree-Fock Integrals for 4fNIV.

F2 F4 F6 MO M2 M4

4f1 Ce IV - - - 696.41 - - -

4f2 Pr IV 98723 61937 44564 820.22 1.991 1.110 0.752

4f3 Nd IV 102720 64462 46386 950.51 2.237 1.248 0.846

4f4 Pm IV 106520 66856 48111 1091.46 2.492 1.391 0.943

4f5 Sm IV 110157 69143 49758 1243.60 2.756 1.540 1.044

4f6 Eu IV 113663 71373 51342 1407.71 3.031 1.694 1.149

4f7 Gd IV 117058 73470 52873 1584.45 3.318 1.855 1.258

4f8 Tb IV 120366 75541 54361 1774.46 3.615 2.022 1.372

4f9 Dy IV 123592 77558 55810 1998.44 3.924 2.195 1.490

4f10 Ho IV 126751 79530 57227 2197.06 4.246 2.376 1.612

4f11 Er IV 129850 81462 58615 2431.00 4.580 2.563 1.739

4f12 Tm IV 132897 83361 59978 2680.97 4.928 2.758 1.872

4f13 Yb IV - - - 2947.69 - - -

AF0

o0

0890 0o a o

- 6 0

0

I I I I I I I I 1Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm

Fig. 1. Variation of the differences between the pseudo relativistic

Hartree-Fock (HFR) values of the Slater integrals Fk and those determined

experimentally, Fk(HFR)-Fk(EXP) AFk, .ith lanthanide atomic number.

34

30E-

28-

24-

20F-

161-

EC)

0

121

8

41-

0

E

SM* /M*EXP HFR

Pr Nd Pm Sm Eu

(SPIN OTHER ORBIT)

Gd Tb Dy Ho

Fig. 2. Variation of the differences between the pseudo relativistic

Hartree-Fock (HFR) values of the spin-orbit Slater integral , and those

determined experimentally, i(HFR)-4(EXP) a A; and the ratios of the

corresponding spin-other orbit values MEXP/MHFR with lanthanide atomic

number.

100

80

60I-

40-

201-

0

'A5 0 o 0

00

i.0II

0.8

0 0I I I I II0

I I I I I I. I I I0

&l0.Er Tm

Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho

- -- _ -l i

l / l _ 1 1 1 t 1 1 1 _

Er Tm Yb

-rL

1

F

v

-6-

be found for the Ti and a1 (or alternately Mk and Pk) which satisfy all known

spectra, whether free ion or crystal, in terms of only a few constants. It

follows therefore that the same model can be used for preliminary estimates for

other spectra such as LaF3-doped crystals.

As to the parametrization of the crystal-field itself, the general spec-

ification that the number of possible two-particle operators is equal to the

number of available independent cells in the Hamiltonian requires 366 additional

ones besides the ten single-particle ies (of which only four are formally used).

Fortunately, the single-particle mode , Judd (1963), Wybourne (1965), Dieke

(1968), works very well, although we ust recognize tiit these parameters repre-

sent much more complicated effects th n the simple Coulomb model visualized by

the early theories. Newman and Tayl (1971) give a discussion of the physical

significance of these parameters.

APPENDIX II

APPENDIX II - TABLE 1

Atomic Parameters for Ln3+:LaF3 .

EAVr2

F6

a

B

Y

9

T7

T8

MO(b)

M4(b)

p2(c)p4(c)

p6(c)

139

16.6

aValues in parenthesis were not freelybRelativistic Hartree-Fock values were

10163.

69305.50675.

32813

(21)

-842

1625

750.8

(1.99)

(1.11)

(0.75)

(200)

(150)

(100)

18016.7

varied.assumed.

649

(26)d36

20122

cOnly P2 was varied, the ratios P4/P2 = 0.75, P6/P2 = 0.5 were maintained.

dFree-ion Hamiltonian only

No. ofLevel sa

fit 1141

(27)d

32

117

12.1

Tm

18000.-

102459

7242451380

(17.)

-737

(1700)

2640

24490.

73036

52624

35793

21.28

-583.

1443

884.9

306

41

59

-283

326

298

(2.237)

(1.248)

(0.84)

213

160

106.5

(12)0

76

_

(37272)

(77000)

(55000)

(37500)

(21.00)

(-560)

(1400.)

(1022.)

(330.)(41.5)

(62.)

(-295.)

(360)

(310)

(2.49)

(1.39)(0.94)

(440)(330)

(220)

Sin47760.

79915

57256

40424

20.07

-563

1436

1177.2

28836

56-283

333

342(2.76)(1.54)(1.04)344258172

Eu(64263)(84000)

(60000)(42500)(20.)

(-570)(1450)

('327)

(330)(41.5)

(62)(-295)

(360)(310)

(3.03)

(1.69)(1.15)(300)(200)(150)

Gd87847.85587

6136145055

(20.)

(-590)(1450)

1503.5(330)

(+41.5)

(+62)

-(295)

(+360)

(+310)

(3.32)

(1.85)(1.26)

611458306

Tb68608.

91220

65796

43661

19.81(-600.)

(1400)

1702(+330)

(+41.5)

(+62)

-(295)

(+360)(+310)

(3.61)

(2.02)(1.37)

583

437291

56492.

94877

6747045745

17.64-6081498

1912+423

+50

+117

-334

+432

+353

(3.92)

(2.19)(1.49)771578

386

Ho

48453.

97025

68885

47744

18.98-5791570

2144(+330)

(+41.5)

(+62)

-(295)

(+360)(+310)

(4.25)

(2.38)(1.61)

843632421

Er

35915.

1002747055549900

17.88

-599

1719

2381+441

+42

+64

-314

+387+363

(4.58)

(2.56)

(1.74)

852

639426

(4.93)

(2.72)(1.37)

729.6

547364

1

....

1

-2-

APPENDIX II - TABLE 2

Crystal-field Parameters for Ln3+:LaF 3

Nd Sm Gd Dy Er

B2 216 209 (210) 218 2290

B4 1225 1042 (1050) 1099 9650

B6 1506 1415 (1250) 1129 9090

B6 770 659 (600) 553 484

APPENDIX III

PAGE 1

APPENDIX III

PR+3: LAF3

OBSERVED

200

4487

521565687031

10001

17047

2092721514

22746

46986

TABLE 1CENTERS OF

CALC 0-C

191 92303 .4495 -7

5196 196595 -267009 22

10012 -10

17052 -4

20935 -721555 -4021743 ..22690 56

46986 0

GRAVITY

STATE

3H43H53H 6

3F23F33F4

1G4

1D2

3P03P11I63P2

1S0

PAGE 2APPENDIX III

TABLE 2U(K)*2 FOR PR+3

J1 LEVEL 1 J2 LEVEL 2

00000

11111111

2222222

222222

33333

4444

444

55

6 4499 6

20931 220931 220931 420931 420931 6

21552 121552 221552 221552 321552 421552 421552 521552 6

5194 25194 25194 35194 45194 45194 55194 6

17054 217054 317054 417054 417054 517054 6

6595 36595 46595 46595 56595 6

189 4189 4189 5189 6

7008 4.7008 57008 6

2305 52305 6

519417054

18970084499

215525194

170546595189

700823054499

5194170546595189

700823054499

170546595

189700823054499

6595189

700823054499

189700823054499

700823054499

2305 0.9192 0.3668 0.12144499 0.1080 0.2327 0.6420

4499 1.2383 0.7108 0.7878

(U2)*2 (U4)*2 (U6)*2

0.2954 0.0 0.00.0152 0.0 0.00.0 0.1729 0.00.0 0.1079 0.00.0 0.0 0.0726

0.1607 0.0 0.00.2685 0.0 0.00.0799 0.0 0.00.5714 0.1964 0.00.0 0.1702 0.00.0 0.2636 0.00.0 0.2857 0.08920.0 0.0 0. 1246

0.0618 0.0065 0.00.0140 0.0866 0.00.0209 0.0509 0.00.5090 0.4030 0.11730.0014 0.0012 0.09050.0 0.2977 0.65900.0 0.0164 0.3038

0.3745 0.3295 0.00.0319 0.0177 0.00.0027 0.0174 0.05340.6015 0.0000 0.02010.0 0.0019 0.00040.0 0.0686 0.0066

0.0625 0.0030 0.06250.0653 0.3465 0.69820.0252 0.0731 0.00540.6285 0.3467 0.00.0 0.3182 0.8459

0.7782 0.4645 0.26420.0189 0.0503 0.48890.1095 0.2012 0.61150.0000 0.0333 0.1392

0.0146 0.2427 0.03580.0296 0.3116 0.44070.5668 0.6095 0.4623

APPENDIX IV

PAGE 1

APPENDIX IV

TABLE 1ND+3: LAF3

OBSERVED CALC 0-C STATE J

045

136296500

338

142294500

1978 19632037 20422068 20752091 20982187 22032223 2227

3918 39013978 39834038 40434076 41024118 41264208 42054278 4275

5816 58205874 58385986 59976141 61716167 61876323 62936454 64206556 6545

11592 1159611634 11626

12596 1258512614 1258912622 1263012676 1267812694 1270412754 1276312843 1285412902 12873

13514 1351413590 1358313671 1367313676 1369313711 1369513715 13711

-2 417 41

-5 412 410 41

15 41-4 41-6 41-6 41

-15 41-3 41

17 41-4 41-4 41

-25 41-7 41

3 413 41

-3 4136 41

-10 41-29 41-19 41

30 4134 4111 41

9/2 -9/29/2 5/29/2 3/29/2 1/29/2 -7/2

11/2 -11/211/2 5/211/2 3/211/2 1/211/2 -7/211/2 -9/2

13/2 13/213/2 5/213/2 3/213/2 1/213/2 -7/213/2 -9/213/2 -11/2

15/2 15/215/2 -7/215/2 -9/215/2 5/215/2 1/215/2 3/215/2 -11/215/2 13/2

-3 4F 3/2 1/28 4F 3/2 3/2

11 2H225 4F-7 4F-1 4F-9 4F-8 2H2

-10 2H229 2H2

0 4F7 4F

-1 4F-16 4S

16 4S4 4F

9/2 -7/25/2 3/25/2 1/25/2 5/25/2 1/29/2 3/29/2 -9/29/2 5/2

7/2 3/27/2 -7/27/2 1/23/2 1/23/2 3/27/2 5/2

MJ

PAGE 2

APPENDIX IV

TABLE 1ND+3: LAF3

OBSERVED CALC 0-C STATE J

1483414861148921492614959

159971603316060160461610016165

17316173061736317510175201757117605

19147192351925219324

1956719615196511970419686

197411979919835

S...

19960

2115521176211982123221252

1484714861148861492414957

160281604616059160601609516140

17308173111736017491175051756417611

19139192451927119324

195671963219645196871969319737197381979019845199171992719971

2115021183211992123521267

21338 2133921353 21352

-120622

-30-12

1-13

525

8-43

19157-5

8-9

-180

0-16

617-6

39-9

-10

5-60-2

-14

01

4F4F4F4F4F

9/2 1/29/2 3/29/2 3/29/2 5/29/2 -7/2

2H2 11/2 5/22H2 11/2 -11/22H2 11/2 3/22H2 11/2 -7/22H2 11/2 1/22H2 11/2 -9/2

4G4G4G4G2G 14G4G

4G4G4G4G

2K4G2K4G4G2K4G2K4G2K2K2K

2G12G 12G 12G 12G1

5/25/25/27/27/25/25/2

7/27/27/27/2

13/29/2

13/29/29/213/29/2

13/29/213/213/213/2

9/29/29/29/29/2

2D1 3/22D1 3/2

3/21/25/25/23/25/21/2

5/21/23/2

-7/2

13/25/21/2

-7/2-9/2

-11/23/23/21/25/2

-9/2-7/2

-7/23/2-7/21/2

-9/2

3/21/2

M(I

PAGE 3

APPENDIX IV

TABLE 1ND+3: LAF3

OBSEPVED CALC

2163321718

218 46

"""

S...

21992

23473

2399124080

2153521620216212173121774217792179021824218272185721901219312194621995

23455

239962399924057

0-C STATE J

12-12

-10

-2

4G2K4G4G4G2K4G2K2K2K2K2K2K4G

11/215/211/211/211/215/211/215/215/215/215/215/215/211/2

MJ

-7/215/25/2

-9/2-11/2

13/2-9/21/25/23/2

-11/2-9/2-7/2

1/2

18 2P 1/2 1/2

-723

2D12D12D1

5/25/25/2

3/21/25/2

26378 26394 -15 2P26426 26416 10 2P

28341 28361 -19 4D28374 28369 5 4D

S...

28525...

28676289622946329489295682964429773

30275...

S...

...

...

...

30576...

30631306823071930807

28495285282863428686289382945929475295652965929767

302713034630411304513053330534306023061530646307013071230792

-2

-9244

143

-146

4."..

...

S...

...

...

-25.0..

-14-18715

4D4D214D4D2121212121

2L2L2L2L2L2L4D2L4D2L4D4D

3/23/2

3/23/2

5/25/2

11/25/21/2

11/211/211/211/211/2

15/215/215/215/215/215/27/2

15/27/2

15/27/27/2

1/23/2

1/23/2

5/21/2

-11/23/21/2

-9/2-7/2

5/23/21/2

13/215/21/23/2

-11/25/25/2

-9/2-7/2-7/23/21/2

PAGE 4

APPENDIX IV

TABLE 1ND+3: LAF3

OBSERVED CALC 0-C STATE J

.3.0 .3089330933

309943103031068

3178131859

S...

.5..

S...

3303033107331813322833255

30850308953095531002310413107031079

317673183631926319683199032013320483209332126

3303533137331683322633258

33619 3361233649 33631

34292343803441934521

...

...

3467834706

...

386903873538841

4010340155

.. S

40288

342743437434445345193455134573346863470934818

387233877838815

40113401264018740254

-1-21

-7-10

-1

1423

..

-4-29

132

-2

718

186

-252

...

-7-2

...

-32-42

26

-929

-25

34

21212121212121

2L2L2L2L2L2L2L2L2L

2H12H12H12H12H1

13/2 -11/213/2 -9/213/2 -7/213/2 13/213/2 5/213/2 3/213/2 1/2

17/217/217/217/217/217/217/217/217/2

9/29/29/29/29/2

2D2 3/22D2 3/2

2H1 11/22H1 11/22D2 5/22H1 11/22H1 11/22H1 11/22H1 11/22D2 5/22H1 11/2

2F2 5/22F2 5/22F2 5/2

2F2 7/22F2 7/22F2 7/22F2 7/2

15/217/21/23/2

13/25/2

-11/2-9/2-7/2

-7/21/2

-9/25/23/2

3/21/2

-9/21/25/2

-7/21/23/2

11/"

3/25/2

5/21/23/2

-7/23/21/25/2

PAGE 5

APPENDIX IV

TABLE 1ND+3: LAF3

OBSERVED CALC

47894479374799948043

48839489084897749088

...

4787147888479644800648055

48861488694897949065

66548667056679366859

678576785868075

0-C STATE J

6-26

-6-11

-2139-123

... .

.. 0

2G 22G22G22622G2

2G22G22G22G2

2F12F12F12F1

... 2F1

... 2F1

... 2F1

9/29/29/29/29/2

7/27/27/27/2

7/27/27/27/2

5/25/25/2

5/2-9/2

3/2-7/21/2

-7/23/25/21/2

5/2-7/23/21/2

5/23/21/2

PAGE 6

APPENDIX IV

TABLE 1AND+3:LAF3 CENTERS OF GRAVITY

CALC CENTER STATE

235 41 9/22114 4111/24098 4113/26148 4115/2

11621 * 4F 3/212660 4F 5/212768 2H 9/213619 4F 7/213691 4S 3/214899 4F 9/216105 2H11/2

17428 4G 5/217469 4G 7/219293 4G 7/219709 4G 9/219785 2K13/221425 2D 3/221714 4G11/221780 2K15/223458 2P 1/224004 2D 5/226424 2P 3/2

PAGE 7APPENDIX IV

TABLE 2U(K)*2 FOR ND+3

J1 LEVEL 1 J2 LEVEL 2

1/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/2

3/23/23/23/23/23/23/23/23/23/23/23/23/23/23/23/23/23/23/23/2

3/23/23/23/23/23/23/23/23/23/23/23/2

23458234582345823458234582345823458234582345823458234582345823458234582345823458

1162111621116211162111621116211162111621116211162111621116211162111621116211162111621116211162111621

136911369113691136911369113691136911369113691136911369113691

3/23/23/23/25/25/25/27/27/27/29/29/29/29/2

11/213/2

3/23/23/25/25/25/27/27/27/29/29/29/29/2

11/211/211/213/213/215/215/2

3/23/25/25/27/27/29/29/29/29/2

11/211/2

11621136912142526424126601742824004136191746919293

2351276814899197091610519785

116212142526424126601742824004136191746919293

2351276814899197092114

1610521714

4098197856148

21780

214252642417428240041746919293

235127681489919709

211416105

(U2)*2 (U4)*2 (U6)*2

0.01310.01750.02910.00560.01010.03460.02600.00.00.00.00.00.00.00.00.0

0.06120.00500.00240.07730.48560.00090.00630.07350.10620.00.00.00.00.00.00.00.00.00.00.0

0.00730.00600.00070.00710.00060.00210.00.00.00.00.00.0

0.00.00.00.00.00.00.00.02000.00970.00260.03960.08710.00330.00100.00.0

0.00.00.00.05330.04330.00070.08000.04000.06290.22830.01490.00460.05700.14230.00010.00150.00.00.00.0

0.00.00.17750.00000.07310.20230.00250.00440.00230.19220.00000.0563

0.00.00.00.00.00.00.00.00.00.00.00.00.00.00. 16430. 1746

0.00.00.00.00.00.00.00.00.00.05540.02230.11150.14510.40830. 00810.09920.20930.00760. 02800. 0094

0.00.00.00.00.00.00.23470.00010.00110.00090.20990.0016

PAGE 8APPENDIX IV

TABLE 2U(K)*2 FOR ND+3

J1 LEVEL 1 J2 LEVEL 2

3/23/23/23/2

3/23/23/23/23/23/23/23/23/23/23/23/23/23/23/23/23/23/2

3/23/23/23/23/23/23/23/23/23/23/23/23/23/23/23/23/2

5/25/25/25/25/25/25/25/2

13691136911369113691

214252142521425214252142521425214252142521425214252142521425214252142521425214252142521425

2642426424264242642426424264242642426424264242642426424264242642426424264242642426424

1266012660126601266012660126601266012660

11/213/215/215/2

3/23/25/25/25/27/27/27/29/29/29/29/2

11/211/211/213/215/215/2

3/25/25/25/27/27/29/29/29/29/2

11/211/211/213/213/215/215/2

5/25/25/27/27/27/29/29/2

2171440986148

21780

2142526424126601742824004136191746919293

2351276814899197092114

1610521714197856148

21780

264241266017428240041361919293235

127681489919709

211416105217144098

197856148

21780

126601742824004136191746919293

23512768

(U2)*2 (U4)*2 (U6)*2

0.00.00.00.0

0.01610.00520.00380.00020.17440.00250.01040.02060.00.00.00.00.00.00.00.00.00.0

0.08380.00220.00580.00740.00030.00630.00.00.00.00.00.00.00.00.00.00.0

0.04620.26710.00050.06550.25040.25690.00060.0062

0.32450.00.00.0

0.00.00.00000.00390.00110.00010.08390.04180.02020.02590.00030.00230.00160.10780.03000.00.00.0

0.00.00330.00000.00010.00070.00520.00100.01000.05270.06010.01590.01940.00430.00.00.00.0

0.02180.13010.00060.05400.00750.00090.23370.0308

0.00040. 32950. 33060.0030

0.00.00.00.00. 00.00.00.00.00010. 08720.01390. 03250. 03260. 18070.00030. 08600.00830.3482

0.00.00.00.00.00.00.00050.08130.05780. 06470.00050.01360. 00000.00980.23190.00290. 0076

0.00.00.00.08720.07500.11950.39830.0052

PAGE 9APPENDIX IV

TABLE 2U(K)*2 FOR ND+3

JI LEVEL 1 J2 LEVEL 2

5/25/25/25/25/25/25/25/25/2

5/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/2

5/25/25/25/25/25/25/25/25/25/25/25/25/25/2

7/27/27/27/27/27/27/27/2

126601266012660126601266012660126601266012660

17428174281742817428174281742817428174281742817428174281742817428174281742817428

2400424004240042400424004240042400424004240042400424004240042400424004

1361913619136191361913619136191361913619

9/29/2

11/211/211/213/213/215/215/2

5/25/27/27/27/29/29/29/29/2

11/211/211/213/213/215/215/2

5/27/27/27/29/29/29/29/2

11/211/211/213/213/215/2

7/27/27/29/29/29/29/2

11/2

14899197092114

1610521714

4098197856148

21780

1742824004136191746919293

235127681489919709,21141610521714

4098197856148

21780

24004136191746919293

2351276814899197092114

1610521714

40981978521780

136191746919293

2351276814899197092114

(U2)*2 (U4)*2 (U6)*2

0.01050.19120.00.00.00.00.00.00.0

0.00240.00140.03820.00020.00000.89750.00120.00260.00000.00.00.00.00.00.00.0

0.29770.00050.00640.00030.00000.00780.00030.00000.00.00.00.00.00.0

0.15250.12670.17470.00110.00560.09340.55480.0009

0.05080.09950.16980.00300.06110. 18170.00260.00.0

0.18830.00080.10470.13210.23910.41260.01340.00700.10350.28670.00020.00940.03420.00180.00.0

0.00450.01710.06880.04810.00020. 19940.01100.00770.00010. 25230.03140.00370.00330.0

0.00820.05890.07320.04060.03440.09120.00010.2335

0.10910.00220. 03690.02390.19760.40100.00480.23000.0051

0.00.00.16260. 08450.05710.03460.00180. 13030.24770.09610.01450.09140. 04850.00770. 00460. 0051

0.00.00640. 17560.05580.00170.07910.01320.00640.00290.01840. 00060.01690. 18280.4889

0;10330.,01040.00230. 42720.00400.07830.08250. 3076

PAGE 10APPENDIX IV

TABLE 2U(K)*2 FOR ND+3

J1 LEVEL 1 J2 LEVEL 2

7/27/27/27/27/27/2

7/27/27/27/27/27/27/27/27/27/27/27/27/2

7/27/27/27/27/27/27/27/27/27/27/27/2

9/29/29/29/29/29/29/29/29/29/29/2

9/29/29/29/2

136191361913619436191361913619

17469174691746917469174691746917469174691746917469174691746917469

192931929319293192931929319293192931929319293192931929319293

235235235235235235235235235235235

12768127681276812768

11/211/213/213/215/215/2

7/27/29/29/29/29/2

11/211/211/213/213/215/215/2

7/29/29/29/29/2

11/211/211/213/213/215/215/2

9/29/29/29/2

11/211/211/213/213/215/215/2

9/29/29/2

11/2

16105217144098197856148

21780

1746919293

2351276814899197092114

16105217144098

197856148

21780

19293235

1276814899197092114

16105217144098

197856148

21780

2351276814899197092114

16105217144098

197856148

21780

1276814899197092114

(U2)*2 (U4)*2 (U6)*2

0.03360.17290.00.00.00.0

0.01110.06330.07070.00880.02610.07300.39960.00680.01190.00.00.00.0

0.09370.05960.05080.00060.03750.66840.00110.00230.00.00.00.0

0.11950.00950.00090.00440.01940.00000.00000.00000.00710.00.0

0.11560.04870.03900.0028

0.01540.16230.33140.06640.15530.0000

0.00020.00080.17200.02920.05040.13950.17640.00000.12330.08750. 70620.00100.0383

0.04330. 17090.05430.01120.14490.10750.01970.03410.24070.51790.02730.0186

0.17270.00820.00920.05840. 10720.00270.00520.01350.00020.00000.0052

0.00160.00290.04780.0004

0. 12870. 14470. 00010.00030.61660.0077

0.01890. 03590.02740. 19890. 26650.00340. 05220. 35930.00150.03450.00120. 10640.0122

0.07720.05660. 36850.00990.14170. 00990.10200.38110.06130.00140. 00450.0018

0.68920. 11950.04060.03831. 16390.01040.00790.45490.03300.04520.0149

0.27280.00180. 09820.0254

PAGE 11APPENDIX IV

TABLE 2U(K)*2 FOR ND+3

J1 LEVEL 1 J2 LEVEL 2

9/29/29/29/29/29/2

9/29/29/29/29/29/29/29/29/2

9/29/29/29/29/29/29/2

"9/2

11/211/211/211/211/211/211/2

11/211/211/211/211/211/2

11/211/211/211/211/2

13/213/213/2

127681276812768127681276812768

148991489914899148991489914899148991489914899

1970919709197091970919709197091970919709

2114211421142114211421142114

161051610516105161051610516105

2171421714217142171421714

409840984098

11/211/213/213/215/215/2

9/29/2

11/211/211/213/213/21 5/215/2

9/211/211/211/213/21 3/215/215/2

11/211/211/213/213/215/215/2

11/211/213/213/215/215/2

11/213/213/215/215/2

13/213/215/2

16105217144098

197856148

21780

1469919709

21141610521714

4098197856148

21780

197092114

1610521714

4098197856148

21780

21141610521714

4098197856148

21780

1610521714

4098197856148

21780

217144098

1 97856148

21780

4098197856148

(U2) *2 (U4) *2 (U6) *2

0.06870.17960.03890.12110.00.0

0.14070.11810.00010.08730.94230.00290.04620.00.0

0.00420.14030.00320.05210.95520.02100.00.0

0.13210.00430.00440.02560.00020.00000.0020

0.01070.00090.00430.00140. 12930.1556

0.00150.12830.00011.53010.0099

0. 16930.00320.0195

0.00370.00010.00640.00630.21550.4675

0.09600.15930.03280.02390. 20600.21480.00420. 50000.0003

0.00630.34950.03080.51020.38430.04400. 14260.1408

0.11590.00940.06450.13520.00000.01090.0003

0.00090.08090.01680.00440. 06870.0017

0.63440.35140.00770.89150.0162

0.17290.00010.1187

0.27550.00370.12171.12920. 07750.4145

0.00030.02070.37020.01720.04110.51020.29310.46280.2037

0.01220.05050.23100. 12060.01570.11310. 23980. 0246

0.06730. 00620. 05631.23760.01680.41800.0039

0.02840.01090. 00290.47950.00001.5224

0. 18580. 16090. 02330. 15900.0912

0.23310.00241.4522

2.34 2.38 2.42 2.46 2.50 2.54I I ' I '

2.5 -

01.5-

orZr)

.00

4200 4100 4000 3900CM'

Fig. 1. X Group ( I13/2)

i

1.52 1.56 1.60 1.64 1.68 1.72p.I I I I I I I

LODO

0.2

D CDz (0-

<0 .. ')

z II

0.10.0

6600 6400 6200 6000 5800

CM-'

Fig. 2. W Group (4I15/2)

I

0.50.6O.7

v

TN

IMP

UZ

0m,

11700 11600 11500

CM'

Fig. 3. R Group (4F3/2)

2.0-

1.51-

I

I.0F-

0.51-

0.0 - 1 r

i

0.85 0.86 O.87,.I I II I

I I

0

0.670 0.672 0.674 0.676u

V.4

r' )

OO

0.3

( N C

0.0 -

CCM-

WOB.2G (z I)

0Cv)

0.I

0.0

14950 14900 14850 14800CM'

Fig. 6. B Group (4F9 2)

^0.668

0.622

0

- O

oc00

16100

CM'

Fig. 7. C Group (2H11/2

0.250.620

0.20-

0.624 0.626p

Oa

x .15WVE

0N'm4

0.051

0.00

(D

16150 16050 16000

I

T

IT

T I I i

II

0.57

ILC0t0

I

17600

CM~Fig. 8. D Group (4G7/2 + 4G5/2

02.O i

.56

1.5f-

058 uI

r

OD

1 .0 -D

WVZ

U)

4

0.51

0.0-

17800 17400

IV

17200

-

W

.. " r ".

I I -

i

-'V

0

I

0

-

ti

TDi

II

11 i iI

0.517 0.519 O.52I~

1.21-

1915019250

CM~

Fig. 9. E Group (4G7/2)

1.4- NI),N

to

N

rti

01.

.O-

- VCJ.N

-o

I-z

0.8W

00.6

0.4

0.2

0.0I I - I

0.521p0.517 0.519

TT

iI I I

i

.5 .505 .5IOg

19835119960

1919799

I

IC

'97

113686

1965119615

19567

19800 19600

CM~Fig. 10. F Group (2K13/2 + 4G9/2)

wV-z4

0Co)mn4

20000

.5 .505 .510 sT

1 i

.455 .460 .465 .470

21877Q

0 21992 21 8a 21807

21768211

21718I2168121633 21542 21398

215192147 21353 21321 221474 21321 22

21338 21245

2155

22000 21800 21600 21400 21200

CM-,

Fig. 11. G Group (2K15/2 + 4G11/2 + 2D3/2 + 2G9/2)

0.415 0.416 0.417 0.418/L0.06 - - - - - - - - 7 -

N

cn2O.O4-0

002-

2 N

24100 24000 23900

CM~

Fig. 12. I Group (2D5/2 + 2P1/2

0.375 0.380k

ft)

N

N

N

I _t

26600 26200CM 1

Fig. 13. K Group (2P3/2 )

0.02

NoI-Z

Wz

0Nom4

0.00I I

0.375 O.380p

, 1

O.01}

I

2.0 0.344 0.348 0.352

2854528650-

1.5 28659

28676 28501z

v 1.0za

0 2896228703

a 0.5 28597

2852528435

28374

0.0 -2834129100 28700 28300

CM1

Fig. 14. L Group ( z+ 4D512 + 4D3/2 )

0.336 0.338 0.340 p

0.1-

(1)

z

w

U)

m

29773 29644 2956829489

29463

29800 29600 29400

Fig. 15. L. Group (2X11/2)

rvu. 020 0325 330

30576

S3O68 '130275 -0 30893S3 30933 3071930318

0.0- 31859 31155 3099807 30631 30346

31781

31900' 31700 31500 31300 31100 30900 30700 30500 30300

CM'

Fig. 16. O,N,M Groups (2L17/ + 213/2 + 4D7/2 + 2 L15/2)

0.330 90320 0.3250315

0.296 0.298 0.300 0.302 0.304I.IP

33649

33619

3325533228

3318133107

33030

33800 33600 33400 33200

(2D3/2 + 2H9/2Fig. 17. Q and P Groups

0.31

z

C3

0Ul)CIn 0.I

0.0133000

i

0.298 0.300 0.302 0.304 Iu0.296

T I I i

1 ii i

0.286 0.288 0.290 0.292~0.10

0.08

0.06

zw0

0.040CD,

0.02

0.00

34700 34500 34300

Fig. 18. R' Group (2H11/2 + 2D5/2

I-

467834893 370

34635.34521

344193438 I

34292

35100 34900I I I

0.286 0.288 0.290 0.292

i I I I

i i I I I

0.6-0 60.257

0.258

O.259

4

(0Q4z

0 0.2

I I 0

38800 38600

CM- 1

Fig. 19. T Group (2F5/2)

0.247 0.248 0.249 0.250~

In

0

N0

40200

U Group (2 F7/2)

Z,

W.zw

0Cm

Mf0

lo

40400 40000

0.247 0.248 0.249 0.250

r T 1

I _- -L

Fi g . 20.

0.20916 plI

I1

0D0

0YOD.

0M

0 I

O I

o I

IrbO

I/

49200 48800 48400 48000

CM~'

Fig. 21. 2 G7/2 and 2G9/2 Groups

Iz

wVM

z4

04

'

I I I I I I

0.20316 0.20515 0.20716ri

T 'T T T I I I

iI

or)I I I II I1

APPENDIX V

PAGE 1

APPENDIX V

PM +3: LAF3

OBSERVED

...

...

... 0

... 0

... 0

TABLE 1CENTERS OF GRAVITY

CALC 0-C STATE

120161232394951671s4

126381308013933144861488716223

1693918053180751825518565198622030720554219352247522807231402377224216247022484024907258112589525907

... "

... "

... "

... "

... "

... "

... "

... 0

514515516517518

5F15F25F35S25F45FS

3K65G23H43K75G 33K83HS5G43G 35G55G63D23L73P13H63G43L 83P03D33L9

PAGE 3APPENDIX V

TABLE 2U(K)*2 FOR PM+3

J1 LEVEL 1 J2 LEVEL 2

000000

11111111111111111111

111111111111111

222222

25810258102581025810258'25P

12: 8126581265812658126581265812658126581265812658126581265812658126581265812658126581265812658

242212422124221242212422124221242212422124221242212422124221242212422124221

130941309413094130941309413094

244666

12223334444555566777

222334455566677

222333

231421488218083

32471695722787

12658130941807023142139391857721946

161148821808320565

1639162082030322476

324722787

49371825823796

13094180702314218577219461808320565

16392030322476

32471695722787

493716258

130941807023142139391857721946

(U2) *2 (U4)1*2 (U6) *2

0.00540.00.00.00.00.0

0.02680.05290.26820.00320.01770.19650.02250.00.00.00.00.00.00.00.00.00.00.00.00.0

0.00130.00770.00820.01000.05390.00.00.00.00.00.00.00.00.00.0

0.02020.28600.00010.08990.14860.0339

0.00.00190.01420.00.00.0

0.00.00.00.00.06410.04690.01050.14050.03220.02770.06040. 5320.00160.00900.00660.00.00.00.00.0

0.00.00.00.01010.05370.01370.02360.00470.04020.03400.00.00.00.00.0

0.07940.07220.00110.00030.03700.0127

0.00.00.00.00370.08200.0168

0.00.00.00.00.00.00.00.00.00.00.00. 15440.13030.04620.08560. 29860.03030.06800.00460. 0016

0.00.00.00.00.00.00.00.01320.01210.00540.00490.00190.06540.00190. 2964

0.00.00.00.00.00.0

PAGE 4APPENDIX V

TABLE 2U(K)*2 FOR PM+3

J1 LEVEL 1 J2 LEVEL 2 (U2) *2 (U4) *2 (U6) *2

2 130942 130942 130942 130942 130942 130942 130942 130942 130942 130942 130942 130942 130942 13094

2 144862 144862 144862 144862 144862 144862 144862 144862 144862 144862 144862 144862 144862 14486

2 180702 180702 180702 180702 180702 180702 180702 180702 180702 180702 180702 180702 180702 180702 180702 180702 180702 180702 180702 18070

4 1614 148824 180834 205655 16395 162085 203035 224766 32476 169576 227877 49378 66748 19845

2 180703 185773 219464 1614 180834 205655 16395 162085 203035 224766 32476 227877 49378 6674

2 180702 231423 139393 185773 219464 1614 148824 180834 205655 16395 162085 203035 224766 32476 227877 49377 182587 237968 66748 19845

0.00160.03850.09160.20710.00.00.00.00.00.00.00.00.00.0

0.00300.00330.00000.00000.00030.00070.00.00.00.00.00.00.00.0

0.00110.00350.06100.00130.00130.72930.00240.00060.00270.00.00.00.00.00.00.00.00.00.00.0

0.2042 0.12580.0705 0.05370.0097 0.01740.0209 0.09290.0594 0.28780.0172 0.14670.0733 0.01350.0634 0.01250.2200 0.08250.0032 0.00330.0032 0.11470.0 0.30570.0 0.07430.0 0.0130

0.1411 0.00.2032 0.00.0308 0.00.0011 0.22960.0794 0.00030.2059 0.00220.0000 0.19080.0035 0.00040.1422 0.00140.1509 0.00230.0023 0.23940.3099 0.00370.0 0.36830.0 0.3463

0.0345 0.00.0013 0.00.1119 0.00.2434 0.00.0203 0.00.2412 0.00490.0212 0.20000.1209 0.00810.1409 0.11330.2702 0.04070.0003 0.09880.0365 0.08020.0235 0.09230.0372 0.07810.0008 0.02030.0 0.02030.0 0.00970.0 0.01580.0 0.00130.0 0.0111

PAGE 5APPENDIX V

TABLE 2U(K)*2 FOR PM+3

J1 LEVEL 1 J2 LEVEL 2

222222222222222222

3333333333333333333

3333333333

231422314223142231422314223142231422314223142231422314223142231422314223142231422314223142

13939139391393913939139391393913939139391393913939139391393913939139391393913939139391393913939

18577185771857718577185771857718577185771857718577

233344445555666778

33344

s

4555566677789

3344445555

23142139391857721946

161148821808320565

1639162082030322476

32471695722787182582379619845

139391857721946

161148821808320565

1639162082030322476

32471695722787

49371825823796

667425888

1857721946

161148821808320565

1639162082030322476

(U2)*2 (U4)*2 (U6)*2

0.05210.01040.01610.00020.00440.00100.08780.00670.00.00.00.00.00.00.00.00.00.0

0.02210.37400.03440.00010.12950.04070.22480.00000.02850.18190.15600.00.00.00.00.00.00.00.0

0.00000.00700.15380.06230.01840.00480.70980.00110.00000.0020

0.08300.00000.00780.00660.00200.00450.07910.04120.00030.00030.02010.06700.01970,00810.03310.00.00.0

0.05890.04150.00070.10790.03490.00070.03790.24290.06730.09280.05950.07770.00560.07270.24880.00460.00080.00.0

0.04710.00000.28550.07920.11970.09220.02760.00320. 19070.0568

0.00.00.00.00.00250.00240.01270.00000. 03420. 00090. 13290.09270.01840.07230.06470.01260.01370.4641

0.04090. 12230. 03440.42260.11350.00400.03160.02600. 08100.02850.01100.20990.00120.17610. 25510.00320.00080.33260.0094

0.08760.05790.05120.01560.00020.05700.01730.22120.09100.0397

PAGE 6APPENDIX V

TABLE 2U(K)*2 FOB PM+3

J1 LEVEL 1 J2 LEVEL 2

333333333

333333333333333333

44444444444444444

444

185771857718577185771857718577185771857718577

219462194621946219462194621946219462194621946219462194621946219462194621946219462194621946

161161161161161161161161161161161161161161161161161

148821488214882

666777889

344445555666777889

44445555666777889

444

324716957227874937

1825823796

66741984525888

21946161

148821808320565

1639162082030322476

32471695722787

49371825823796

66741984525888

161148821808320565

1639162082030322476

32471695722787

49371825823796

66741984525888

148821808320565

(U2) *2 (U4) *2 (U6) *2

0.00.00.00.00.00.00.00.00.0

0.10570.01370.01080.02980.04950.16230.00040.00420.02240.00.00.00.00.00.00.00.00.0

0.11560.00040.00790.00810.02470.00000.00010.00020.00170.00210.00000.00.00.00.00.00.0

0.01800.22260.2695

0.32980.00080.02380.05330.02450.06060.00.00.0

0.02280.04280.02690.02640.16010.03140.01000.00050.13410.04220.01890.00220.00480.08060.26600.00.00.0

0.13930.02900.03130.08030.11720.00200.00790.00820.03000.00230.00030.00240.00180.00140.00000.00020.0

0.00000.11790.0467

0. 03880. 00730.11780.06110.00700.00880.00480.01050.0362

0.00940.00560.00000.11620.00460.02010.01060.01630. 08250.00030.04980.00970.01740.03100.16140.00860.00260.0110

0.34950. 24000.02780.07640.97020.03410.01020.03300.68910.01010.00160. 15730.01910.00950.01010.00800.0017

0.06930.04530.0319

PAGE 7APPFNDJ:X V

TABLE 2U(K)*2 FOR PM+3

J1 LEVEL 1 J2 LEVEL 2

44444444444444

4444444444444444

444444444444444

55

1488214882148821488214882148821488214882148821488214882148821488214882

18083180831808318083180831808318083180831808318083180831808318083180831808318083

205652056520565205652056520565205652056520565205652056520565205652056520565

1639 51639 5

5555666777889

10

445555666777889

10

45555666777889

10

(U2)*2 (U4)*2 (U6)*2

1639162082030322476

32471695722787

49371825823796

6674198452588830230

1808320565

1639162082030322476

32471695722787

493718258237966674

198452588830230

205651639

162082030322476

32471695722787

49371825823796

6674198452588830230

163916208

0.00250.13800.25520.33710.00560.00090.20400.00.00.00.00.00.00.0

0.07710.12870.05150.02450.00170.03020.21660.02850.01460.00.00.00.00.00.00.0

0.01290.17730.02630.00720.00110.76270.02560.00000.00.00.00.00.00.00.0

0.1078 0.04980.0003 0.0229

0.14680.14090.02130.00000.29310.00030.22080. 15890.02540.00120.20250.01180.00.0

0.28550.09250.04490.00250.01220.07440.00010.04000.01330.19030.06170.07830.02490.06520.00.0

0.07440.27200.03560.26390.08040.05060.02570.12900.31060.02470.01060.02520.0)010.00.0

0.36790.01430. 13510.05870. 18300.01580.09150.02390.02570.01370.72320.00500.01860.0041

0.03960.03030. 09790.09460.01960.06890.00000.38930. 16060.04930.77510.19010.00040.40240.00950.0141

0.05200.00520. 16790.00650.01160.01590.26290.15740.03530.34750.06620.03250. 10900.04980.0097

0.05140.1551

PAGE 8APPENDIX V

TABLE 2U(K)*2 FOR PM+3

J1 LEVEL 1 J2 LEVEL 2

555555555555

5555555555555

555555555555

5555555555

163916391639163916391639163916391639163916391639

16208162081620816208162081620816208162081620816208162081620816208

203032030320303203032030320303203032030320303203032030:;20303

22476224762247622476224762247622476224762247622476

55666777889

10

555666777889

10

55666777889

10

5666777889

2030322476

32471695722787

49371825823796

6674198452588830230

162082030322476

32471695722787

49371825823796

6674198452588830230

2030322476

32471695722787

49371825823796

6674198452588830230

224763247

1695722787

49371825823796

66741984525888

(U2)*2 (U4)*2 (U6)*2

0.01200.00720.03520.00000.00010.00230.00030.00140.00.00.00.0

0.14480.29120.14310.00270.00001.08150.01000.00030.00000.00.00.00.0

0.01990.13680.10140.00090.02760.56690.00920.00000.00.00.00.0

0.05570.13770.00240.00190.73520.03880.00050.00.00.0

0.06730.07920.13780.00160.00960.03720.00030.00020.00180.00040.00140.0

0.17990.10710.06740.11180.00000.27770.32470.00260.00030.53680.03050.01520.0

0.04790.22240.15600.00440. 10580.05400.00970.03110.34150.02260.14050.0

0.04450.25520.00780.33390. 15350.02160.02300.18880.04180.0243

0. 08670.037u0.71600.02190.01810.78620.00240.00880. 12480.00340. 00930.0010

0.00110.06340. 08280. 39660.01230.01810.67480.00690.01350.59270.00320.00130.0516

0.08910.06460.14620. 29030.29840.01860.03970.30410.03940.36260.23780.0.20

0.02260.00600. 34940.07750.00290.07800.28240.04310.49740.2139

PAGE 9APPENDIX V

TABLE 2U(K)*2 FOR PM+3

J1 LEVEL 1 J2 LEVEL 2

5 22476 10 30230

666666666

666666666

66666666

7777777

777777

7777

324732473247324732473247324732473247

169571695716957169571695716957169571695716957

2278722787227872278722767227872278722787

4937493749374937493749374937

182581825818258182581825818258

23796237962379623796

66677788

10

66777889

10

6777889

10

777889

10

77889

10

7889

32471695722787

49371825823796

66741984530230

1695722787

49371825823796

6674198452588830230

227874937

1825823796

6674198452588830230

493718258237966674198452588830230

1825823796

6674198452588830230

237966674

1984525888

(U2) *2 (U4) *2 (U6) *2

0.0

0.12160.00170.00990.03650.00010.00130.00140.00000.0

0.01390.00000.00000.04040.19360.00000.00010.00.0

0.01440.13200.00000.00001.36280.00030.00.0

0.15350.00720.00000.02660.00040.00090.0

0.07930.08400.00000.05210.00000.0

0.34010.00000.00010.0010

0.0

0.05420.00060.08510.15280.00430.00010.02430.00010.0033

0.00640.00000.00140.02270.11510.00000.01820.00460.0000

0.24780.36480.00000.00060.76290.00000.04590.0617

0,12250.00260.00050. 13630.00850.00300.0005

0.00540.03510.00360.08060.03880.0168

0.73100.00000.04760.1302

0.0109

0.05620.00050.08461.02110.03420.00000.62240.01090.0024

0.00000.02990.00720.09531.15230.00460. 14700.25820. 0159

0. 19070.18180. 10990.05510. 14950.12120. 15450. 1374

0.03310.00000.01161.55290.04880.03250. 0044

0.35510.53250.01810.00001.29920. 3257

0.00330.00870.06450.0867

PAGE 10APPENDIX V

TABLE 2U(K)*2 FOR PM+3

J1 LEVEL 1 J2 LEVEL 2

7 23796 10 30230

6674667466746674

198451984519845

2588825888

889

10

6674198452588830230

8 198459 25888

10 30230

q 2588810 30230

(U2) *2 (U4) *2 (U6) *2

0.0 0.0215 0.0057

0.20t50.01740.00820.0000

0.14020.19420.0000

0.33660.00940.01560.0354

0.00560. 58840.5142

1.67410.02510.03210. 0577

0.34640.67291.5495

1.3240 0.1557 0.48920.3074 1.2875 1.3587

10 30230 10 30230 3.3000 0.0004 1.5336

8888

888

99

APPENDIX VI

PAGE 1

APPENDIX VI

TABLE 1SM+3: LAF3

OBSERVED CALC

048

115

-152

126

1000 10001044 10171185 12031280 1258

2210 21912245 22362343 23352409 24082473 2461

3520 35293568 35313651 36513676 36583727 37313791 3794

4972 4971

0-C STATE J MJ

2 6H 5/2 3/2-3 6H 5/2 1/2

-10 6H 5/2 5/2

0 6H 7/2 1/227 6H 7/l 5/2

-17 6H 7/2 -7/222 6H 7/2 3/2

19 6H 9/2 1/29 6H 9/2 -7/27 6H 9/2 -9/21 6H 9/2 3/2

12 6H 9/2 5/2

-8 6H 11/2 1/237 6H 11/2 -9/2

0 6H 11/2 -7/218 61H 11/2 3/2-3 6H 11/2 -11/2-2 6H 11/2 5/2

1 6H 13/2 -11/24983 4995 -11 6H 13/2 3/25007 5006 1 6H 13/2 1/25046 5014 32 6H 13/2 5/25056 5038 18 6H 13/2 -9/25123 5116 7 6H 13/2 -7/25160 5183 -22 6H 13/2 13/2

6309 6299 10 6H 15/2 -7/26342 6335 7 6H 15/2 -9/26406 6417 -10 6F 1/2 1/26450 6464 -13 bH 15/2 5/26461 6493 -31 6H 15/2 -11/26567 6565 2 6H 15/2 j/26567 6580 -12 6F 3/2 1/2... 6588 ... 6H 15/2 15/2... 6660 ... 6H 15/2 -11/26691 6710 -18 6F 3/2 3/26707 6736 -28 6F 3/2 1/2

7176 7168 8 6F 5/2 1/27184 7179 5 6F 5/2 5/27223 7228 -4 6F 5/2 3/2

PAGE 2

APPENDIX VI

TABLE 1SM+3: LAF3

OBSERVED CALC 0-C STATE J MJ

7992 7995 -2 6F 7/2 3/28041 8020 21 6F 7/2 -7/28060 8048 12 6F 7/2 1/28092 8096 -3 6F 7/2 5/2

9170 9170 0 6F 9/2 1/29178 9171 7 6F 9/2 -7/29228 9209 19 6F 9/2 -9/29252 9240 12 6F 9/2 3/29268 9268 0 6F 9/2 5/2

10561 10560 1 6F 11/2 -7/210584 10577 7 6F 11/2 -9/210593 10581 11 6F 11/2 5/210603 10616 -12 6F 11/2 -11/210613 10618 -4 6F 11/2 3/210644 10640 4 6F 11/2 1/2

17858 17874 -15 4G 5/2 1/217949 17969 -19 4G 5/2 5/218045 18077 -31 4G 5/2 3/2

18924 18921 3 6F 3/2 3/218942 18934 8 6F 3/2 1/2

20037 20050 -12 4G 7/2 1/220093 20094 0 4G 7/2 -7/220112 20120 -7 4G 7/2 3/220164 20159 5 4G 7/2 5/2

20416 20413 3 4M 15/2 -9/220473 20483 -9 41 9/2 1/220499 20517 -17 41 9/2 5/220526 20528 -1 41 9/2 3/2

... 20541 ... 41 9/2 -7/2... 20653 ... 4M 15/2 -11/2... 20790 ... 4h 15/2 -7/2... 20793 ... 4M 15/2 13/2... 20870 ... 4H 15/2 5/2... 20874 ... 48 15/2 -9/2... 20909 ... 4H 15/2 1/2... 20916 ... 4K 15/2 3/2... 20941 ... 41 11/2 5/2... 20987 ... 4K 15/2 15/2... 21103 ... 41 11/2 -11/2... 21124 ... 41 11/2 3/2... 21148 ... 41 11/2 -7/2... 21249 ... 41 11/2 1/2... 21272 ... 41 11/2 -9/2

PAGE 3

APPENDIX VI

TABLE 1SM+3:LAF3

OBSERVED CALC

...

21663216742170621736

221642220722240

"...

.

..

. .

"...

"...

.

..

...

"..

S...

.. S

2...S...

S...

S...

S...

24084... S

241 19

24153.5..

...

...

21539216072162321638216592166521681

221712222322242

224862253922546225592257022631226782272722734

22794228162285422902229432298123018230252303523045230772311123146

23973240322407424079241002411524118241402414724160241652417224178

0-C STATE J

.. 0

25154055

4I414141414141

13/213/213/213/213/213/213/2

-6 4F 5/2-15 4G 5/2

-1 4F 5/2

S...

S...

S...

...

.. S

S...

S...

S...

S...

S...

...

-6

S...

" "

S...

...

... "

-6"

... "

... "

... "

4M4M4M4M4M4M4M4M4M

4G4G4G414G4141414141414141

4M4M4K4M4M4M6P6P4M4M6P4M4M

17/1.17/217/217/217/217/217/217/217/2

9/29/29/2

15/29/2

15/215/215/215/215/215/215/215/2

19/219/219/219/219/219/25/25/2

19/219/25/2

19/219/2

-7/213/2-9/25/2

-11/21/23/2

3/25/21/2

13/215/2

-11/2-9/2-7/25/23/21/2

17/2

-9/25/23/2

-11/2-7/2

-11/2-7/2-9/213/25/23/21/2

15/2

15/213/217/2

-11/2-9/2-7/2

1/23/21/23/25/2

-19/25/2

PAGE 4

APPENDIX VI

TABLE 1SM+3:LAF3

OBSERVED CALC

246082462924631246442467824682247 10

2491124993250072506425081

2516625182

252042521625248

... e

25282...

..

S. S

.

..

..

..

..

..

.. .

...

.. S

S...

... 5

... S

...

... 0

24620246402464124659246852469324724

249132497424997250552506225093

25153251592519825202252042523825255252752531225335254042542025444255222555525582256032563625641256892569125692256982576425765257782581225812258492586625896

0-C STATE J

-11-10-9

-14

-6-10

-13

-11910919

7-15

0-21-6

-29

..

..

..

..

...

...

..

...

...

...

.. 0

5..

....

..

S...

... S

4L4L4L4L4L4L4L

4G4F4G6P6P4F

4M4K4K4K4K4K4M4K4M4M4M4M4M4M4M4M4L4L4L4M4L4L4L4L4G4L4G4G4G4G4G

13/213/213/213/213/213/213/2

7/27/27/23/23/27/2

21/211/211/211/211/211/221/211/221/221/221/221/221/221/221/221/215/215/215/221/215/215/215/215/211/215/211/211/211/211/211/2

... 4D 1/2 1/2

1/23/2

13/2-11/2-7/25/2

-9/2

5/23/2

-7/21/23/21/2

17/21/23/25/2

-9/2-11/2

15/2-7/2

-19/21/23/25/2

13/2-11/2-9/2-7/2

1/23/2

13/2-21/2-11/2

5/215/2-9/25/2

-7/23/2

-7/2-11/2

1/2-9/2

... 26492

PAGE 5

APPENDIX VI

TABLE 1SM+3:LAF3

OBSERVED CALC

S...

...

S...

S...

S...

S...

... S

... 0

... @

...

2736327417274322744827508

26691266932670726735267472677126792267992680726809268392684526852

269262695326983269902701 12707027110

2736127432274682748427533

0-C STATE J

.. *

...

... @

... 6

... 2

... 4

... 5

... 5

... 4

... "

... "

... "

... "

... "

2

-14

-35

-35

-24

2764E 27655 -627658 27664 -5

276912773427758

28248282622834328409

287152872628753

...

2877828790

...

28810

277192776527773

28250282532834928403

2873828742287532875428781287942880628811

-27-30-14

-19-5

b

-22-150

...

-2-3

...0

4K2K4L4K6P2K4L6P4K6P6P4L4K

4K4K4K4K4K4K4K

4F4F4F4F4F

4D4D

6P6P6P

6P4H4H6P

4K4K4K4K4K4K4K4K

17/217/217/217/27/2

17/217/27/2

17/27/27/217/217/2

13/213/213/213/213/213/213/2

9/29/29/29/29/2

3/23/2

5/25/25/2

7/27/27/27/2

15/215/215/215/215/215/215/215/2

MJ

1/23/25/2

13/2-7/215/217/25/2

-11/23/21/2

-7/2-9/2

1/2-9/2

3/2-11/2

5/2-7/213/2

-7/25/23/21/2

-9/2

. 1/23/2

1/25/23/2

1/25/2

-7/23/2

5/21/23/2

-11/213/215/2-9/2-7/2

PAGE 6

APPENDIX VI

TABLE 1SM+ 3: LAF3

OBSERVED CALC

28938

2898129037290552908629094

S...

..

S...

. .

.. .

S...

S...

.6..

. ..

. . .

. . .

. . .

. . .

. ..

. . .

. . .

. . .

. ..

... 0

289352895629007290632907629103291032910729117291362913729143291382915129168291842920 529223

2929829307293122933129347293562936129365294122944829456294762950329572295822958729590296322964029683297082971229761

0-C STATE J

3...

-25-25-20-16

-8

...

...

...

...

...

...

.@..

.. .

"...

* 0 0

... 6

... 6

... 0

...

4H4H4H6P4H6P6P4K4K4K4K4K6P4K4K4H4K4K

4M4M4L4H4L4H4H4H4H4M4L4M4H4M4H4L4M4H4H4H4H4H*4H

9/29/29/27/29/27/27/2

17/217/217/217/217/27/2

17/217/29/2

17/217/2

19/219/219/211/219/211/211/211/211/219/219/219/211/219/213/219/219/213/213/213/213/213/213/2

MJ

-7/21/2

-9/25/23/23/21/25/2

-7/2-9/2

-11/217/2-7/213/23/25/21/2

15/2

1/23/25/2

-9/217/2

-11/21/2

-7/23/2

15/2-19/2

13/25/2

-11/2-11/2-7/2-9/23/21/25/2

-9/213/2-7/2

PAGE 7

APPENDIX VI

TABLE 1SM+3:LAF3

OBSERVED CALC 0-C STATE J [J

30027 30025 2 4G 7/2 5/230125 30139 -13 4G 7/2 -7/230141 30142 0 4G 7/2 1/2... 30191 ... 4G 9/2 3/2... 30207 ... 4G 9/2 -9/2

30219 30217 2 4G 5/2 5/230235 30262 -26 4G 9/2 -7/230286 30275 11 4G 9/2 1/230329 30346 -16 4G 7/2 3/2

... 30444 ... 4G 5/2 1/2

... 30489 ... 4G 5/2 5/2... 30539 ... 4G 5/2 3/2

... 31222 ... 4P 1/2 1/2

... 31341 ... 2K 15/2 13/2

... 31358 ... 2K 15/2 -11/231420 31412 8 2L 15/2 -9/231442 31454 -11 2L 15/2 5/231473 31474 0 2L 15/2 3/2

... 31488 ... 2K 15/2 1/2

... 31515 ... 4G 11/2 -11/2

... 31518 ... 4G 11/2 5/2

... 31525 ... 2L 15/2 -7/2... 31552 ... 4P 3/2 3/2

31627 31608 19 2K 15/2 1/2... 31615 ... 4G 11/2 3/2... 31628 ... 4G 11/2 -7/2... 31676 ... 2L 15/2 15/2

31716 31710 6 4G 11/2 -9/231761 31735 26 4G 11/2 1/2

32800 32808 -7 4P 5/2 1/232822 32829 -6 4P 5/2 5/232853 32861 -7 4P 5/2 3/2

33608 33583 25 2F 5/2 1/233681 33658 23 2F 5/2 5/233765 33742 23 2K 13/2 -11/2... 33795 ... 2K 13/2 -9/2... 33824 ... 2F 5/2 3/2... 33891 ... 2K 13/2 1/2... 33903 ... 2K 13/2 -7/2... 33942 ... 4F 9/2 -7/2... 33961 ... 2K 13/4 3/2... 33966 ... 4F 9/2 5/2... 33996 ... 4F 9/2 -9/2... 34033 ... 4F 9/2 1/2... 34048 ... 2K 13/2 5/2... 34056 ... 4F 9/2 3/2... 34090 ... 2K 13/2 13/2

PAGE 8

APPENDIX VI

TABLE 1SM+3:LAF3

OBSERVED CALC

S...

5..

S...

S...

5...

...

358903590535954... 963600736055

... "

... "

... "

... "

... "

... "

... "

35 9 "

359" "

359"4"

35996"

360"7"

3"0" "

3433134347343973440234407344533445534474344873453434557345883459934630

355753565135666356783569935701357183572835737357593576235762357943586435874

359023592035955359833600936052

0-C STATE J

S...

.. S

5...

S...

... 1

.. 1

... 0

... 3

... 1

... 3

... "

... "

... "

... "

... "

... "

... "

... e

... "

... "

... "

... "

... "

... "

-11

-14

0

13

-1

3

2L2L2L2L2L2L2L2L412L41414141

2N4F2N4F2N2N2N2N2N2N2N2N4F2P4F

4141414I4141

17/217/217/217/217/217/217/217/29/2

17/29/29/29/29/2

19/27/2

19/27/2

19/219/219/219/219/219/219/219/27/21/27/2

11/211/211/211/211/211/2

MJ

13/215/2

-11/2-7/23/25/2

-11/2-9/25/2

17/25/23/2

-9/21/2

-19/25/2

17/21/2

-9/2-11/2-7/217/213/25/23/21/2

-7/21/23/2

-7/2-9/25/2

-11/23/21/2

PAGE 9

APPENDIX VI

TABLE 1SM+3:LAF3

OBSERVED CALC

..

S...

S..

..

..

..

. .

S..

36517

...

. .

. .

. .

..

..

36755

... S

5...

...

5...

3655

S...

S...

.5..

...

3762337.34376.4376 5

376279

36310363393640436461364963652036526365263655036571365723660236632366433664536648366513665936680367003670236709367263673536738367523678136839368393684036845

369033692236960370373705137072371313721637268

3759937614376193763837661

0-C STATE J

...

..

17

.. .

.. .

...

..

.

..

. .

..

...

*..

...

.. .

..

.. .

...

2420151918

41412K414F2K414I4F2K41412N2N2N4F414F2N2N414141412N4F2N2N2N2N2N

2M2M2M2M2(12M2M2(12M

2H2H2H2H2H

15/215/215/215/25/2

15/215/215/25/215/213/213/221/221/221/23/213/25/2

21/221/213/213/213/217/221/23/2

21/221/221/221/221/2

17/217/217/217/217/217/217/217/217/2

9/29/29/29/29/2

MJ

13/2-11/2-9/2-7/2

1/23/21/25/25/215/2

-11/2-9/2

-19/2-21/2

17/23/2

-7/23/21/23/25/21/23/2

13/25/21/2

15/2-7/213/2-9/2

-11/2

17/217/2-9/21/25/23/2

-11/215/213/2

- 7/21/2

-9/23/25/2

PAGE 10

APPENDIX VI

TABLE 1SM+3:LAF3

OBSERVED CALC

...

3846738492

6.

.

...

....

.

.

.

.

.

..

.

...

.

...

...

.

.

.

.

... 6

... 6

... 6

... 6

... 6

... 6

... 0

... 0

... 6

... 0

... 6

... 6

... 6

... 0

... 0

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 0

381843825638280384673848438499

389023897139000391173914239171392143924539316393613940039446394553946839475395063954139581

403164048640505405554069640698407194076240781408064084940883408954090940972409814106141087411394117841250412524135941382

0-C STATE J

...

09

.

...

.

...

.

...

...

...

.

.

...

...

.

.

.

.... 6

...

... 6

... 6

... 6

... 6

... 0

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 6

... 6

2F2F2F2P2P2F

2K2K2G2K2G2G2K2K2G2K2H2H2H2K2H2H2G2K

2M2M2M2M2M2M12D2D2D2D2D2I2M21212112M

212K212K2K2K2K

7/27/27/23/23/27/2

15/215/27/2

15/27/27/2

15/215/27/2

15/211/211/211/215/211/211/27/2

15/2

19/219/219/219/219/219/219/25/25/2

5/211/219/211/211/211/219/211/213/211/213/213/213/213/2

MJ

3/2-7/2

1/23/21/25/2

13/2-11/2

5/23/21/2

-7/21/25/23/2

-9/2-7/2-9/2

-11/215/25/21/23/2

-7/2

-19/2-7/2-9/2

-11/21/25/23/21/21/25/23/2

-11/217/23/21/2

-9/215/2-7/2

-11/25/2

-9/21/25/23/2

PAGE 11

APPENDIX VI

TABLE 1SM+3: LAF3

OBSERVED CALC

... 41397

... 41534

... 41737

... 41756

4206642124421354217642227

423784246242486

42616

4265842711

..

...

429594299043040

43074.

.

.

.

.

.

.. .

...

4..

43258

43324... 5

4205442102421124215842199

423904246542481

42616426254264242700

427154287542923429384296342973429934302343025430454304543079430884311843144431464316743169431964321543226432284323743266432664327243276432944330243368

0-C STATE J

... 2K

... 2K

... 2D

... 2D

1222231828

-11-2

5

05...

1611

.

.. 6

-13-217

29..

..

..

...

...

.5..

...

...

5..

-7.. 5

.. 5

22...

2G2G2G2G2G

4G4G4G

4G4G4G4G

202K2K4H4H4H4H4H204H2N2K202K2K2K4H2K20414H2020204H4H4H2K2020

13/213/2

3/23/2

9/29/29/29/29/2

5/25/25/2

7/27/27/27/2

21/215/215/29/29/29/29/29/2

21/29/2

21/215/221/215/215/215/211/215/221/211/211/221/221/221/211/211/211/215/221/221/2

MJ

13/2-7/2

1/23/2

-9/25/23/21/2

-7/2

1/25/23/2

-7/21/23/25/2

-21/213/2

-11/23/25/2

-9/21/2

-7/2-9/2-7/2-9/2

1/2-11/2

5/215/2-7/2

1/2-9/213/2

-11/23/2

-19/23/21/2

-7/2-9/2

5/2-7/217/215/2

PAGE 12

APPENDIX VI

TABLE 1SM+3: LAF3

OBSERVED CALC

.

.

..

44597

45122

.. .

45366

...

.

.

..

...

...

...

... e

... e

... e

... e

... 0

... e

... g

... 0

44ee7

434464354243545

4354543565435764359043703

4372843780437894382943888

43953440034400544016

4448144559446224469244710

45031450654507145146

4526245320453244537945440

45668457324577045812458174582845837458454586145869458914591745930

0-C STATE J

...

-24

-23

...

-12

.

.

.

..

...

...

.

.

.

... e

... g

... 0

... g

... e

.. g

4H4H4H4H4H4H2H21

2121212121

4H4H4H4H

2G2G2G2G2G

4G4G4G4G

4G4G4G4G4G

2121214G214G4G4H4G4G214H4G

13/213/213/213/213/213/213/211/2

11/211/211/211/211/2

7/27/27/27/2

9/29/29/29/29/2

7/27/27/27/2

9/29/29/29/29/2

13/213/213/2

11/213/211/211/213/211/211/213/213/211/2

MJ

13/23/2

-7/2-11/2

1/25/2

-9/25/2

-11/23/2

-7/21/2

-9/2

-7/23/21/25/2

-7/21/2

-9/25/23/2

1/2-7/23/25/2

1/2-7/2-9/2

3/25/2

13/21/23/25/2

-11/23/21/25/2

-11/2-7/2-9/2-7/2-9/2

PAGE 13

APPENDIX VI

TABLE 1SM+3:LAF3

OBSERVED CALC

46284

46402

... "

464023

47336

47374..

... 0

4620446222462674626946298463004633046334463414634946393464124642246422464354644546448464734649246512465314655046587465924696246972470574726647327473974747147620

0-C STATE J

-15

-19

16

9-22

2H2H4H2L2L2H2L2L2L2L2H4H4H4H2L2L2L4H4H4H4H4P4P2H2D2D2D2H2H2H2H2H

11/211/213/217/217/211/217/217/217/217/211/213/213/213/217/217/217/213/213/213/213/23/23/211/25/25/25/2

11/211/211/211/211/2

MJ

1/2-9/213/217/2-7/2

-11/23/21/2

15/25/23/21/25/23/2

-11/2-9/213/2-7/2-7/2

-11/2-9/23/21/2

-7/25/23/21/2

-11/25/23/2

-7/2-9/2

PAGE 14

APPENDIX VI

TABLE 1ASM+3:LAF3 CENTERS OF GRAVITY

CALC CENTER STATE

101 6H 5/21135 6H 7/22341 6H 9/23667 6H11/25072 6H13/26387 6F 1/26520 6H15/26637 6F 3/27141 6F 5/28009 6F 7/29189 6F 9/2

10583 6F11/2

18031 4G 5/218982 4F 3/220161 4G 7/220660 41 9/220825 4M15/221147 4111/221644 4113/222301 4F 5/222612 4M17/222873 4G 9/223048 4115/224132 6P 5/224138 4M19/224676 4L13/224995 4F 7/225064 6P 3/225201 4K11/225434 4M21/225667 4L15/225829 4G11/226446 4D 1/226762 4L17/226786 6P 7/2

PAGE 15APPENDIX VI

TABLE 2U(K)*2 FOR SM+3

J1 IEVEL 1 J2 LEVEL 2

1/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/2

1/21/21/21/21/21/21/21/21/21/21/21/21/21/21/21/2

3/23/23/23/23/23/23/23/23/23/23/23/23/23/2

638763876387638763876387638763876387638763876387638763876387638763876387

26446264462644626446264462644626446264462644626446264462644626446264462644626446

66376637663766376637663766376637663766376637663766376637

3/23/25/25/25/25/27/27/27/27/29/29/29/2

11/211/211/213/213/2

3/23/25/25/25/27/27/27/29/29/2

11/211/211/211/213/213/2

3/23/23/25/25/25/25/25/27/27/27/27/27/29/2

663725064

1017141

2230124132

11358009

2499526786

23419189

206603667

1058325829

507224676

189822506418031223012413220161249952678620660228733667

2114725201258292164424676

66371898225064

1017141

180312230124132

11358009

201612499526786

2341

(U2)*2 (U4) *2 (U6) *2

0.01740.19200.19380.01400.00110.01060.00.00.00.00.00.00.00.00.00.00.00.0

0.00790.00610.01800.05320.01060.00.00.00.00.00.00.00.00.00.00.0

0.00290.00110.29220.14440.02220.00100.00360.07110.24340.01830.00230.00100.00240.0

0.00.00.00.00.00.00.13860.03060. 00300.09660. 15210.00710.00120.00.00.00.00.0

0.00.00.00.00.00.02260.07890.00280.08050.02940.00.00.00.00.00.0

0.00.00.00.13650.02820.00000.00030.02370.11740.00800.00090.00630.15920.1183

0.00.00.00.00.00.00.00.00.00.00.00.00.00. 33650.02870.00150.10260. 0026

0. 00.00.00.00.00.00.00.00.00.00.00290. 09920.05270.08160.02970.1446

0.00.00.00.00.00.00.00.00.00.00.00.00.00.3793

PAGE 16APPENDIX VI

TABLE 2U(K)*2 FOR SM+3

J1 LEVEL 1 J2 LEVEL 2 (U2)*2 (U4)*2 (U6)*2

3/2 6637 9/2 9189 0.0 0.0301 0.00983/2 6637 9/2 22873 0.0 0.0000 0.00263/2 6637 11/2 3667 0.0 0.2119 0.02873/2 6637 11/2 10583 0.0 0.0041 0.04893/2 6637 11/2 21147 0.0 0.0062 0.00013/2 6637 11/2 25201 0.0 0.0013 0.00013/2 6637 11/2 25829 0.0 0.0000 0.00133/2 6637 13/2 5072 0.0 0.0 0.40363/2 6637 15/2 6520 0.0 0.0 0. 06153/2 6637 15/2 20825 0.0 0.0 0.00993/2 6637 15/2 23048 0.0 0.0 0.00283/2 6637 15/2 25667 0.0 0.0 0.0021

3/2 18982 3/2 18982 0.0023 0.0 0.03/2 18982 3/2 25064 0.0037 0.0 0.03/2 18982 5/2 18031 0.0763 0.0440 0.03/2 18982 5/2 22301 0.0609 0.0018 0.03/2 18982 5/2 24132 0.0015 0.0771 0.03/2 18982 7/2 1135 0.0083 0.0000 0.03/2 18982 7/2 8009 0.0001 0.0013 0.03/2 18982 7/2 20161 0.0289 0.1525 0.03/2 18982 7/2 24995 0.0149 0.0001 0.03/2 18982 7/2 26786 0.0013 0.0158 0.03/2 18982 9/2 2341 0.0 0.0029 0.00493/2 18982 9/2 20660 0.0 0.1075 0.02043/2 18982 9/2 22873 0.0 0.0401 0.37293/2 18982 11/2 21147 0.0 0.2990 0.09853/2 18982 11/2 25201 0.0 0.1483 0.00663/2 18982 11/2 25829 0.0 0.0016 0.23963/2 18982 13/2 5072 0.0 0.0 0.00143/2 18982 13/2 21644 0.0 0.0 0.05563/2 18982 13/2 24676 0.0 0.0 0.07543/2 18982 15/2 20825 0.0 0.0 0.35773/2 18982 15/2 23048 0.0 0.0 0.13863/2 18982 15/2 25667 0.0 0.0 0.0019

3/2 25064 3/2 25064 0.0096 0.0 0.03/2 25064 5/2 101 0.0000 0.1630 0.03/2 25064 5/2 7141 0.2638 0.0094 0.03/2 25064 5/2 18031 0.0008 0.0001 0.03/2 25064 5/2 22301 0.0012 0.0310 0.03/2 25064 5/2 24132 0.1220 0.0013 0.03/2 25064 7/2 1135 0.0000 0.1589 0.03/2 25064 7/2 8009 0.1450 0.0469 0.03/2 25064 7/2 20161 C.0018 0.0087 0.03/2 25064 7/2 24995 0.0326 0.0070 0.03/2 25064 7/2 26786 0.4736 0.0000 0.03/2 25064 9/2 2341 0.0 0.1125 0.00113/2 25064 9/2 9189 0.0 0.1448 0.00003/2 25064 9/2 20660 0.0 0.0101 0.0023

PAGE 17APPENDIX VI

TABLE 2U(W)*2 FOR SM+3

J1 LEVEL 1 J2 LEVEL 2

3/23/23/23/23/23/23/23/23/23/2

5/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/2

5/25/25/25/25/25/25/25/25/25/25/25/25/25/2

25064250642506425064250 642506425064250642506425064

101101101101101101101101101101101101101101101101101101101101101101101101

71417141714171417141714171417141714171417141714171417141

9/211/211/211/211/211/213/213/215/215/2

5/25/25/27/27/27/27/27/29/29/29/29/2

11/211/211/211/211/213/213/213/215/215/215/217/2

5/25/25/25/27/27/27/27/27/29/29/29/29/2

11/2

228733667

10583211472520125829

507221644

652025667

1017141

2413211358009

201612499526786

23419189

2066022873

366710583211472520125829

50722164424676

6520208252566722612

7141180312230124132

11358009

201612499526786

23419189

2066022873

3667

(v2) *2 (U4) *2 (U6) *2

0-. 00.00.00.00.00.00.00.00.00.0

0.38810.03310.00000.20620.00200.00030.00020.00000.02570.00000.00220.00000.00.00.00.00.00.00.00.00.00.00.00.0

0.00070.00620.00720. 18840.41180.03970.00040.00140.02430.33740.01400.00000.00230.0

0.03880.04650.37020.00400.00390.00490.00.00.00.0

0.05670.28440.02440.19630.14290.00170.00110.00150.13970.02050.00050.00090.02400.00060.00000.00040.00010.00060.00320.00800.00.00.00.0

0.01170.00140.00420.0b740.05070.00370.00080.00740.18020.10130.02980.00110.00040.2305

0.00160.01180.00000.04700.00000.01180.05700.05740.14180.0316

0.00.00. 00.09520.43010.00230. 00050. 06630.32620.34160.00140.00270.26490.05160.01110. 00240.00100.06620.02370.00920.00430.03190. 00560.0054

0.00.00.00.00.41920.00450.00190.00030.00000.04750. 03560.00190.00040.2087

PAGE 18APPENDIX VI

TABLE 2U(K)*2 FOR SM+3

J1 LEVEL 1 J2 LEVEL 2

5/25/25/25/25/25/25/25/25/25/2

5/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/2

5/25/25/25/25/25/25/25/25/25/25/25/25/25/2

7141714171417141714171417141714171417141

180311803118031180311803118031180311803118031180311803118031180311803118031180311803118031180311803118031180311803118031

2230122301223012230122301223012230122301223012230122301223012230122301

11/211/211/211/213/213/213/215/215/217/2

5/25/25/27/27/27/27/27/29/29/29/29/2

11/211/211/211/213/213/213/215/215/215/217/217/2

5/25/27/27/27/27/27/29/29/29/29/2

11/211/211/2

105832114725201258295072

21644246766520

2082522612

18031223012413211358009

20161249952678623419189

20660228733667

211472520125829

507221644246762082523048256672261226762

223012413211358009

20161249952678623419189

206602287336671058321147

(U2)*2 (U4)*2 (v6)*2

0.00.00.00.00.00.00.00.00.00.0

0.34240.00980.01060.00000.00000.00000.00220.00180.00960.00160.03690.00320.00.00.00.00.00.00.00.00.00.00.00.0

0.00030.04440.00040.01230. 18500.02030.01250.00640.00990.02290.02220.00.00.0

0.02250.00270.00160.00000.21160. 00940.00000.00.00.0

0.24830.03690. 03910.00780.00150.00830.00570.02500.00610.00020.11010.01870.00450.00090.00740.00130.00000.30280. 14590.00.00.00.00.0

0.03450.00160.00590.00870.08080.03190.01280.00080.00370.00910.11560.01410.00440.0054

0.05110.00000.00040.00750.28670.00010.00180.34000. 01790.0193

0. 00.00.00. 00750.00010.40940.02810.00080.00190.00020.01880. 03090.00180. 23500.02890.13140.00140.72180.07670.83540.04350.00570.02030. 0092

0.00.00. 00080.00000.03730.11240.00000.00000.00000.11220.07210.00800.00010.3369

PAGE 19APPENDIX VI

TABLE 2U(K)*2 FOR SM+3

J1 LEVEL 1 J2 LEVEL 2

5/25/25/25/25/25/25/25/25/2

5/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/25/2

7/27/27/27/27/27/27/27/27/27/27/27/27/27/27/27/2

223012230122301223012230122301223012230122301

2413224132241322413224132241322413224132241322413224132241322413224132241322413224132241322413224132241322413224132

1135113511351135113511351135113511351135113511351135113511351135

11/211/213/213/213/215/215/215/217/2

5/27/27/27/27/27/29/29/29/29/2

11/211/211/211/211/213/213/213/215/215/215/217/217/2

7/27/27/27/27/29/29/29/2

11/211/211/211/211/213/213/215/2

2520125829

5072216442467620825230482566722612

2413211358009

201612499526786

23419189

2066022873

366710583211472520125829

50722164424676

652020825230482261226762

11358009

201612499526786

23419189

206603667

1058321147252012Ca29

507224676

6520

(U2)*2 (U4)*2 (06)*2

0.00.00.00.00.00.00.00.00.0

0.23860.00000.30460.00620.06510.27370.00040.32460.00070.05770.00.00.00.00.00.00.00.00.00.00.00.00.0

0.28310.04200.00000.00000.00000.29380.00170.00020.03820.00000.00020.0016G.0'0300.00.00.0

0.14950.00560.00350. 11860.13410.00.00.00.0

0.00050.07060.11990.03190.01100.00770.13020.17150.01850.01950.15270.16160.02050.02650.11190.12240.00020.00470.00.00.00.00.0

0.01430.29550.00240.00000.00880. 16520.13280.00070.17920.01040.00000.0 2i50.00.5

0.0245

0.00000.0003

0.01160. 20320. 00700.11110. 06390. 00820.01720. 19340. 6557

0.00. 00500.00000.00170.00050. 00010.01780.00000. 13260. 00520.04670.00000.00090.02170. 06270.08240.04760. 00150. 07400. 02490.20110.01860. 1009

0.29510. 00040.00480.00260.08620.00800.45550.00660.25420.22810.0104

0.28110.00950.0467

PAGE 20APPENDIX VI

TABLE 2U(K)*2 FOR SM+3

1 J2 LEVEL 2J1 LEVEL

7/2 11357/2 11357/2 11357/2 11357/2 1135

7/2 80097/2 80097/2 80097/2 80097/2 80097/2 80097/2 80097/2 80097/2 80097/2 80097/1 80097/2 80097/2 80097/2 80097/2 80097/2 80097/2 80097/2 80097/2 80097/2 80097/2 8009

7/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 201617/2 20161

15/215/217/217/219/2

7/27/27/27/29/29/29/29/2

11/211/211/211/211/213/213/215/215/215/215/217/219/2

7/27/27/29/29/29/29/2

11/211/211/211/211/213/213/213/215/215/215/217/217/219/2

(U2) *2 (U4) *2 (U6) *2

2304825667226122676224138

8009201612499526786

23419189

2066022873

366710583211472520125829

5072216446520

2082523048256672261224138

201612499526786

23419189

2066022873

366710583211472520125829

50722164424676208252304825667226122676224138

0.00.00.00.00.0

0.00340.00700.00350.11280.24930.05500.00000.00000.48810.00650.00000.00000.00120.00.00.00.00.00.00.00.0

0.33270.00660.00030.00000.00040.00990.00060.00990.00100.02400.00000.00790.00.00.00.00.00.00.00.00.0

0.00190.00760.00.00.0

0.02170.00000.00410.17130.13200.00510.00230.00070.03080.04970.00060.00180.00000.43720.00490.13060.00120.00380.00190.00.0

0.11970.05560.04360.00900.00000.07680.00050.00140.00000.07410.00810.00220.00590.01330.06850.14410. 12230.28060.00.00.0

0.01040. 00710. 02200.00330.0043

0.02900.00040.00000. 00000. 26250. 05860.00090.00310. 32760.03570.00130.00130.00170.00020.00170.71210.00430.00150.00310.02300.0215

0.00010.02660.00620.00380.00060. 10860.19040.00000.00010. 16620. 14390.06600.00000. 01690. 30550.00000.97510.16340.50220.07690.0330

PAGE 21APPENDIX VI

TABLE 2U(K)*2 FOR SM+3

J1 LEVEL 1 J2 LEVEL 2 (U2)*2 (U4)*2 (U6)42

7/2 24995 7/2 24995 0.0048 0.0668 0.01357/2 24995 7/2 26786 0.0000 0.0030 0.01067/2 24995 9/2 2341 0.0009 0.0011 0.01037/2 24995 9/2 9189 0.0137 0.0010 0.00007/2 24995 9/2 20660 0.0092 0.0009 0.20287/2 24995 9/2 22873 0.2300 0.1546 0.00497/2 24995 11/2 3667 0.0074 0.0002 0.01217/2 24995 11/2 10583 0.0230 0.0010 0.00017/2 24995 11/2 21147 0.0122 0.0098 0.03737/2 24995 11/2 25201 0.0003 0.0576 0.13357/2 24995 11/2 25829 0.0137 0.0245 0.09917/2 24995 13/2 5072 0.0 0.0121 0.00167/2 24995 13/2 21644 0.0 0.0230 0.11647/2 24995 13/2 24676 0.0 0.0061 0.13917/2 24995 15/2 6520 0.0 0.0185 0.00077/2 24995 15/2 20825 0.0 0.0001 0.00747/2 24995 15/2 23048 0.0 0.1892 0.00097/2 24995 15/2 25667 0.0 0.0196 0.13907/2 24995 17/2 26762 0.0 0.0 0.16597/2 24995 19/2 24138 0.0 0.0 0.7863

7/2 26786 7/2 26786 0.1550 0.0042 0.00057/2 26786 9/2 2341 0.0000 0.0361 0.09687/2 26786 9/2 9189 0.3402 0.1343 0.00007/2 26786 9/2 20660 0.0002 0.0002 0.10087/2 26786 9/2 22873 0.0000 0.0714 0.02407/2 26786 11/2 3667 0.0004 0.1080 0.09337/2 26786 11/2 10583 0.8957 0.0658 0.00007/2 26786 11/2 21147 0.0002 0.0072 0.09757/2 26786 11/2 25201 0.0000 0.0075 0.06907/2 26786 11/2 25829 0.0765 0.1692 0.02817/2 26786 13/2 5072 0.0 0.2295 0.07067/2 26786 13/2 21644 0.0 0.0122 0.07217/2 26786 13/2 24676 0.0 0.0082 0.04817/2 26786 15/2 6520 0.0 0.4309 0.03057/2 26786 15/2 20825 0.0 0.0002 0.00597/2 26786 15/2 23048 0.0 0.0000 0.05947/2 26786 15/2 25667 0.0 0.0049 0.02957/2 26786 17/2 26762 0.0 0.0 0.03987/2 26786 19/2 24138 0.0 0.0 0.0394

9/2 2341 9/2 2341 0.3.86 0.0210 0.26329/2 2341 9/2 9189 0.0336 0.3340 0.18099/2 2341 9/2 20660 0.0002 0.0001 0.00349/2 2341 9/2 22873 0.0000 0.0026 0.00289/2 2341 11/2 3667 0.3432 0.2317 0.00009/2 2341 11/2 10583 0.0007 0.0691 0.52069/2 2341 11/2 21147 0.0000 0.0006 0.00159/2 2341 11/2 25201 0.0010 0.0000 0.00439/2 2341 11/2 25829 0.0001 0.0001 0.0011

PAGE 22APPENDIX VI

TABLE 2U(K)*2 FOR SM+3

J1 LEVEL 1 J2 LEVEL 2

9/29/29/29/29/29/29/29/29/29/2

9/29/29/29/29/29/29/29/29/29/29/29/29/29/29/29/29/29/2

9/29/29/29/29/29/29/29/29/29/29/29/29/29/29/29/2

9/29/29/2

2341234123412341234123412341234123412341

918991899189918991899189918991899189918991899189918991899189918991899189

20660206602066020660206602066020660206602066020660206602066020660206602066020660

228732287322873

13/213/213/215/215/'215/217/217/219/221/2

9/29/29/2

11/211/21 1/211/211/213/213/213/215/215/215/217/217/219/221/2

9/29/2

11/211/211/211/213/213/213/215/215/215/215/217/217/219/2

9/211/211/2

507221644246766520

208252566722612267622413825434

918920660228733667

10583211472520125829

50722164424676

6520208252304822612267622413825434

206602287310583211472520125829

50722164424676

65202082523C4825667226122676224138

228733667

21147

(U2) *2 (U4) *2 (U6) *2

0.03370.00000.00020.00.00.00.00.00.00.0

0.03680.00000.00720.24170.05030.00000.00000.00020.70850.00050.00010.00.00.00.00.00.00.0

0. 22b70.00290.00000.14710.03160.00070.00000.03060.01810.00.00.00.00.00.00.0

0.31430.00000.0176

0.17440.00010.00140.01170.00080.00000.00020.00660.00.0

0.00940.00090.00010.35310.06520.00350.00100.00080.03320.00130.00180.57430.00100.00370.00020.00230.00.0

0.15800.03560.00000.10090.05540.01380.00000.12250.19040.00000.02220.02880.0b300.00110.00290.0

0.02240.00950.0879

0. 39050.01290.00020.20430.00560.00690.00180. 00950.01250.0016

0.04820.00030.00030.00920.01490. 00000.00000.00040. 43120.00200.00000.74570.00020.00900.00480.00610.02310.0160

0.00480. 14000.00090.07820.22560. 06870.00170.27100.45060.00101.00550.02650.24170.46360.00270.0253

0. 12910.00160.0067

PAGE 23APPENDIX VI

TABLE 2U(K)*2 FOR SM+3

J1 LEVEL 1 J2 LEVEL 2

9/29/29/29/29/29/29/29/29/29/29/29/29/2

11/211/211/211/211/211/211/211/211/211/211/211/211/211/211/2

11/211/211/211/211/211/211/211/211/211/211/211/211/211/2

11/211/211/211/211/2

22873228732287322873228732287322873228732287322873228732287322873

366736673667366736673667366736673667366736673667366736673667

1058310583105831058310583105831058310583105831058310583105831058310583

2114721147211472114721147

11/211/213/213/213/215/215/215/215/217/217/219/221/2

11/211/211/211/213/213/213/215/215/215/215/217/217/219/221/2

11/211/211/211/213/213/213/215/215/215/215/217/219/221/2

11/211/211/213/213/2

2520125829

50722164424676

652020825230482566722612267622413825434

3667105832114725829

50722164424676

652020825230482566722612267622413825434

10583211472520125829

50722164424676

6520208252304825667267622413825434

2114725201258292164424676

(U2)*2 (UL4)*2 (U6)*2

0.00150.01000.00440.01200.00020.00.00.00.00.00.00.00.0

0.48710.01580.00170.00020.34280.00000.00050.01710.00000.00000.00020.00.00.00.0

0.13820.00000.00000.00660.16680.00020.00001.01030.00030.00180.00020.00.00.0

0.11810.01410.00930.26400.0437

0.00410.00150.00050.04970.03190.00160.01610.00070.07930.21560.43470.00.0

0.00040.25590.00000.00080.35600.00160.00020.10850.00160.00010.00310.00180.00010.00180.0

0.04830.00080.00010.00010.59490.00380.00150.77890.00120.00790.00340.00120.00010.0

0.04950.03740.07390.04810.1422

0.20430. 00050.00000. 14060. 02720. 00020. 07060.16300. 28810. 03571.10740.28490.0467

0. 28370.76100.00500.00680.10860.00110. 01010.49780.00000.02090.00250.01370.00290.00320. 0051

0.00260.00330.00120. 00050.71120.00420.00350.34200.00040.00290.00470.00410.00270.0200

0.32970.33880.16360. 14490. 1556

PAGE 24APPENDIX VI

TABLE 2U(K)*2 FOR SM+3

J1 LEVEL 1 J2 LEVEL 2

11/211/211/211/211/211/211/211/2

11/211/211/211/211/211/211/211/211/211/211/211/211/2

11/211/211/211/211/211/211/211/211/211/211/211/2

13/213/213/213/213/213/213/213/213/213/213/2

2114721147211472114721147211472114721147

25201252012520125201252012520125201252012520125201252012520125201

258292582925829258292582925829258292582925829258292582925829

50725072507250725072507250725072507250725072

15/21 5/21 5/21 5/217/217/219/221/2

11/211/213/213/213/21 5/215/215/215/217/217/219/221/2

11/213/213/213/215/215/215/215/217/217/219/221/2

13/213/213/215/215/215/215/217/217/219/221/2

652020825230482566722612267622413825434

2520125829

507221644246766520

20825230482566722612267622413825434

258295072

21644246766520

20825230482566722612267622413825434

507221644246766520

20825230482566722612267622413825434

(U2) 2 (U4)*2 (06)*2

0.00000.02780.01290.03160.00.00.00.0

0.16730.00050.00000.02510.43100.00000.00520.00480.00010.00.00.00.0

0.14540.00170.04090.01200.00000.00040.00100.02980.00.00.00.0

0.79780.00480.00070.25040.00010.00000.00150.00000.00000.00.0

0.00000.59350.00920.05930.01220.03860.00780.0

0.00330.00510.00000.00200.20640.00000.55040.19940.05490.22080.08010.00630.0

0.00610.00320.11230.02240.00110.00700.05690.00860.04150.00390. 36230.0

0.07450.00000.00000.41480.00210.00100.00090.00570.00360.00240.0039

0.00280.04270. 09500.31420.89060.11260. 45680.0141

0. 00600. 09090.00480.24420. 04270. 00290. 16590. 39660.19690. 02890. 09200.06480.0038

0. 06520.00630.04870. 08200.00120.01260.00010. 09730. 08280.09350. 07590. 0548

0. 10340.00220.00140. 69920.00390.00740.01790.00380.03100.03140.0028

PAGE 25APPENDIX VI

TABLE 2U(K)*2 FOR SM+3

J1 LEVEL 1 J2 LEVEL 2

13/213/213/213/213/213/213/213/213/213/2

13/213/213/213/213/213/213/213/213/2

15/215/215/215/215/215/215/215/2

15/215/215/215/215/215/215/2

15/215/215/215/215/215/2

15/215/215/215/215/2

21644216442164421644216442164421644216442164421644

246762467624676246762467624676246762467624676

65206520652065206520652065206520

20825208252082520825208252082520825

230482304823048230482304823048

2566725667256672566725667

13/213/215/215/215/215/217/217/219/221/2

13/215/215/215/215/217/217/219/221/2

15/215/215/215/217/217/219/221/2

15/215/215/217/217/219/221/2

15/215/217/217/219/221/2

15/217/217/219/221/2

2164424676

652020825230482566722612267622413825434

246766520

20825230482566722612267622413825434

652020825230482566722612267622413825434

20825230482566722612267622413825434

230482566722612267622413825434

2566722612267622413825434

(U2) *2 (U4) *2 (U6) *2

0.04930.07270.00010.32840.03880.06490.05290.03460.00.0

0.00810.00000.51110.24410.00070.01130.02580.00.0

1.33580.00220.01030.00090.00040.00340.00030.0

0.34610.10930.03340.05010.00710.01540.0

0.06080.03360.13580.08550.04690.0

0.00380.76040.00010.02760.0

0.05290.13630.00010. 14090.16820.09262.52930.02080.02280.0055

0.08180.00000.10210.00140.07170.01240.04520.02450.0000

0. 64970.00000.00010.00000.00190.00260.00950.0030

0.97390.01220.07230.01370.01560.16310.0009

0.42220.16000. 37010.00820.44400.0003

0.08580.22000.14370.21820. 1058

0.08710.05500.00410.07200. 20020. 16370.01380.29621. 07420. 3034

0.00010.00340. 42090. 10290.00320. 88550. 66030.03160.0681

0.46290.00010. 00380. 00370.00150.00030.00560. 0649

0.02400.20400.27130.35810. 25980.15810.3139

0.01390.09330. 00020.00130.01381.1875

0. 07670.04170.04840. 99320. 0364

PAGE 26APPENDIX VI

TABLE 2U (K)*2 FOR SM+3

J1 LEVEL 1 J2

22612 17/222612 17/222612 19/222612 21/2

26762 17/226762 19/226762 21/2

24138 19/224138 21/2

21/2 25434 21/2

LEVEL 2 (U2)*2 (U4)*2 (U6)*2

22612267622413825434

267622413825434

2413825434

25434

0.60200.00070.04990.0056

0.00070.93570.0674

0. 84420.06290.06760.0853

0.13500.55520.5015

0.00080.41900. 49430.3317

0. 04590. 06700.7585

0.6771 0.9087 0.02750.0437 0.0943 0.8814

0.6674 1.2562 0.9348

17/217/217/217/2

17/217/217/2

19/219/2

APPENDIX VII

PAGE 1

APPENDIX VII

EU+3: LAF3

OBSERVED

...

.0..

.. .

.. .

. ..

. ..

. ..

. . .

. . .

. . .

... .

TABLE 1CENTERS OF GRAVITY

CALC 0-C STATE

0372

10261866282338494907

172931902721483243552532526357263922662226735267522676327244275862796028427

.S.

S..

...

.5..

... 5

...

... "

... "

... "

... "

... "

... "

... "

... "

... "

... "

... "

... "

7F07F17F27F37F47F57F6

5D05D15D 25D35L65L75G 25G35G45G65G 55L85D45L95L10

PAGE 3APPENDIX VII

TABLE 2U(K)*2 FOR EU+3

J1 LEVEL 1 J2 LEVEL 2

000000

00000000

1111111111111

111111111111

2222222

000000

1729317293172931729317293172931729317293

372372372372372372372372372372372372372

190271902719027190271902719027190271902719027190271902719027

1026102610261026102610261026

244666

22244466

1123334556667

122333445667

2233445

10262823

275864907

2532526752

10262148326392

282326735275862532526752

37219027

10261866

2435526622

28233849

267634907

253252675226357

190272148326392

18662435526622

28232758626763253252675226357

102621483

186624355

282326735

3849

(U2)*2 (U4)*2 (U6)*2

0.13740.00.00.00.00.0

0.00320.01420.01460.00.00.00.00.0

0.15410.00250.05180.20920.00040.00020.00.00.00.00.00.00.0

0.01330.01220.02090.00380.01830.01640.00.00.00.00.00.0

0.10000.00180.18630.00020.22260.00000.0

0.00.14020.00110.00.00.0

0.00.00.00.00230.03590.01340.00.0

0.00.00.00.12810.00120.00120.17410.11920.00040.00.00.00.0

0.00.00.00.00190.00590.05940.00280.00780.04840.00.00.0

0.12190.00150.21240.00200.00620.00070.3153

0.00.00.00.14500.01530. 0037

0.00.00.00.00.00.00. 23840.2212

0.00.00.00.00. 00.00. 00.05440.00970. 37740. 00910.00490.0181

0.00.00.00.00.00.00.00.00.23320. 14790. 57170.2020

0.00.00.00.00.03290.00780.2089

PAGE 4APPENDIX VII

TABLE 2U(K)*2 FOR EU+3

J1 LEVEL 1 J2 LEVEL 2

22222

22222222222222

22222222222

333333333333

3333

10261026102610261026

2148321483214832148321483214832148321483214832148321483214832148321483

2639226392263922639226392263922639226392263922639226392

1866186618661866186618661P6613661866186618661866

24355243552435524355

56678

22333444556678

23344456678

333445566689

3344

267634907

267522635727244

2148326392

18662435526622

28232673527586

38492676325325267522635727244

263922435526622

282326735275862676325325267522635727244

18662435526622

282326735

384926763

490725325267522724427960

2435526622

282326735

(U2) *2 (U4) *2 (U6) *2

0.00.00.00.00.0

0.00110.00860.00230.03510.03050.00200.02670.00420.00.00.00.00.00.0

0.07840.00170.10970.00000.02250.00050.00.00.00.00.0

0.02750.00100.00000.38800.00020.17540.00000.00.00.00.00.0

0.01490.01150.00390.0215

0.00030.04770.00000.00.0

0.00690.04820.00260.01260.00310.00030.03200.00030.00160.00080.00410.03430.00.0

0.00320.03730.02150.00000.02790.00400.00670.04800.00040.00.0

0.02600.00050.00040.13520.00010.25270.00110.23100.00000.00040.00.0

0.00230.02290.00020.0005

0.00460.46 960.00190.01100.0193

0.00.00.00.00.00.00000.22100.00400. 00000. 38290. 15860. 15860.24790.2481

0.00.00.00.00190.00570.42820. 10560. 45990.01920.51300.0853

0.02810.00000.00480. 15880.00300.38360. 00050.41350.00130.01020.00920.0168

0.00310.26560.00000.3633

PAGE 5APPENDIX VII

TABLE 2U(K)*2 FOR EU+3

J1 LEVEL 1 J2 LEVEL 2

33333333

33333333333

444444444444

44444444444

4444

2435524355243552435524355243552435524355

2662226622266222662226622266222662226622266222662226622

282328232823282328232823282328232823282328232823

2673526735267352673526735267352673526735267352673526735

27586275862758627586

45566789

34455666789

44455666789

10

4455666789

10

4556

275863849

267632532526752263572724427960

266222673527586

384926763

49072532526752263572724427960

28232673527586

384926763

4907253252675226357272442796028427

2673527586

384926763

4907253252675226357272U.2796028427

275863849

267634907

(U2)*2 (U4)*2 (U6)*2

0.05920.00010.03280.00.00.00.00.0

0.02270.14140.00210.00000.02920.00.00.00.00.00.0

0.01170.00020.00050.56840.00010.08560.00000.00010.00.00.00.0

0.04070.01560.00090.09820.00000.00940.02790,.00.00.00.0

0.08170.00340.08250.0012

0.00630.00140.01740.00000.00970.00670.00.0

0.00170.07450.01290.00000.05080.00000.01550.00190.11060.00.0

0.28410.00090.00060.01280.00000.51450.00020.00200.00000.00000.00.0

0.00000.03810.00000. 12410.00030.02970.05380.00700.18970.00.0

0.00220.00040.03220.0000

0.00300.00000.20230.01640.05000.13490.32560.3477

0.06440.15080.21310.00100.00860.00200.36770.12490. 12480.56650. 1166

0.35280.00050.00020.44120.00590.26910.00460.00480.00800.00400.00520. 0093

0.01160.11080.00360.05050.00200. 15690.00010. 26620. 12320.82110.0161

0.00680.00030.03840.0000

PAGE 6APPENDIX VII

TABLE 2U(K)*2 FOR EU+3

J1 LEVEL 1 J2 LEVEL 2

444444

55555555

55555555

666666

666666

66666

7777

275862758627586275862758627586

38493849384938493849384938493849

2676326763267632676326763267632676326763

490749074907490749074907

253252532525325253252532525325

2675226752267522675226752

26357263572635726357

66789

10

5566789

10

5666789

10

66789

10

66789

10

6789

10

789

10

253252675226357272442796028427

3849267634907

2675226357272442796028427

267634907

253252675226357272442796028427

49072675226357272442796028427

253252675226357272442796028427

2675226357272442796028427

26357272442796028427

(U2) *2 (U4) *2 (U6) *2

0.00050.03630.00.00.00.0

0.27620.00200.54100.00010.00000.00.00.0

0.05110.00100.00700.07430.01630.00.00.0

1.20290.00850.00000.00000.00.0

0.00500.00190.01470.00760.00.0

0.10470.00040.01770.00.0

0.01520.02490.00830.0

0.00200.00280.00230.00010.00.0

0.20630.00020.64510.00160.00020.00000.00020.0

0.00080.00110.04030.17160.06920.01340.25960.0

0.39400.00220.00010.00030.00000.0020

0.93550.03280.33100.07250.00000.0067

0.05780.06120.04800.03840.2938

0.67400.48560.08040.0000

0.01660.00000.06080. 15750. 15470.6379

0.32250. 00470. 12130.00260. 00730.01760.01760.0015

0. 01860.00120.04050.00250.15180.30160.48880.6377

0.02940.00030.00130.00740.02420.0524

0. 17580.00090. 14680.10200. 00460.0563

0.00940.05090.06010. 30091.4460

0.02690. 16710.10320.0313

PAGE 7APPENDIX VII

TABLE 2U(K)*2 FOR EU+3

LEVEL 2 (U2)*2

27244 0.017627960 0.034528427 0.0055

27960 0.011028427 0.0379

28427 0.0002

(U4)*2 (U6)*2

0.7179 0.01750.5312 0.19170.0466 0.1105

1.0574 0.05130.4237 0.2855

1.8415 0.4018

J1

888

99

10

LEVEL 1

272442724427244

2796027960

28427

J2

89

10

9'10

10

APPENDIX VIII

PAGE 1APPENDIX VIII

TABLE 1GD+3:LAF3

OBSERVED

0.00.00.00.0

32177.1132185.6232199.6132228.57

32771.7532791.9632809.29

33352.0033370.00

35923.0035945.2435969.0335996.14

36275.2536286.0836306.2436314.2636333.45

36340.8136343.0336347.1836351.6936354.8036364.5136371.7136377.8636384.90

36551.4336563.3336573.1836586.1436594.8636613.04

36661.8136670.9936679.9836690.1736700.50

CALC 0-C STATE J

1111

32183321893220932232

327873278832799

3335133366

35929359393596235979

3626836276362973630536316

363423634436344363463634836350363553635836359

365563656736575365933659336611

3667436686366903670636707

0 8S 7/20 8S 7/20 8S 7/20 8S 7/2

-5 6P 7/2-3 6P 7/2-8 6P 7/2-3 6P 7/2

-14 6P 5/24 6P 5/2

10 6P 5/2

1 6P 3/24 6P 3/2

-5 61 7/26 61 7/27 61 7/2

17 61 7/2

8 61 9/210 61 9/2

9 61 9/210 61 9/217 61 9/2

MJ

-7/25/23/21/2

-7/25/23/21/2

3/25/21/2

1/23/2

5/23/2

-7/21/2

5/2-7/23/2

-9/21/2

-1 61 17/2 -9/20 61 17/2 -11/23 61 17/2 -7/25 61 17/2 13/27 61 17/2 15/2

15 61 17/2 5/217 61 17/2 3/220 61 17/2 17/226 61 17/2 1/2

-4 61 11/2 -7/2-3 61 11/2 -9/2-1 61 11/2 5/2-6 61 11/2 3/2

2 61 11/2 -11/22 61 11/2 1/2

-11 61 15/2 -9/2-14 61 15/2 -11/2

-9 61 15/2 -7/2-14 61 15/2 5/2

-5 61 15/2 13/2

PAGE 2APPENDIX VIII

TABLE 1GD+3:LAF3

OBSERVED CALC 0-C STATE J MJ

36703.65 36714 -10 61 15/2 15/236713.00 36722 -8 6I 13/2 -7/236715.52 36722 -6 61 15/2 3/236720.08 36725 -3 61 15/2 15/236724.82 36732 -6 6I 15/2 1/236734.46 36737 -2 6I 13/2 -11/236738.99 36737 2 61 13/2 5/236752.82 36759 -5 61 13/2 13/236763.02 36769 -5 61 13/2 3/236772.50 36770 2 61 13/2 1/2

39667.00 39660 7 6D 9/2 -9/239686.00 39694 -7 6D 9/2 1/239719.00 39729 -9 6D 9/2 3/239742.00 39741 1 6D 9/2 -7/239758.00 39763 -4 6D 9/2 5/2

... 40644 ... 6D 1/2 1/2

40734.00 40732 2 6D 7/2 5/240740.00 40737 3 6D 7/2 3/240744.00 40743 1 6D 7/2 1/240751.00 40754 -2 6D 7/2 -7/2

... 40895 ... 6D 3/2 3/2... 40909 ... 6D 3/2 1/2

... 41004 ... 6D 5/2 3/2

... 41049 ... 6D 5/2 5/2

... 41061 ... 6D 5/2 1/2

49170.00 49159 11 6G 7/2 -7/2... 49232 ... 6G 7/2 3/2

49221.00 49233 -11 6G 7/2 5/249240.00 49248 -7 6G 7/2 1/2

49533 ... 6G 11/2 -11/249539 ... 6G 9/2 -9/249608 ... 6G 9/2 -7/2

... 49633 ... 6G 9/2 1/2... 49643 ... 6G 5/2 5/2... 49670 ... 6G 9/2 3/2... 49674 ... 6G 9/2 -7/2... 49695 ... 6G 9/2 3/2... 49732 ... 6G 11/2 1/2... 49735 ... 6G 11/2 3/2... 49741 ... 6G 11/2 5/2... 49815 ... 6G 11/2 -7/2... 49849 ... 6G 11/2 -9/2

PAGE 3

APPENDIX VIII

TABLE 1AGD+3:LAF3 CENTERS OF GRAVITY

CALC CENTER STATE

2 8S 7/2

32232 6P 7/232827 6P 5/233398 6P 3/2

35949 61 7/236291 61 9/236345 6117/236582 6111/236704 6115/236742 6113/2

39723 6D 7/240654 6D 1/240901 6F 3/241037 6D 5/2

PAGE 5APPENDIX VIII

TABLE 2U(K)*2 FOR GD+3

J1 LEVEI 1 J2 LEVEL 2 (U2) *2 (U4) *2 (U6) *2

1/2 40654 3/2 33398 0.0028 0.0 0.01/2 40654 3/2 40901 0.0090 0.0 0.01/2 40654 5/2 32827 0.0572 0.0 0.01/2 40654 5/2 41037 0.0301 0.0 0.01/2 40654 9/2 36291 0.0 0.0031 0.01/2 40654 9/2 39723 0.0 0.0088 0.01/2 40654 11/2 36582 0.0 0.0 0.0033

3/2 33398 3/2 33398 0.0109 0.0 0.03/2 33398 3/2 40901 0.0222 0.0 0.03/2 33398 5/2 32827 0.0430 0.0002 0.03/2 33398 5/2 41037 0.0725 0.0013 0.03/2 33398 7/2 322.2 0.0090 0.0004 0.03/2 33398 9/2 36291 0.0 0.0002 0.23103/2 33398 9/2 39723 0.0 0.0022 0.00003/2 33398 11/2 36582 0.0 0.0016 0.43823/2 33398 13/2 36742 0.0 0.0 0.57193/2 33398 15/2 36704 0.0 0.0 0.5483

3/2 40901 3/2 40901 0.0323 0.0 0.03/2 40901 5/2 32827 0.0742 0.0018 0.03/2 40901 5/2 41037 0.0130 0.0061 0.03/2 40901 7/2 32232 0.0276 0.0022 0.03/2 40901 7/2 35949 0.0000 0.0045 0.03/2 40901 9/2 36291 0.0 0.0005 0.00113/2 40901 11/2 36582 0.0 0.0116 0.03623/2 40901 13/2 36742 0.0 010 0.08373/2 40901 15/2 36704 0.0 0.0 0.0355

5/2 32827 5/2 32827 0.0292 0.0013 0.05/2 32827 5/2 41037 0.0424 0.0073 0.05/2 32827 7/2 32232 0.0180 0.0001 0.00015/2 32827 7/2 35949 0.0000 0.0023 0.70745/2 32827 9/2 36291 0.0001 0.0015 0.41095/2 32827 9/2 39723 0.2003 0.0014 0.00025/2 32827 11/2 36582 0.0 0.0000 0.14285/2 32827 13/2 36742 0.0 0.0052 0.00265/2 32827 15/2 36704 0.0 0.0 0. 14355/2 32827 17/2 36345 0.0 0.0 1.1037

5/2 41037 5/2 41037 0.0005 0.0247 0.05/2 41037 7/2 2 0.0026 0.0000 0.00005/2 41037 7/2 32232 0.0868 0.0005 0.00005/2 41037 7/2 35949 0.0002 0.0123 0.04455/2 41037 9/2 36291 0.0001 0.0088 0.09455/2 41037 9/2 39723 0.0035 0.0120 0.00025/2 41037 11/2 36582 0.0 0.0000 0.06605/2 41037 13/2 36742 0.0 0.0245 0.00155/2 41037 15/2 36704 0.0 0.0 0.07155/2 41037 17/2 36345 0.0 0.0 0.1368

PAGE 6APPENDIX VIII

TABLE 2U(K)*2 FOR GD+3

J1 LEVEL 1 J2 LEVEL 2

7/27/27/27/27/27/27/27/21/27/27/27/27/27/27/27/27/2

7/27/27/27/27/2

9/29/29/29/29/2

9/29/29/29/29/2

11/211/211/211/2

13/213/213/2

15/215/2

(U2)*2 (Uf)*2 (U6)*2

2 7/2 3223.22 7/2 359492 9/2 362912 9/2 397232 11/2 365822 13/2 367422 15/2 367042 17/2 363450 1/2 - 0

32232 7/2 3223232232 7/2 3594932232 9/2 3629132232 9/2 3972332232 11/2 3658232232 13/2 3674232232 15/2 3670432232 17/2 36345

35949 7/2 3594935949 9/2 3629135949 9/2 3972335949 11/2 3658235949 15/2 36704

36291 9/2 3629136291 9/2 3972336291 11/2 3658236291 13/2 3674236291 15/2 36704

39723 9/2 3972339723 11/2 3658239723 13/2 3674239723 15/2 3670439723 17/2 36345

36582 11/2 3658236582 13/2 3674236582 15/2 3670436582 17/2 36345

36742 13/2 3674236742 15/2 3670436742 17/2 36345

36704 15/2 3670436704 17/2 36345

0.00110.00000.00000.00600.00000.00.00.0

0.12020.00000.00030.18790.00000.00.00.0

0.00470.00080.00020.00080.0

0.00990.00080.00210.00160.0

0.08540.00110.00020.00.0

0.01290.00450.00190.0

0.01220.00820.0013

0.0067 0. 0189 0.00400.0115 0.0088 0.0008

0.0000 0.00000.0000 0.00440.0000 0.01100.0001 0.00000.0000 0.01880.0000 0.02570.0000 0.02870.0 0.0228

0.0017 0.00010.0018 0.17630.0048 0.37590.0210 0.00000.0051 0.56390.0019 0.71860.0037 0.81860.0 0.7818

0.0087 0.00070.0013 0.00250.0011 0.00460.0005 0.00000.0001 0.0009

0.0141 0.00050.0060 0.00990.0019 0.00180.0013 0.00010.0019 0.0002

0.0722 0.00170.0191 0.00750.0326 0.00020.0318 0.00950.0140 0.0409

0.0191 0.00070.0009 0.00160.0033 0.00000.0018 0.0011

0.0235 0.00150.0001 0.00100.0055 0.0012

APPENDIX IX

PAGE 1

APPENDIX IX

TABLE 1TB+3:LAF3 CENTERS OF GRAVITY

OBSERVED

4423507455605814

205662631726529271112791928247

28489290692934329595

30750...

3149232998339423446635063

... S

35344

3665737275

.

..

..

.. .

39287

4091341447

..

CALC 0-C STATE

124217234394418510655615784

205682636026547270952789128231284112853229101293142958129655297943073431348

315033301533891344633505835060352553549836674367133726037606377223773238110

39297395154030940939414584147341817

5-31

030

-1-42-17162816

-42-312914

16

-10-165135

89.

-5515

...

..

-9..

-25-10... S

... 5

7F67F57F47F37F27F17F0

5D45D35G 65L 105G 55D25G45L 95G 35L 85L7SG 25L65D 1SDO

5H75H 65HS5H45F55H35185F45F35175F25F1516514SIS

5K95D25G65K 85K55G65K7

PAGE 3APPENDIX IX

TABLE 2U(K)*2 FOR TB+3

J1 LEVEL 1 J2 LEVEL 2

00000

111111111

222222222222

22222222222

333333333

57845784578457845784

556155615561556155615561556155615561

510651065106510651065106510651065106510651065106

2823128231282312823128231282312823128231282312823128231

441944194419441944194419441944194419

2 51062 282314 34394 205696 125

1 55612 51063 44193 263604 34394 205695 21736 1257 29582

2 51062 282313 44193 263604 34394 205694 284125 21735 278926 1257 295828 29315

2 282313 44193 263604 34394 205694 284125 21735 278926 265487 295828 29315

3 44193 263604 34394 205694 284125 21735 278926 1258 29315

(U2) *2 (U4) *2 (U6) *2

0.1391 0.0 0.00.0010 0.0 0.00.0 0.1389 0.00.0 0.0017 0.00.0 0.0 0.1441

0.1562 0.0 0.00.0513 0.0 0.00.2102 0.1273 0.00.0011 0.0013 0.00.0 0.1719 0.00.0 0.0025 0.00.0 0.1188 0.05370.0 0.0 0.37610.0 0.0 0.0093

0.1002 0.1211 0.00.0010 0.0010 0.00.1829 0.2101 0.00.0014 0.0026 0.00.2224 0.0060 0.03240.0011 0.0004 0.00010.0016 0.0000 0.00050.0 0.3135 0.20710.0 0.0001 0.00140.0 0.0481 0.46950.0 0.0 0.01570.0 0.0 0.0199

0.0168 0.0080 0.00.0028 0.0024 0.00.0269 0.0056 0.00.0009 0.0005 0.00000.0362 0.0048 0.00590.0602 0.0008 0.02170.0 0.0027 0.00060.0 0.0166 0.14660.0 0.0255 0.35100.0 0.0 0.35790.0 0.0 0.2572

0.0272 0.0253 0.02780.0007 0.0009 0.00000.3782 0.1343 0.1575).0022 0.0005 0.00060.0025 0.0000 0.00260.1767 0.2504 0.38160.0019 0.0001 0.00200.0 0.2323 0.41290.0 0.0 0.0152

PAGE 4APPENDIX IX

TAi3LE 2U(K)*2 FOR TB+3

J1 LEVEL 1 J2 LEVEL 2

3 4419 9 28532

33333333333

44444444444

4444444444

444444444

555

2636026360263602636026360263602636026360263602636026360

34393439343934393439343934393439343934393439

20569205692056920569205692056920569205692056920569

284122841228412'284122841228412284122841228412

217321732173

34445566789

4445566789

10

445566789

10

45566789

10

556

263603439

2056928412

217327892

12526548295822931528532

34392056928412

217327892

1252654829582293152853227096

2056928412

217327892

1252654829582293152853227096

284122173

27892125

2654829582293152853227096

217327892

125

(U2) *2 (U4) *2 (U6) *2

0.00000.00650.05350.00580.00050.08370.00.00.00.00.0

0.01200.00020.00000.55410.00420.08880.00230.00.00.00.0

0.04630.00040.01420.00460.00090.11310.00.00.00.0

0.06820.00050.02850.00010.01130.00.00.00.0

0.27640.00010.5377

0.0

0.00210.00010.02910.02280.00280.03030.00020.03410.00350.00.0

0.27920.00220.00060.01200.00090.51590.00000.00290.00140.00.0

0.03100.00510.00130.06310.00080.20670.03330.01040.00.0

0.01480.00000.15640.00030.09030.24160.00160.00.0

0.20710.00020.6420

0.0255

0.00270.00020.00850. 13980.00160. 29810.00140.24120.02210.25460.4161

0. 35000.00140.00680.43740.00120.26580.00250.00690.00040.01500. 0322

0.00380.51150.00220.39640.00130. 14570.03360. 07230.27580. 8669

0.00620.00210.06650. 00870.06770.39200.00360.27360.0540

0. 317,90.00650.1178

PAGE 5APPENDIX IX

TABLE 2U(K)*2 FOR TB+3

J1 LEVEL 1 J2 LEVEL 2

55555

5555555

666666

66666

7777

888

99

21732173217321732173

27892278922789227892278922789227892

125125125125125125

2654826548265482654826548

29582295822958229582

293152931529315

6789

10

566789

10

66789

10

6789

10

789

10

89

10

2654829582293152853227096

27892125

2654829582293152853227096

1252654829582293152853227096

2654829582293152853227096

29582293152853227096

293152853227096

(U2) *2 (U4) *2 (U6) *2

0.00400.00020.00.00.0

0.25780.00120.02030.01150.00.00.0

1.21250.00160.00060.00000.00.0

0.58020.04640.00590.00.0

0.08320.08290.00030.0

0.05090.07710.0000

0.00280.00030.00190.00070.0

0.00010.00180.14660.10910.28770.00010.0

0.38810.00440.00010.00010.00190.0003

0.02880.00020.02560.25200.0068

0.44930.40470.03420.1037

0.61740.48090.0698

0.00930.00660.02100.01360.0015

0.05400.01310. 06480.30750. 33960. 05320.4457

0.02750.01160.01210. 02280.04550.0580

0. 34990.32640.56070.71230.9143

0.21580.08770. 11390.0029

0.17990. 12040. 0803

28532 9 28532 0.0236 0.9206 0.176628532 10 27096 0.0727 0.4668 0.1548

0.0002 1.8580 0.377710 27096 10 27096

APPENDIX X

PAGE 1

APPENDIX X

TABLE 1DY+ 3: LAF3

OBSERVED CALC 0-C STATE J

017E9

124184208307

3502

35763618363036453695

588259085924594459756020

76337665772877587801781478387842793379988077899290879074914491819235928293439435

102221028510345

222679

106195216323335

3493357936163636363736713679

587359105913593159686021

76137654771977777809783978407852793880098094897490699072913991769240927793309447

102111026510336

MJ

-21 6H-8 6H-9 6H18 6H

-10 6H-7 6H

-15 6H... 6H

9 6H... 6H

-39 6H-17 6H

-6 6H-25 6H

16 6H

9 6H-1 61111 6H13 6H1

7 6H0 6H

20 61111 6H

9 6H-18 6F

-7 6H-24 6F

-1 6F-9 6F-4 6H

-10 6F-16 6F

18 6F18 6H

2 6F5 6F5 6F

-4 6F5 6H

13 6H-11 6H

11 6H20 6H

9 6H

15/2 1/215/2 15/215/2 13/215/2 3/215/2 5/215/2 -11/215/2 -9/215/2 -7/2

13/2 13/213/2 -7/213/2 3/213/2 5/213/2 1/213/2 -9/213/2 -11/2

11/2 -7/211/2 -11/211/2 -9/211/2 5/211/2 3/211/2 1/2

9/2 5/29/2 -9/29/2 3/2

11/2 1/29/2 -7/2

11/2 1/211/2 5/211/2 3/29/2 1/2

11/2 -7/211/2 -9/29/2 5/27/2 3/29/2 -7/29/2 3/29/2 1/29/2 -9/27/2 -7/27/2 1/27/2 5/2

5/2 5/25/2 1/25/2 3/2

PjAGE 2

APPENDIX X

TABLE 1DY+3: LAF3

OBSERVED CALC 0-C STATE J MJ

11037 11028 9 6F 7/2 5/211109 11101 8 6F 7/2 1/211140 11141 0 6F 7/2 -7/211153 11143 10 6F 7/2 3/2

12456 12455 1 6F 5/2 3/212504 12493 11 6F 5/2 5/212520 12512 8 6F 5/2 1/2

13271 13277 -5 6F 3/2 3/213285 13286 0 6F 3/2 1/2

... 13827 ... 6F 1/2 1/2

21057 21071 -13 4F 9/2 -9/221142 21137 5 4F 9/2 5/221159 21163 -3 4F 9/2 3/221205 21205 0 41 9/2 1/221395 21393 2 4F 9/2 -7/2

22022 21956 66 4I 15/2 15/222132 22111 21 41 15/2 5/222175 22155 20 41 15/2 13/222189 22184 5 41 15/2 1/222213 22209 4 4F 15/2 -7/222292 22272 20 41 15/2 -9/222342 22332 10 41 15/2 -11/222379 22357 22 4I 15/2 13/2

23468 23457 11 4G 11/2 -9/223497 23512 -14 4G 11/2 -7/223513 23531 -17 4G 11/2 -11/223537 23554 -16 4F 11/2 5/223551 23570 -18 4G 11/2 3/2

... 23616 ... 4G 11/2 1/2

24865 ... 4M 21/2 -21/2... 24955 ... 4M 21/2 -7/2

24990 24969 21 4M 21/2 3/225008 24986 22 4M 21/2 1/225073 25097 -23 4M 21/2 -9/2

... 25097 ... 4M 21/2 -11/225098 25102 -3 4M 21/2 5/2... 25145 ... 4M 21/2 13/2

25195 25227 -31 4M 21/2 15/2... 25229 ... 4M 21/2 -19/2

25303 25272 31 4M 21/2 17/2

PAGE 3

APPENDIX X

TABLE 1DY+3:LAF3

OBSERVED CALC 0-C STATE J MJ

... 25588 ... 4I 13/2 13/225661 25647 14 41 13/2 5/225691 25713 -21 41 13/2 3/225740 25716 24 4I 13/2 -7/225748 25748 0 4F 7/2 1/225778 25811 -32 4K 17/2 17/2

... 25814 ... 41 13/2 -9/225824 25819 5 41 13/2 -11/2... 25831 ... 4K 17/2 -7/2

25849 25859 -9 4I 13/2 1/225867 25899 -31 4K 17/2 5/2... 25900 ... 41 13/2 -9/2... 25907 ... 4F 7/2 -7/2... 25910 ... 4F 7/2 3/2

25903 25920 -16 4K 17/2 1/225918 25929 -10 4K 17/2 3/2

... 25958 ... 4F 7/2 5/225940 25968 -27 4K 17/2 15/225953 25974 -20 41 17/2 -11/225990 25986 4 4I 17/2 13/2

... 26178 ... 4M 19/2 -19/2.. 26188 ... 4M 19/2 -9/2... 26194 ... 4M 19/2 -7/2

26260 26228 32 4M 19/2 -11/226358 26348 10 4M 19/2 13/226429 26396 33 4M 19/2 5/226448 26421 27 4M 19/2 3/226509 26481 28 4M 19/2 1/226571 26517 54 4M 19/2 17/226583 26527 56 4M 19/2 15/2

27482 27508 -25 4P 3/2 1/227536 27556 -19 6P 3/2 3/2

27581 27586 -4 6P 5/2 5/227624 27626 -1 6P 5/2 3/227665 27659 6 6P 5/2 1/2

... 27841 ... 41 11/2 -11/227919 27944 -24 4I 11/2 5/2

... 27959 ... 41 11/2 3/227988 28001 -12 4I 11/2 1/228036 28006 30 4I 11/2 -7/228074 28035 39 41 11/2 -9/2

PAGE 4

APPENDIX X

TABLE 1DY+3: LAF3

OBSERVED CALC 0-C STATE J MJ

28347 28409 -61 4M 15/2 15/228381 28448 -66 4M 15/2 5/2... 28515 ... 4M 15/2 3/2

28536 28574 -37 4M 15/2 1/228577 28600 -22 4M 15/2 -7/228613 28632 -18 6P 7/2 3/228636 28653 -16 6P 7/2 1/228658 28662 -3 6P 7/2 3/228674 28667 7 6P 7/2 5/228715 28707 8 6P 7/2 -7/228734 28769 -34 4M 15/2 -11/2... 28797 ... 4M 15/2 13/2

... 29467 ... 4F 5/2 3/229535 29574 -38 4I 9/2 1/229638 29620 18 4I 9/2 -9/229667 29622 45 4F 5/2 5/229684 29688 -3 4F 5/2 1/229752 29732 20 4I 9/2 -7/229787 29825 -37 4I 9/2 3/229855 29855 0 4I 9/2 5/2

29890 29917 -26 4G 9/2 -7/2... 29945 ... 4M 17/2 5/2... 29949 ... 4M 17/2 3/2... 29984 ... 4M 17/2 17/2... 29986 ... 4M 17/2 -9/2

29982 29988 -5 4K 17/2 1/2... 30037 ... 4M 17/2 15/2... 30047 ... 4K 17/2 -11/2

30075 30066 9 4K 17/2 13/230141 30131 10 4M 17/2 -7/2

... 30193 ... 4G 9/2 1/230243 30220 23 4G 9/2 3/2... 30269 ... 4G 9/2 5/2

30302 30293 9 4G 9/2 3/2

30887 30868 19 6P 3/2 1/230924 30897 27 6P 3/2 3/2

PAGE 5

APPENDIX X

TABLE 1DY+ 3:LAF3

OBSERVED CALC

31055 31061... 31101

31134 3113731169 3115231194 3117131211 3121931225 31232... 31234

31259 3125031282 3128231294 31315

... 3131931328 31329

... 31344

... 3134831379 3138231452 31431

... 31434

31580 3157031660 31655

... 3171231716 31717

32060320893216932182

33146 33149... 33179

33202 33186... 33196

33218 3320733236 3321233256 33225

... 3349233527 3351533527 3351933527 3352033527 3352333623 3357833558 33579

33666 3364533652 33646

0-C STATE J

-5 4K4K

-2 4K17 4L23 4K-7 4K-6 4L

... 4M

9 4M0 2L

-20 4L... 4M

0 4M... 4K

... 4L

-2 2L21 2L

... 4K

10 4G5 4G

... 4G

0 4G

... 4D

... 4D

... 4D

... 4D

-2 4K... 4K

16 4K... 4K

11 4K24 4K31 4K

... 4H12 4H

8 4H7 4H4 4H

45 4K-20 4H

MJ

15/2 1/215/2 13/215/2 3/219/2 15/215/2 -11/215/2 5/219/2 -9/219/2 17/219/2 17/219/2 1/219/2 -9/219/2 5/219/2 -19/215/2 -7/219/2 15/219/2 13/219/2 -11/215/2 -9/2

7/2 5/27/2 3/27/2 1/27/2 -7/2

5/25/21/25/2

1/25/21/23/2

13/2 -11/213/2 -7/213/2 13/213/2 3/213/2 1/213/2 -9/213/2 5/2

13/2 -9/213/2 13/213/2 5/213/2 3/213/2 13/213/2 -7/213/2 1/2

21 4F 3/26 4F 3/2

1/23/2

... 0

PAGE 6

APPENDIX X

TABLE 1DY+3: LAF3

OBSERVED CALC

34029 3402134040 3402234052 3403634080 34058

... 34236

... 34237

... 3426734247 3427234250 34284... 34284... 34293... 3417

34298 3432234313 34327

... 34328

... 34345

... 34354

... 34361

... 3436734366 3438434393 3441834418 3443734426 3443934477 3447034466 34470

... 3448234525 34502

34854 3485134873 3487434904 3490234914 3490634935 3492934973 34998

... 35780

... 35781

... 3591035940 3595235966 3596335994 3598836006 3600836055 3605436055 3609536080 36162

0-C STATE J

8 4D18 4D16 4D22 4D

... 4H

... 4H

... 4H

-24 4L-33 4L... 4H

... 4F

... 4H-23 4L-13 4L... 4L... 4L... 4L... 4H1

... 4L-17 4L-24 4H-18 4H-12 4H

7 4H-3 4H

... 4F23 4H

3 4G0 4H2 4H8 4G6 4G

-24 4G

... 4K

... 4K

... 4K

-11 4K3 4K6 4G

-1 4G1 4G

-39 4K-81 4G

MJ

7/2 1/27/2 3/27/2 -7/27/2 5/2

11/2 -9/211/2 -7/211/2 -11/217/2 3/217/2 1/211/2 -7/25/2 3/2.

11/2 1/217/2 5/217/2 15/217/2 13/217/2 3/217/2 -11/211/2 -7/217/2 -9/217/2 17/29/2 -9/29/2 1/29/2 -7/2

11/2 5/29/2 3/25/2 1/29/2 5/2

11/2 -9/211/2 -7/211/2 5/211/2 1/211/2 3/211/2 -11/2

11/2 -9/211/2 1/211/2 -11/211/2 -7/211/2 3/27/2 3/27/2 -7/27/2 1/2

11/2 -7/27/2 5/2

PAGE 7

APPENDIX X

TABLE 1DY+3:LAF3

OBSERVED CALC 0-C STATE J NJ

... 36467 ... 4L 13/2 5/2

... 36469 ... 4G 5/2 1/2

... 36469 ... 4L 15/2 3/236494 36502 -7 4L 15/2 1/236512 36526 -13 4L 13/2 -7/236536 36543 -6 4L 15/2 13/2... 36544 ... 4L 13/2 -9/2... 36564 ... 4L 15/2 15/2... 36564 ... 4L 15/2 5/2... 36565 ... ' 4L 13/2 1/2... 36582 ... 4L 13/2 13/2... 36586 .... 4L 13/2 3/2... 36604 ... 4G 5/2 5/2

36634 36610 24 4G 5/2 3/2... 36622 ... 4L 13/2 -11/2

36668 36639 29 4L 15/2 -7/236653 36640 13 4L 15/2 -9/236686 36677 9 4L 13/2 1/2

36752 36750 2 4G 9/2 -7/236780 36785 -4 4G 9/2 3/2

... 36812 ... 4G 9/2 -9/2

... 36840 ... 4G 9/2 -7/2... 36854 ... 4G 9/2 1/2

... 37638 ... 4G 7/2 3/2

... 37672 ... 4P 1/2 1/2

... 37681 .o, 4L 7/2 5/2

... 37777 ... 4L 7/2 -7/2.

... 37794 ... 4G 7/2 1/2

37944 37948 -3 2L 15/2 15/2... 37977 ... 4F 3/2 1/2

37978 38011 -32 2L 15/2 -7/2... 38021 ... 4F 3/2 3/2... 38112 ... 2L 15/2 -9/2... 38214 ... 2L 15/2 5/2... 38223 ... 2L 15/2 1/2... 38296 ... 2L 15/2 -9/2... 38416 ... 2L 15/2 -11/2... 38476 ... 2L 15/2 13/2

38937 38908 29 4P 5/2 3/238996 38994 2 4P 5/2 5/239090 39080 10 4P 5/2 1/2

39152 39169 -16 4P 3/2 3/239176 39192 -15 4P 3/2 1/2

PAGE 8

APPENDIX X

TABLE 1ADY+3:LAF3 CENTERS OF GRAVITY

CALC CENTER STATE

175 6H15/23626 6H13/25952 6H11/27806 6H 9/27853 6F11/29166 6F 9/29223 6H 7/210273 6H 5/211070 6F 7/212471 6F 5/213267 6F 3/213814 6F 1/2

21228 4F 9/222222 4H15/223563 4G11/225109 4M21/225794 4113/225856 4F 7/225890 4K17/226334 4M19/227543 6P 3/227624 6P 5/2

PAGE !APPENDIX

9X

TABLE 2U (K)*2 FOR DY+3

31 LEVEL 1 J2

1/21/21/21/21/21/21/21/21/21/21/21/21/21/21/2

3/23/23/23/23/23/23/23/23/23/23/23/23/23/23/2

3/23/23/23/23/23/23/23/23/23/23/23/23/23/23/23/23/2

138141381413814138141381413814138141381413814138141381413814138141381413814

132671326713267132671326713267132671326713267132671326713267132671326713267

2754327543275432754327543275432754327543275432754327543275432754327543275432754327543

3/23/25/25/25/27/27/27/29/29/2

11/211/211/213/213/2

3/25/25/25/27/27/27/29/29/2

11/211/213/213/215/215/2

3/25/25/25/27/27/27/29/29/29/2

11/211/211/213/213/215/215/2

LEVEL 2

1326727543102731247127624

922311070258567806916659527853

235633626

25794

275431027312471276249223

1107025856

78069166595278533626

25794175

22222

27543102731247127624

9223110702585678069166

2122859527853

235633626

25794175

22222

(U2) *2 (U4) *2 (U6) *2

0.01750.06650.19180.01000.01470.00.00.00.00.00.00.00.00.00.0

0.13910.13960.02220.09640.24210.01340.00000.00.00.00.00.00.00.00.0

0.05720.00070.11510.01850.00040.04200.19070.00.00.00.00.00.00.00.00.00.0

0.00.00.00.00.00.13940.02940.01560.15120.00600.00.00.00.00.0

0.00.13460.02600.02270.11790.01040.02120.12240.03110.19350.0166

0.00.00.0

0.00.07430.00230.00000.04940.01790.04790.03930.07570.10310.00720.17310.04170.00.00.00.0

0.00.00.00.00.00.00.00. 00.00.00.34780.00500. 00200.10010.0013

0.00. 00.00.00.00.00.00. 36360.01120. 03900.03530. 39500. 00760.06110.0014

0.00. 00.00.00. 00.00.00.00890.00090.01140.00700.00000.02960.00160. 15310.05080.0051

PAGE 10APPENDIX X

TABLE 2U(K)*2 FOR DY+3

J1 LEVEL 1 J2 LEVEL 2

5/25/25/25/25/25/25/25/25/25/25/25/25/25/25/2

5/25/25/25/25/25/25/25/25/25/25/25/25/2

5/25/25/25/25/25/25/25/25/25/25/25/25/25/2

7/27/2

.7/27/27/2

102731027310273102731027310273102731027310273102731027310273102731027310273

12471124711247112471124711247112471124711247112471124711247112471

2762427624276242762427624276242762427624276242762427624276242762427624

92239223922392239223

5/25/25/27/27/27/29/29/29/2

11/211/211/213/215/217/2

5/25/27/27/27/29/29/29/2

11/211/213/213/215/2

5/27/27/27/29/29/29/2

11/211/211/213/213/215/217/2

7/27/27/29/29/2

102731247127624

9223110702585678069166

2122859527853235633626175

25890

12471276249223

110702585678069166

2122859527853362625794

175

276249223

1107025856

78069166

2122859527853

235633626

25794175

25890

9223110702585678069166

(U2)*2 (U4)*2 (U6)*2

0.37360.03790.00000.21430.00440.00070.02370.00000.00000.00.00.00.00.00.0

0.00070.28680.19430.04390.00590.33580.00950.00620.00.00.00.00.0

0.25700.00000.46840.09100.00110.49810.00870.00.00.00.00.00.00.0

0.26180.05110.00310.30350.0039

0.06590.27270.03020.19190.13150.00590.13000.01820.00350.02360.00310.00070.00120.00.0

0.01710.08900.05830.00330.01880.10450.03330.00080.19710.05030.21030.00080.0

0.00040.09970.18660.01460. 16210.25510.00460.32300. 16380.01990.17220.00980.00.0

0.00870.29300.00450.15340.1255

0.00.00.00. 07580. 43930.01130. 31270.35510.00110.29930. 01770.00140. 05900. 00260.0024

0.00.00. 41920. 00870.00050.03900.03010. 00040. 15570. 08980.28520. 00070. 3446

0. 00.01440. 00000.00050. 02790.00000.01100.06640. 00290.02830.08200.00100.07210.0633

0. 28590.00000. 02310. 01320.4498

PAGE 11APPENDIX X

TABLE 2U(K)*2 FOR DY+3

J1 LEVEL 1 J2 LEVEL 2

7/27/27/27/27/27/27/2.7/27/27/2

7/27/27/27/27/27/27/27/27/27/27/27/27/2

7/27/27/27/27/27/27/27/27/27/27/27/27/2

9/29/29/29/29/29/29/29/29/29/29/2

9223922392239223922392239223922392239223

11070110701107011070110701107011070110701107011070110701107011070

25856258562585625856258562585625856258562585625856258562585625856

78067806780678067806780678067806780678067806

9/211/211/211/213/213/215/215/217/219/2

7/27/29/29/29/2

11/211/211/213/213/215/217/219/2

7/29/29/29/2

11/211/211/213/213/215/215/217/219/2

9/29/29/2

11/211/211/213/213/215/215/217/2

2122859527853

235633626

25794175

222222589026334

1107025856

78069166

2122859527853

235633626

25794175

2589026334

2585678069166

2122859527853

235633626

25794175

222222589026334

78069166

2122859527853

235633626

25794175

2222225890

(U2) *2 (U4) *2 (U6)*2

0.00080.03480.00040.00010.00.00.00.00.00.0

0.00160.01020.22670.06300.00020.44490.03440.01070.00.00.00.00.0

0.05120.00270.05440.04650.04050.10140.08720.00.00.00.00.00.0

0.30810.04240.00210.34460.00370.00020.03380.00570.00.00.0

0.00790.14650.03060.00440.03090.00340.00060.00160.00.0

0.02740.02430.13870.00520.00430.05030.03440.00000.42170.00150.13520.00.0

0.02750.00740.02660.08380.03180.01980. 14100.01110.37560.06950.03890.00.0

0.01600.33270.00240.16620.11590.00740.17120.00000.01660.00770.0202

0.00670.34260.14380.00130.27630.00010.03930.00030.00210.0082

0.02670.00040. 25080.04350.00290.25210. 08430.00000.00000.00760.71380.00730.0156

0.01230.01120. 00080.33070.00200.00010. 25880. 03210. 00070.02600.00450. 03930. 2900

0.26840.17160.00320.01370. 49690.00000. 39210.00400. 20170.00000.0000

PAGE 12APPENDIX X

TABLE 2U(K)*2 FOR DY+3

J1 LEVEL 1 J2 LEVEL 2 (U2)*2 (U4)*2 (U6)*2

9/2 7806 21/2

9/29/29/29/29/29/29/29/29/29/29/29/2

9/29/29/29/29/29/29/29/29/29/29/2

11/211/211/211/211/211/211/211/211/211/2

11/211/211/211/211/211/211/211/211/2

11/211/2

916691669166916691669166916691669166916691669166

2122821228212282122821228212282122821228212282122821228

5952595259525952595259525952595259525952

785378537853785378537853785378537853

23563 11/223563 13/2

9/29/211/211/211/213/213/215/215/217/219/221/2

9/211/211/211/213/213/215/215/217/219/221/2

11/211/211/213/213/215/215/217/219/221/2

11/211/213/213/215/215/217/219/221/2

25109

91662122859527853235633626

25794175

22222258902633425109

2122859527853

235633626

25794175

22222258902633425109

59527853

235633626

25794175

22222258902633425109

7853235633626

25794175

22222258902633425109

0.0

0.03130.00070.25610.02390.00610.66700.00120.00.00.00.00.0

0.02260.00930.00320.47080.04900.00680.00.00.00.00.0

0.49890.00210.00010.25470.00320.09120.00490.00.00.0

0.14280.01560.25180.00040.93940.00200.00.00.0

0.0

0.01060.00550.39420.01900.00010.02360.01670.57460.01260.03210.00.0

0.25720.00180.00320.16860.01640.04180.00460.55840.45570.00.0

0.03280.19190.00010.49330.00310.03690.00260.00420.02130.0

0. 13850.00010.42480.00680.82990.02700.00370.00010.0

23563 0.3577 0.0555 0.01273626 0.0012 0.0299 0.0065

0. 0226

0.04280.00040.01140.01540.00850. 37780.00820.71860.00630.01320.04250.0919

0. 38270.00330.00240.15110.05450. 39710. 02920.01820.03150. 14100.9115

0. 04980.83360.00190.03000.02210.63920.00060.01130. 03300.0376

0.06240. 00830.77510.00380.20610.00220.00360.02080.0318

PAGE 13APPENDIX X

TABLE 2U(K)*2 FOR DY+3

J1 LEVEL 1 J2 LEVEL 2

11/211/211/211/211/211/2

13/213/213/213/213/213/213/2

13/213/213/213/213/213/2

15/215/215/215/215/2

15/215/215/215/2

17/217/217/2

19/219/2

235632356323563235632356323563

3626362636263626362636263626

257942579425794257942579425794

175175175175175

22222222222222222222

258902589025890

13/215/215/217/219/221/2

13/213/215/215/217/219/221/2

13/215/215/217/219/221/2

15/215/217/219/221/2

15/217/219/221/2

17/219/221/2

26334 19/226334 21/2

21/2 25109 21/2

25794175

22222258902633425109

362625794

17522222258902633425109

25794175

22222258902633425109

17522222258902633425109

22222258902633425109

258902633425109

(U2)*2 (U4)*2 (U6)*2

0.00000.00040.13560.00.00.0

0.79450.00140.24540.00490.00220.00.0

0.50600.00390.02200.00090.00.0

1.33100.00710.01010.00020.0

0.47380.27410.00140.0

0.12630.80450.0018

0.00530.01410.08120.09810.41170.0

0.07380.00030.41360.00410.00440.00490.0034

0.21080.00120.01030.17910.27610.0878

0.64520.00030.00420.01390.0101

0.43630.05480.49090.2399

0.07330. 03611.2616

0.59580.00030.90260. 76940.53930. 6624

0. 10640.00040.68340. 01070.00010. 00260.0037

0. 18770.02490.08860.00810.00290. 4522

0.42660.06590. 09050.09680.0808

0.56341.28891.02510.8452

0. 13460. 02650. 0075

26334 0.0428 0.0612 1.682925109 0.7683 0.3865 0.1317

25109 0.6719 1.2335 0.8622

APPENDIX XI

PAGE 1

APPENDIX XI

TABLE 1HO+3:LAF3 CENTERS OF GRAVITY

OBSERVED CALC 0-C STATE

0 9 -8 5185146 5064 82 5178568 8578 -9 516

11123 11145 -21 515... 13212 ... 514

15420 15456 -35 5F518397 18381 16 5S218524 18538 -13 5F420596 20594 2 5F321056 21039 17 5F221279 21267 12 3K822187 22179 8 5G6

... 22255 ... 5F123942 23987 -44 5G525872 25859 13 5G426087 26058 29 3K727672 27652 20 5G527672 27678 -5 3H6

... 28234 ... 5F228878 28875 3 5G328878 28895 -16 3L929943 29941 2 3K6.29943 29947 -3 3F430795 30799 -3 5G2

... 33063 ... 3D3... 33247 ... 3P1

34100 34072 28 3M1034100 34156 -55 3L834811 34812 0 5G435206 35203 3 3G3

... 36008 ... 3P0.. 36009 ... 5D4

.. G 36294 .. 3F2

36314 ... 1L836724 36720 4 3H5

... 37794 ... 3P2.. 37900 ... 3L7

38378 38339 39 317... 38515 ... 3F4'

PAGE 3APPENDIX XI

TABLE 2U(K)*2 FOR HO+3

J1 LEVEL 1 J2 LEVEL 2

000000

111111111111111

2222222222222222

2222222222

360083600836008360083600836008

222552225522255222552225522255222552225522255222552225522255222552225522255

18381183811838118381183811838118381183811838118381183811838118381183811838118381

21040210402104021040210402104021040210402104021040

244666

122344455556677

2234445555666778

2344455556

210401321218538

85792218027679

2225518381210402059513212185382585911146154572398827653

857922180

506526059

18381210402059513212185382585911146154572398827653

85792218027679

506526059

9

210402059513212185382585911146154572398827653

8579

(U2)*2 (U4)*2 (U6)*2

0.00270.00.00.00.00.0

0.03020.01090.05390.00840.00.00.00.00.00.00.00.00.00.00.0

0.00000.00160.00650.00140.00000.03280.00.00.00.00.00.00.00.00.00.0

0.00770.05210.00050.00850.26080.00.00.00.00.0

0.00.00890.00130.00.00.0

0.00.00.00.05860.14420.04680.05640.13960.00000.00700.00740.00.00.00.0

0.00160.00350.00000.03020.01590.28110.00520.01230.10620.05700.02400.31280.04370.00.00.0

0.02710.00000.20110.08050.00120.04730.00520. 14230.12330.1365

0.00.00.00.01750. 03510. 0805

0.00.00.00.00.00.00.00. 17040.1.1470.05370. 05480.23830. 05280.05680.0020

0.00.00.00. 28390.00330.02160. 09680. 00500. 00040. 00060. 14580. 00470. 00650.41950. 05530.2145

0. 00.00. 02930.03170. 12040. 30240.14660.01150.00120. 1604

PAGE 4APPENDIX XI

TABLE 2U(K)*2 FOR HO+3

J1 LEVEL 1 J2 LEVEL 2

222222

3333333333333333

444444444444444

4444444444

210402104021040210402104021040

20595205952059520595205952059520595205952059520595205952059520595205952059520595

132121321213212132121321213212132121321213212132121321213212132121321213212

18538185381853818538185381853818538185381853818538

667788

3444555566677889

44455556667788

10

4455556667

2218027679

506526059

921268

2059513212185382585911146154572398827653

85792218027679

506526059

92126828896

13212185382585911146154572398827653

85792218027679

506526059

92126834072

185382585911146154572398827653

85792218027679

5065

(U2)*2 (U4)*2 (U6)*2

0.00.00.00.00.00.0

0.03580.00020.09730.21470.00000.03960.18000.15440.00.00.00.00.00.00.00.0

0.12220.00020.01520.03100.00010.00000.00360.00230.00050.00000.00.00.00.00.0

0.07700.39620.00160.19800.27510.29740.00110.25260.01030.0

0.02250.00680.00.00.00.0

0.06370.09820.02980.01760.21730.08160.10050.04740.08970.06550.01120.24630.00680.00.00.0

0. 13080.02410. 10720.12370.00610.00910.00970.02820.001.30.00000.00340.00060.00000.00460.0

0.00850.10520.13340.09200.02380.00250.25740.23800.02100.1965

0.10100. 00990. 07890.05500. 20410. 0036

0.02750.39490. 09660.05090.01750.08510.03250.00590.21720.15260.05090.22790.00490.34640.00670.0012

0. 34560.25760.05550.91030.00360.04180.00080.66390.00000.02000.15680.00680.00760.00120. 0019

0.08860.02170.46660.00710. 13950.05170. 17040.12890.02590.0320

PAGE 5APPENDIX XI

TABLE 2U(K)*2 FOR HO+3

J1 LEVEL 1 J2 LEVEL 2

44444

44444444444444

5555555555555

555555555555

55

1853818538185381853818538

2585925859258592585925859258592585925859258592585925859258592585925859

11146111461114611146111461114611146111461114611146111461114611146

154571545715457154571545715457154571545715457154571545715457

23988 523988 5

7889

10

4555566677889

10

555566677889

10

55566677889

10

(U2) *2 (U4) *2 (U6) *2

260599

212682889634072

2585911146154572398827653

85792218027679

506526059

9212682889634072

11146154572398827653

85792218027679

506526059

9212682889634072

154572398827653

85792218027679

506526059

9212682889634072

2398827653

0.00.00.00.00.0

0.00980.22540.03450.01280.04150.68960.00360.00300.00.00.00.00.00.0

0.10230.00710.00310.00400.04350.01710.00040.00280.00720.00.00.00.0

0.07290.34250.09070.01131.13050.11130.01940.00150.00.00.00.0

0.01560.23850.00850.00.0

0.01380.27000.01380.14840.14170.02260.20470.00670.28570.01520.03510.09190.00.0

0.03640.02810. 06420.04870.17030.03120.00110.02260.00420.01020.00040.02530.0

0.18150.03530.17520. 12420.36160.00000.33090.01120.42010.02580.06080.0

0.0706 0.0473 0.01390.0263 0.0132 0.2050

0.00910.70900.00070.00930.0417

0.18120.03290.21470.02680.01820.00030. 29090.00930.06420.00440.03320.00130.04920.2687

0.01760. 16300.05680.04180.57200.01240.01530.88960.03120.09300.00410.02420.0368

0.00500.11450.04830.49720.03210.00020.42980.01420. 57010. 01900. 07410. 1693

PAGE 6APPENDIX XI

TABLE 2U(K)*2 FOR HO+3

J1 LEVEL 1

555555555

55555

5555

666666666

66666666

6666666

239882398823988239882398823988239882398823988

27653276532765327653276532765327653276532765327653

857985798579857985798579857985798579

2218022180221802218022180221802218022180

27679276792767927679276792767927679

J2 LEVEL 2

66677889

10

56667788q

10

66677889

10

6677889

10

677889

10

85792218027679

506526059

9212682889634072

276538579

2218027679

506526059

9212682889634072

85792218027679

506526059

9212682889634072

2218027679

506526059

9212682889634072

276795065

260599

212682889634072

(U2)*2 (U4)*2 (U6)*2

0.12860.05650.19870.56960.01640.00.00.00.0

0.03600.09490.00050.15010.70030.00360.00.00.00.0

0.12730.00910.00010.03140.00110.00870.00590.00.0

0.00250.05090.14840.00001.48300.00000.00.0

0.14990.03140.00400.25400.10770.00.0

0.16950.25680.06570.02400.00590.52390.00830.33210.0

0.00580.24520.07110.58250.19130.01040.09380.00510.18030.0

0.06810.08190.00510. 13240.00570.03890.00410.00710.0177

0.39680.29920.42840.00030.82010.00480.04020.0455

0.10320.05860.05020.13990.11090.11810.3381

0. 07890. 24560.01680.11710.24870.00000.00070.17020.3522

0.00120.04600. 15800. 00000.05840. 48060. 15960.02140. 17260.2616

0.04020.10940.00240.92950.0722"0.69200.01420.03570.0211

0. 12090. 00690.26330. 12390. 14000. 30760. 28100. 1696

0.04180.00870.68540.00131.59850. 82880.2755

5065 75065 7

5065 0.1502 0.1193 0.029726059 0.0055 0.0059 0.0047

77

PAGE 7APPENDIX XI

TABLE 2U(K)*2 FOR HO+3

J1 LEVEL

7777

77777

8888

8

8

99

5065506550655065

2605926059260592605926059

9999

212682126821268

1 J2 LEVEL 2

8 98 212689 28896

10 34072

7 260598 98 212689 28896

10 34072

8 98 212689 28896

10 34072

8 212689 28896

10 34072

(U2) *2 (U4) *2

0.0249 0.13440.0018 0.00440.0009 0.00410.0 0.0003

0.0016 0.02230.0056 0.00440.0865 0.00020.0788 0.19630.0 0.3112

(U6) *2

1.52310.04020.01030.0059

0.01750.03350.12230.13910.3289

0.1951 0.3117 1.54600.0205 0.0317 0.15350.0179 0.0051 0.14990.0003 0.Ob81 0.0789

0.0260 0.0465 0.20190.3163 0.0568 2.04950.0134 1.0322 0.8500

28896 9 28896 0.6858 1.3547 0.029128896 10 34072 0.7340 1.4855 0.5524

10 34072 10 34072 3.3219 0.0011 1.4872

APPENDIX XII

PAGE 1

APPENDIX XII

TABLE 1ER+ 3: LAF3

OBSERVED CALC 0-C STATE J MJ

0 -9 10 41 15/2 13/251 50 1 41 15/2 -11/2

121 121 0 4I 15/2 3/2200 188 12 41 15/2 5/2219 198 22 4I 15/2 1/2314 310 4 41 15/2 -9/2400 397 3 41 15/2 -7/2443 442 1 41 15/2 15/2

6604 6607 -2 4I 13/2 -11/26630 6639 -8 4I 13/2 -9/26670 6677 -6 41 13/2 -7/26700 6696 4 4I 13/2 1/26723 6730 -6 4I 13/2 3/26754 6765 -10 4I 13/2 5/26823 6835 -11 4I 13/2 13/2

10301 10299 2 41 11/2 -9/210311 10309 2 41 11/2 -7/210330 10336 -5 4I 11/2 1/210344 10348 -3 4I 11/2 3/210358 10359 0 4I 11/2 5/210395 10408 -12 41 11/2 -11/2

12419 12401 18 41 9/2 7/212518 12535 -16 4I 9/2 1/212615 12605 10 4I 9/2 3/212701 12705 -3 41 9/2 5/212730 12725 5 41 9/2 -9/2

15391 15393 -1 4F 9/2 5/215432 15439 -6 4F 9/2 3/215443 15447 -3 4F 9/2 -7/215474 15476 -1 4F 9/2 3/215527 15529 -1 4F 9/2 1/2

18557 18559 -1 4S 3/2 3/218588 18588 0 4S 3/2 1/2

19266 19271 -4 2H2 11/2 1/219307 19296 11 2H2 11/2 -9/219314 19318 -3 2H2 11/2 -11/219359 19344 15 2H2 11/2 -7/219359 19350 9 2H2 11/2 3/219418 19403 15 2H2 11/2 5/2

20656 20655 1 4F 7/2 -7/220703 20698 5 4F 7/2 1/220734 20736 -1 4F 7/2 5/220786 20788 -1 4F 7/2 3/2

PAGE 2

APPENDIX XII

TABLE 1ER+3:LAF3

OBSERVED CALC

22370 2236422374 2236922407 22399

22684 2268922751 22741

24602 2458024680 2471024754 2474824840 2484024862 24858

26530 2653626558 2656726583 2658726647 2665026647 2665026709 26713

276C6 2761027620 2761727631 2762427646 2763927671 27665

27813 2781627820 2782327830 2785127904 2789127935 27939

... 27959

... 2798828127 28135

... 2822128243 2824028257 2824928265 28252

31688 3171131746 31765

... 330943310E 3310533119 3314233167 3315333201 3320333201 33206

... 33317

... 33405

0-C STATE J

6 4F5 4F9 4F

5/25/25/2

MJ

5/21/23/2

-4 4F 3/2 3/210 4F 3/2 1/2

22 4F-29 4F

6 4F0 4F4 4F

-5 4G-8 4G-3 4G-2 4G-2 4G-3 4G

-3 4G3 4G7 4G7 4G6 4G

-2 2K-2 2K

-20 2K13 2K-3 2K

... 2K

... 2K

-7 2K

... 4G

3 4G8 4G

13 4G

-22 2P-18 2P

... 2K

3 2K-22 2K

14 2K-1 2K-4 2K

... 2P

... 2K

9/2 -7/29/2 1/29/2 3/29/2 5/29/2 3/2

11/2 1/211/2 -11/211/2 3/211/2 5/211/2 -9/211/2 -7/2

9/2 -9/29/2 1/29/2 3/29/2 5/29/2 -7/2

15/2 -9/215/2 -7/215/2 -11/215/2 5/215/2 3/215/2 13/215/2 1/215/2 15/2

7/2 5/27/2 -7/27/2 3/27/2 1/2

3/23/2

3/21/2

13/2 -9/213/2 -7/213/2 -11/213/2 5/213/2 1/213/2 3/21/2 1/2

13/2 13/2

PAGE 3

APPENDIX XII

TABLE 1ER+3:LAF3

OBSERVED CALC

34157341963422134280

350263505235085

3652236555366243672136805

3880438814138834

39454395393960639634

412384129741315

4138241497

..

.

.

... 0

...

...

...

334823349433613

34167341973422834284

350423505335092

3652936537366363672136792

387943884338844

39487395283960639638

412354130241330-411714139241492

418094183341874418754190041926419804207142087

0-C STATE J

... 4G 5/2

... 4G 5/2

... 4G 5/2

-90

-6-3

-150

-6

-618

-110

13

10-1-9

-3211

0-3

3-4

-14

-95

...

...

0...

...

42495 42471 2442526 42500 26

4G4G4G4G

2D12D12D1

2H22H22H22H22H2

4D4D4D

4D4D4D4D

212121212121

2L2L2L2L2L2L2L2L2L

4D4D

7/27/27/27/2

5/25/25/2

9/29/29/29/29/2

5/25/25/2

7/27/27/27/2

11/211/211/211/211/211/2

17/217/217/217/217/217/217/217/217/2

3/23/2

43088 43096 -7 4D 3/243126 43121 5 4D 3/2

MJ

1/25/23/2

-7/23/21/25/2

3/25/21/2

-9/25/23/21/2

-7/2

3/25/21/2

1/23/2

-7/25/2

5/23/2

-11/2-7/2

1/2-9/2

-7/2-9/2

1/25/23/2

-11/213/217/215/2

3/21/2

1/23/2

PAGE 4

APPENDIX XII

TABLE 1ER+3: LAF3

OBSERVED CALC

4368743746437464376043834439 15

.. 0

43709437224372843749438214390943968

0-C STATE J

-21241811136

... 47313

492234927249357

...

...

4794147989479994801548035480974819348194

4837148374484274848348513

492104928749349

512955135551370514415151051512

21212121212121

13/213/213/213/213/213/213/2

MJ

5/21/23/2

13/2-7/2-9/2

-11/2

... 4D 1/2 1/2

13-14

8

2L2L2L2L2L2L2L2L

2H12H 12H12H12H1

2D22D22D2

2H12H12H12H12H12H1

15/215/215/215/215/215/215/215/2

9/29/29/29/29/2

5/25/25/2

11/211/211/211/211/211/2

-9/2-7/23/21/25/2

-11/215/213/2

5/23/21/2

-9/2-7/2

3/21/25/2

1/23/2

-11/25/2

-9/2-7/2

PAGE 5

APPENDIX XII

TABLE 1AER+3:LAF3 CENTERS OF GRAVITY

CALC CENTER STATE

217 4115/26712 4113/28583 4S 3/210346 4111/212597 41 9/2

15455 4F 9/219337 2H11/220715 4F 7/222376 4F 5/222712 4F 3/224756 4F 9/2

26631 4G11/227637 4G 9/227922 2K15/228224 4G 7/233319 2P 1/2

PAGE 7APPENDIX XII

TABLE 2U(K)*2 FOR ER+3

J1 LEVEL 1 J2

1/21/21/21/21/21/21/21/21/21/21/2

3/23/23/23/23/23/23/23/23/23/23/23/23/23/23/2

3/23/23/23/23/23/23/23/23/23/23/23/23/2

5/25/25/25/25/25/25/25/2

3331933319333193331933319333193331933319333193331933319

185831858318583185831858318583185831858318583185831858318583185831858318583

22712227122271222712227122271222712227122271222712227122271222712

2237622376223762237622376223762237622376

3/23/25/27/27/29/29/29/2

11/211/211/2

3/23/25/27/27/29/29/29/29/2

11/211/211/213/215/215/2

3/25/27/27/29/29/29/29/2

11/211/211/213/215/2

5/27/27/29/29/29/29/2

11/2

LEVEL 2

1858322712223762071528224125971545527637103461933726631

185832271222376207152822412597154552475627037103461933726631

6712217

27922

2271222376207152822412597154552475627637103461933726631

6712217

2237620715282241259715455247562763710346

(U2)*2 (U4)*2 (U6)*2

0.00570.03530.00730.00.00.00.00.00.00.00.0

0.03710.02660.00770.00000.04750.00.00.00.00.00.00.00.00.00.0

0.07090.0b050.00270.09610.00.00.00.00.00.00.00.00.0

0.01520.07650.38310.01010.00050.00920.16500.0

0.00.00.00.02040.02630.02720.04600.00830.00.00.0

0.00.00.00360.00550.16310.07650.00010.00360.16590.00460.20020. 12820.00.00.0

0.00.03510.05770.03420.23380.00220.01880.17110.09130.00040.02320.00.0

0.00500.04980.00170.06290.23450.02190.08460.0984

0.00.00.00.00.00.00.00.00.03160. 16860.0264

0.00.00.00.00.00. 25690.02280.00140.01030. 07730. 00970. 00400.34190. 22250.0035

0.00.00.00.00. 05450.06160. 00570. 11240.48310.00250. 09070. 03470. 1255

0.00. 09980.03800.11290.34910.00560.00240.0028

PAGE 8APPENDIX XII

TABLE 2U(K)*2 FOR ER+3

J1 LEVEL 1 J2 LEVEL 2

5/25/25/25/25/2

7/27/27/27/27/27/27/27/27/27/27/2

7/27/27/27/27/27/27/27/27/27/27/2

9/29/29/29/29/29/29/29/29/29/2

9/29/29/29/29/29/29/29/29/2

22376 11/222376 11/222376 13/222376 15/222376 15/2

20715 7/220715 7/220715 9/220715 9/220715 9/220715 9/220715 11/220715 11/220715 11/220715 13/220715 15/2

28224 7/228224 9/228224 9/228224 9/228224 9/228224 11/228224 11/228224 11/228224 13/228224 15/228224 15/2

12597 9/212597 9/212597 9/212597 9/212597 11/212597 11/212597 11/212597 13/212597 15/212597 15/2

15455 9/215455 9/215455 9/215455 11/215455 11/215455 11/215455 13/215455 15/215455 15/2

(U2) *2 (U4) *2 (U6) *2

0.00.00.00.00.0

0.15420.12210.01550.01190.08940.62340.00320.12580.08670.00.0

0.05810.03730. 17940.00.0

0.01030.04090.09350.03720.04830.00670.26530.01640. 12640.33930.1465

1933726631

6712217

27922

20715282241259715455247562763710346193372b631

6712217

28224125971545524756276371034619337266316712217

27922

12597154552475627637103461933726631

6712217

27922

1545524756276371034619337266316712

21727922

0.00400.12200.01380.00410.00210.19530.06310.00030.00.0

0.13690.00750.21700.07150.37900.42830.01090.00.0

0.07820.00610.00660.00600.06900.06480.01220.00870. 15870.2101

0.07510.02610.31670.01010.02360.03720.15330.55110.0867

0.0033 .0.00480.1649 0.37030.0000 0.01230.0152 0.00520.0034 0.18870.5073 0.27760.0006 0.03930.0140 0.05440.0 0.09970.0 0.02000.0 0.1206

0. 18470. 08060.34190.22210. 0472

0.10010.00700. 43370.01090. 02730.11940. 15450. 39840.01680. 00010.6272

0. 00050.21680.01630.02440. 14940.16160.27100.01770. 03100.11710. 0048

0.79320.02030. 00320.00490. 15200. 28370. 02280.71000.00720. 0969

0. 05070. 04690.36501. 26710. 00080.01120. 08280. 46210.0142

PAGE '9APPENDIX XII

TABLE 2U(K)*2 FOR ER+3

J1 LEVEL 1 32 LEVEL 2

9/29/29/29/29/29/29/29/2

9/29/29/29/29/29/29/2

11/211/211/211/211/211/2

11/211/211/211/211/2

11/211/211/211/2

13/213/213/2

15/215/2

2475624756247562475624756247562475624756

27637276372763727637276372763727637

103461034610346103461034610346

1933719337193371933719337

26631266312663126631

671267126712

217 15/2217 15/2

9/29/2

11/211/211/213/215/215/2

9/211/211/211/213/215/215/2

11/211/211/213/215/215/2

11/211/213/215/215/2

1 1/213/215/215/2

13/215/215/2

(U2) *2 (U4) *2 (U6) *2

24756276371034619337266316712

21727922

276371034619337266316712

21727922

103461933726631

6712217

27922

1933726631

6712217

27922

266316712

21727922

6712217

27922

0.01650.02510.03810.02850.29510.05900.00.0

0.00180.09370.02370.00001. 10780.00.0

0.07840.03520.00020.03320.02760.0463

0.00210.00040.02350.71580.1010

0.00490.10050.91560.0998

0.17230.01950.0001

0.05460.00040.07530.16350.10680.10590.02430.7139

0.01530.16010.34630.21170. 36720.23370.0056

0.03640.13850.04860.17060.00020.0017

0.07260. 15130.06210.41380.0000

0.25130.26480.52630.0579

0.17310.11720.0015

217 0.2463 0.3803 1.861127922 0.0213 0.0039 0.0735

1.8431 1.0174 0.0676

0.53960.03280. 10470.06100.14140.35310.21470.0822

0.00780.01680. 15700.15160.01060.13680. 0558

0. 02770.03720.01331.09150. 39420.2426

0.10680. 04980.05020. 09271.1445

0. 06690.25700.11670. 6787

0.22981.43250.0257

15/2 279 22 15/2 27922

APPENDIX XIII

PAGE 1

APPENDIX XIII

TABLE 1TM+3:LAF3 CENTERS OF GRAVITY

OBSERVED CALC 0-C STATE

200 175 25 3H65858 5818 40 3F48336 8391 -54 3H5

12711 12721 -9 3H414559 14597 -37 3F315173 15181 -7 3F2

21352 21314 38 1G4

28061 28001 60 1D2

34886 34975 -88 116

35604 35579 25 3P036559 36615 -55 3P138344 38268 76 3P2

... 75300 ... 150

PAGE 3APPENDIX XIII

TABLE 2U(K)*2 FOR TM+3

J1 LEVEL 1 J2 LEVEL 2

0 356210 356210 356210 356210 35621

1 366031 366031 366031 366031 366031 366031 366031 36603

2 151802 151802 151802 151802 151802 151802 15180

2 280282 280282 280282 280282 280282 28028

3 145983 145983 145983 145983 14598

4 58284 58284 58284 5828

4 127354 127354 12735

5 83965 8396

2 151802 280284 58284 127356 153

1 366032 151802 280283 145984 58284 127355 83966 153

2 151802 280283 145984 58284 127355 83966 153

2 280283 14598'4 58284 127355 83966 153

3 145984 58284 127355 83966 153

4 58284 127355 83966 153

4 127355 83966 153

5 83966 153

(U2) *2 (U4) *2 (U6) *2

0.3618 0.0 0.00.0297 0.0 0.00.0 0.2796 0.00.0 0.0235 0.00.0 0.0 0.0756

0.16^l 0.0 0.00.1 1 0.0 0.00. I 0.0 0.00.. 4 0.1964 0.00.C 0.1099 0.00.0 0.4029 0.00.0 0.2857 0.08920.0 0.0 0. 1239

0.1425 0.0443 0.00.0642 0.3073 0.00.0036 0.0745 0.00.3026 0.0562 0.04400.2969 0.1711 0.07630.0 0.2907 0.58440.0 0.0000 0.2550

0.1931 0.0051 0.J0.1638 0.0698 0.00.5689 0.0961 0.02150.1257 0.0124 0.23000.0 0.0012 0.01820.0 0.3131 0.0958

0.0625 0.0030 0.06250.0025 0.0005 0.16880.0817 0.3522 0.28440.6285 0.3467 0.00.0 0.3164 0.8413

0.0104 0.4059 0.26510.1275 0.1311 0.21130.0909 0.1299 0.92640.5395 0.7261 0.2421

0.2672 0.1650 0.57040.0131 0.4762 0.00950.2357 0.1081 0.5916

0.9192 0.3668 0.12140.1074 0.2314 0.6385

6 153 6 153 1.2517 0.6916 0.7759

APPENDIX XIV

-1-

APPENDIX XIV - TABLE 1

Calculated Values of Q for Ln3 +:LaF3

x10- 2 cm2 x1 20cm2 6 -20cm2 Reference

0.12

0.35

0.5

1.0

1.19

1.1

1.1

1.1

1.16

1.07

0.52

1.77

2.57

1.9

0.5

1.16

1.2

1.4

1.4

1.38

0.28

0.59

4.78

2.50

2.2

1.5

0.39

0.5

0.9

0.9

0.88

0.63

0.22

a

a

b

b

c

b

b

b,d

e

f

b,g,h

aKrupke (1966)

bA;proximate values from present work

CWeber (1967a)

dKrupke (1974)

eWeber et al. (1972)

(Weber (1967b)

9Weber (1967c)

hPappalardo (1976)

Pr3+

Nd

Pm

Sm

Eu

Gd

Tb

Dy

Ho

Er

Tm

W "w'

-2-

APPENDIX XIV - TABLE 2

Observed and calculated radiative life-times for Ln3+:LaF3

T

Excited TR ObservedState psec (sec) Reference

Pr 112 902 520 a

3P0 73 47

Nd 4F3/2 635 670 b

Pm 5F1566 - c

Sm 4G5/2 2160 - c

Eu 5D2 9200 5400 d

5D17700 4700

50 6900 6700

Tb 5D3 809 - c

514 1450 -

Dy 4F9/2 896 - c

Ho 5S2 826 - e

5F5 779

Er P3/2 430 290 f

453/2 1020 1000

Tm 112 137 54 a,g

1G4 1560 960 a,g

aWeber (1967c), (1968)

bRiseberg and Weber (1976), Weber (1967c)

CPresent Work

dWeber (1967a), measured at 77*K

eWeber et al. (1972)

(Weber (1967b)

9Compare recent calculations by Pappalardo (1976)

-3-

APPENDIX XIV - TABLE 3

Partial lifetimes and Branching Ratios in the

Relaxation of Excited States in Tb3+:LaF 3.

Transition Partial Electric- Partial Magnetic- ORDipole Lifetime (msec) Dipole Lifetime (msec)

503,5D4 125.4 3.413 0.24

7F0 00 0

7F1 91.01 0.009

7F2 51.93 5.606 0.16

7F3 108.4 109.3 0.015

F4 28.61 1.590 0.54

7F5 30.32 0.027

7F6 87.15

(5D3) TR = 0.809 msec.

5D4+ F 425.6 0.003

F1275.9 0.005

F2 466.4 0.003

7F3 208.3 15.28 0.10

F4 144.1 505.1 0.013

7F5 27.52 1.800 0.86

F6 114.0 0.013

(5D4) TR * 1.45 msec.

aThe intensity parameters used in the calculations are given in Appendix XIV -

Table 1. The matrix elements of U(A) appear in Appendix IX.