I.E.7.: I Nuclear Physics A120 (1968) 369--386;(~North-HollandPublishing Co., Amsterdam

I .D.4 [ Not to be reproduced by photoprint or microfilm without written pefmiuion from the publisher

C O M P R E H E N S I V E C O M P A R I S O N O F N U C L E I D I C M A S S F O R M U L A E

BASED O N D I F F E R E N T N U C L E A R M O D E L S t

JAMES WING Chemistry Division, Argonne National Laboratory Argonne, Illinois 60439

Received 12 July 1968

Abstract: The following quantities have been calculated with the nucleidic masses of the semi- empirical formulae recently developed by Kummel et al., Seeger, Zeldes et aL, and Garvey and Kelson: (i) alpha decay energies as a function of neutron number N; (ii) slopes and inter- cepts of isotone lines representing neutron-pair separation energies as a function of proton number Z; (iii) roughness parameter, defined as E# (Z, Nq-1)q-Ep(Z--1,N)--E#p(Z--1, N-t-l), where E# and E#B are total and double p- decay energies, respectively; (iv) steepness, minima, and separation distances of isobaric parabolas; and (v) energies available for emission of delayed neutron and delayed proton from precursors. These calculated quantities show systematic and random deviations from the empirical data at various mass regions. The present comparison easily reveals the similarities and differences among the nucleidie mass formulae which are based on various nuclear models. A list of errors in the publication by Kummel et aL on their mass equation is included.

1. Introduction

M a n y at tempts have been made in the past thirty years to develop a mathematical formula which describes the binding energies o f the nucleides. We now have some 30 so-called nucleidic mass equations which are formulated f rom various nuclear models. I t is desirable to examine these nucleidic mass equations to see which, if any, best suits our applications. In 1964, a compar ison 2,2) o f 15 mass equations with the experimental masses indicated a need for improvement in these equations. A comprehensive examination 3) was made in 1967 on five semi-empirical equations, which were published between 1964 and 1966 by Hillman 4), Cameron and Elkin 5), Wing and Fong 2), Seeger 6), and Myers and Swiatecki 7). I t was found that these five equations (all based on the l iquid-drop model of the atomic nucleus) reproduced a certain observed mass regularities only at a few selected mass regions, and gave systematic and r a n d o m deviations f rom the empirical data in other mass regions.

The methods o f compar ison adopted in the previous examination a) proved to be useful for showing systematic deviations o f the mass equations f rom the experimental data. They revealed easily similarities and differences among the mass equat ions compared. Other comparat ive methods such as predictions o f charge distribution in nuclear fission and o f elemental abundances in nucleogenesis o f stars involve

t Based on work performed under the auspices of the U.S. Atomic Energy Commission.

369

370 ~. VCING

extrapolations to regions of unknown mass systematics and applications of uncertain nuclear theories. The present work is an extension of the previous systematic comparison to include four newer mass formulae, which were developed from dif- ferent nuclear models.

Specifically, in this work, we compare the systematics of alpha and beta decay energies and of neutron-pair separation energies, which are predicted by four mass formulae, with the empirical ones. The 1964 Atomic Mass Table 8) is the only source of experimental data, in spite of a few incorrect data in it 9). The nucleides involved in these phenomenological comparisons are of known masses and their relationships to each other are schematically shown in fig. 1. We also examine the energetics, as predicted by the four mass formulae, of the reported delayed-neutron and delayed-proton emissions. We assume that the four mass formulae may be extrapolated to the nucleides which are the precursors of the delayed-nucleon emis- sions.

I B2N

Fig. 1, Relationships of the nuc]eides whose mass differences are compared in this work. The symbol B2p denotes separation energy of two protons. The other symbols are defined in the text.

2. Nucleidic mass formulas

We examine the mass formulas developed recently by Kummel et aL x o), Seeger 11), Zeldes et al. 12), and Garvey and Kelson 13). These authors used the 1964 Mass Table 8) to evaluate their adjustable parameters. For simplicity, we identify these formulas by the names of the first authors. Kummel's equation (Formula III only) is based on the liquid-drop model, with the volume and coulomb energies containing shell falling and occupation number dependence. The treatment of deformation energy essentially follows that of Myers and Swiatecki 7), and the pairfng energy is a phenomenological correction. Seeger's equation, not to be confused with his earlier ones 6,14), is based also on the liquid-drop model, with shell and deformation cor- rections given by Nilsson model 15) of level diagram in a manner following Stru- tinsky 16), and pairing energy by Bardeen-Cooper-Schrieffer model 17). Zeldes' latest

NIJCLEIDIC MASS FORMULAE 371

formula 12), like its earlier versions 18,19) and comparable to Levy's mass equation 20),

is based on the concepts of the single-particle model. The nuclear mass surface is divided into sections bound by the magic numbers, and the binding energies of the nucleides within each section are represented essentially by a quadratic equation in terms of the occupation numbers of the nucleons. Garvey's formula is based on the assumption of zero value of "roughness parameter". This roughness parameter 21) R may be defined as

R(Z, N) = Ea(Z, N+ 1 ) + E a ( Z - 1, N)-Eaa(Z- 1, N+ 1), (1)

where Z and N are atomic and neutron numbers, respectively, and E a and Eaa are total and double t - decay energies, respectively. Thus, if the value of R is known, one can obtain the mass of a nucleide when the masses of the other five nucleides involved are substituted into eq. (1).

We calculated the various mass differences to be compared from the mass excesses in the tabulations 12,22-25) by the authors of the mass formulas. We ignored all the nucleides with mass numbers less than 42. The calculations were performed with an IBM-1620 computer, and the results checked with the tabulations. Note that in Zeldes' tabulation x2) the tabulated value of a mass difference, such as separation or decay energy, which involves a nucleide situated on the boundary between two shell regions, may be different from that obtained from the tabulated mass excesses. This discrepancy arises from the fact 24) that the masses of the boundary nucleides were calculated with the coefficients of the lower shell region, whereas, for distant extrapolation along the boundary, the mass differences involving boundary nucleides were obtained with the coefficients of the same shell region.

The mass excesses in Kummel's tabulation 22) are slightly different from those calculated with his mass equation 10). Each time Kummel's mass tabulation is made, the computer generates by iteration a new set of adjustable parameters whose values may be different from the previous ones 22). The following discrepancies are noted among the selected mass regions:

Mass using the equation in ref. 10) Mass region minus mass in tabulation 22)

50 4 keV 100 9 keV 150 16 keV 200 25 keV 250 35 keV

A list of errors in ref. 10) is given in the appendix.

In the following discussion, we shall occasionally mention some of the results o f the previous comparison a). The earlier version 6) of Seeger's mass law is called Seeger's (1965) equation.

372 ~. wn~o

2.0

1.0

v

-_o

o I

-I ,0

Seeger (1968) ~4z

: ', 6 0 8,

I I

l t r ]~x • *+ i " - - - ~ v ~ • i .

/ 9 ~ " 3 r40 / s !I 52

3 7

-2.0 30 40 50 60 70 80 90

Neutron Number

Fig. 2. Deviations of alpha decay energies of Seeger's equation from the empirical values, versus neutron number from 24 to 94.

2.0

1 . 0 o

A

m

O tlj I

O .

== -,.o

-2 .0

I00 I10 120 130 140 150 Neutron Number

Fig. 3. Deviations of alpha decay energies of Sccger's equation from the empirical values, versus neutron number from 92 to 155.

NUCLEIDIC MASS FORMULAE 3 7 3

3. Alpha-decay energies

The deviations of alpha decay energies E~ of the four mass formulae from the empirical values are plotted versus the neutron number of the parent nucleides in figs. 2 to 7. The solid lines and dashes connect isotopes of odd and even atomic numbers, respectively. In ref. 3), we plotted only odd mass parents, which had zero or separated pairing energies, to show the systematic deviations not con- nected with nucleon pairing. The present four mass formulas have non-zero pairing energies for odd mass nucleides, and we included both odd and even mass parents in figs. 2 to 7.

" ,~ " K u m m e l

; I .o z z I 36 i i! 5z 6563 ,,/~<L....~i~3?_,'t 32 33 . A [I r/~--t 5i ,; , ,58 5Z ,Lk,~, - ,.," 't. ,,'\. ," =,, i ; i', .." .

,, .. ,7/~i.~% . 1 - 39 ,' ' t ', ,~ J \ / ~ I ." l ^ '< - ' - " ~ ' ; I "", ' , . . . . i . . , " ~ , ( 2, ~ ..

~ o 1; ~ -~ " "

I "-'.;" ",",'+.

' ' 8 ! 1 / . ; 40 ,,, ~ , 40

3 0 4 0 5 0 6 0 7 0 8 0 9 0

N e u t r o n N u m b e r

Fig. 4. Deviations of alpha decay energies of Kummel's equation from the empirical values, versus neutron number from 24 to 94.

i i i I i i , i I i L i I ' ' i ~ I ' ' i , I i , i i I ' '

',,82 ', K u m m e l

> 1.0 83 \ 8~ ~i 65 9 r ~ - 7, / 7~ , ' - ' f . . . . .

,

_o 70 ^ ,' " \ 9b Ioo \

~'J ~' , .s '..~U)' "~' ~, , ~ TM ~,-\, !~ 'x- 7 , '~ ~ " , Y

~" I - ' i 74 ,,,,-+o,, t ',.'V' ~ - I . 0 1 - \ 6 9 76, 79' \co

I i i i I i ~ ~ i I i i ~ i I i ~ i i I i i i I I i i ~ ~ I ~ i

I 0 0 I 1 0 1 2 0 1 3 0 1 4 0 1 5 0

N e u t r o n N u m b e r

Fig. 5. Deviations of alpha decay energies of Kurnmel's equation from the empirical values, versus neutron number from 92 to 155.

374 I. ~ m o

2 3 4 6 , I i ~ 1 1 . 0 ? 3 2 ~ , ~ 4 2 ',

I A ,, A",Y i ; 2 2 , l / ~ _ 3 3 3 9 ~'" 4 0 ' ' : / ~ / ~ ' t 6 3

; , 44 ~6kj ~z 40

"% -~.o i1~ 3 4 0 5 0 ~7 6 0 Ze ldes 7 0 8 0 9 0

" ~ 1.0

" ' 90 ,oo'°°,,

- I . 0 9 0 I 0 0 I 1 0 120 130 1 4 0 1 5 0

N e u t r o n N u m b e r

Fig. 6. Deviations of alpha decay energies of Z o l d e s ' formula from the empirical values, versus neutron number.

i i ~ I ' ' i i i i ~ i i i l i . ~ ~ j l J i i i i ~ ~ i i i i i i I i

"el "A ~o ~P ~ s ? ~s

[ - 2 4 2 5 3 2 35 3 9 l ~ ~ j , A J~ ' - - 5 5 .~ " / , : : ' , ' t , : i l / " 49 ~ ~,? 5g 6,0

I / ' \ ' ¢o v ~ r - ; , ~ , 4e 5z 2V, 2~ V & ~i

42 if

o I i q I I I i , ~ , I p r 40~'~ 1 ~ i , T I , ~ r , I ~ p , , I uo - I . 0 R I 3 0 4 0 5 0 6 0 7 0 8 0 9 0

G o r v e y !

LO o

8 2 8 4 6s '. ,,,~ 66 ~ /, , ,,

'~ 6 9 " . , ' \ , 9 3 6 6 / \ ,'

8 4 , . J 8 2

9 0 I 0 0 I 1 0 1 2 0 1 3 0 1 4 0 1 5 0 N e u t r o n N u m b e r

Fig. 7. Deviations of alpha decay energies of Garvey's formula from the empirical values, versus neutron number.

NUCLEIDIC MASS FORMULAE 375

All the four mass formulae show relatively large systematic deviations, many in the regions of closed-shell configuration. Seeger's equation has the widest spread of deviations and Garvey's the narrowest. In spite of the shell-model nature in Zeldes' equation, systematic discrepancies due to nuclear shell effects still exist in the regions of 40 neutrons, and 40 and 50 protons. Note in figs. 4 and 6 the almost superimposed peaks in the region of 63 protons (88 neutrons), just after the shell closing of 82 neutrons. We observed 3) these peaks also in Seeger's (1965), Myers', and Wing's equations. In Seeger's equation (fig. 2) these superimposed peaks come right at neutron number 82. There are outstanding systematic deviations for the isotopes of Ce, Tb, and Tm in the present four mass equations and several others in individual formulae.

4. Neutron-pair separation energies

The separation energies of a pair of neutrons B2N generally increase linearly with increasing atomic number of the isotones 19,26). The slopes of the isotone lines representing B~s as a function of atomic number decrease gradually as the isotone number increases. Immediately after a major neutron shell is filled, there is an in- crease in spacing between the isotone lines.

We first determined the slopes Sexy and the X-intercepts Iex p (the non-integral values of Z at B2N = 0) of the isotone lines of all the experimental B2N values, by least squares analysis, each mass excess being weighted by its experimental uncer- tainty. We then determined the corresponding slopes S~,~e and X-intercepts Ie,~o of

f r f r l l f f l f l r J r r r I J ~ l l l J r f t l i l r r l l l r l r j r r l f j l l l z l l r r l i l l l , j l ~ r r i l l

1 .4 - _ j /'-Kummel

1.2 ',',,:, .," :,., / ' i ,

l.O ~v'-.' , ,

/ A:: I j ;

0 . 6 - (19681

u 1.4-

1.2

• I . . . . . . . ! 1 1 \ , - 4 ~ _W" v ",'- ',.-V ~v~.~.. ~ ' i l i ! ' , ; ' ; v i~ l^~ l~"~ . ,Y f 'W,~ v|/IA.I

0 6J- Zeldes_/ ~6orvey _~ " | J l l l l l q l , l l , l l ~ t l l l l l l l l l i l l l i i l l l t l l l i i i l l l l l l l l t l l l t l l l l l l t l i l l J

30 40 ,50 60 70 80 90 I00 I10 120 1 :50 140 150 Neutron Number

Fig. 8. Ratios of the slopes of isotone lines representing neutron-pair separation energies as a func- tion of atomic number, for various isotones.

376 i . WiNG

the isotone lines of B2~ which were constructed from a mass formula, using only the nucleides comprising the experimental B2N. The method of least squares analysis was again used, with a weight factor of 1.0 for each mass excess.

In figs. 8 and 9 we plotted the ratios Scalc(N)/S=,,p(N) and the differences [Ic,lo (N)-Ie,p(N)] versus the neutron number, respectively. In general, the four mass equations exhibit structure and magnitude of deviations fairly similar to each other as well as to those compared in ref. 3) (except Hillman's). The sharp dips at neutron

~ - 4

H-8 I

o

~ 4 H

- 4

- 8

= l ' ' a ' l ; J ' ' l J J a i l i J i l l i , ~ J l , J i i - l l , , , l a i , , l l , , J l , , ~ i l l , J , l , J J = l , i

. I

" ~ ^ A ,i , / l , I

" " ~ t i A ~ F' t-' ;,/ v tS ',,,f kvt ' ;, L i,,,, , <"""

L--Seeger (1968) 7"v,] "

• i

. I i

: Z.,,eI::; " "Oorw, 30 40 50 60 7'0 80 90 I00 I10 120 1 :50 140 150

Neutron Number

Fig. 9. Differences of the intercepts of isotone lines representing neutron-pair separation energies as a function of atomic nttmber, for various isotones. The vertical scale is in units of nuclear charge.

numbers 41, 50 and 82, and the sharp peaks at 126 neutrons are clearly indicative of the shell effects insufficiently accounted for in these mass formulae. In addition, there are outstanding features at neutron numbers 55, 91 and 101 to 103, and 153. Seeger's equation has additional outstanding discrepancies at neutron numbers 59, and, like Kummel's equation, 128 to 140.

For a given mass equation, the curve in fig. 8 has a structure quite similar to that in fig. 9; the former fluctuates around the value of one and the latter around zero. As concluded in ref. 3), the mass equations under consideration describe the neutron- pair separation energies of the most stable isotones more correctly than the other isotones. Kummel's and Seeger's equations, as Myers', Cameron's, and Wing's, predict rather smaller slopes and intercepts than the empirical values for most of the nucleides with neutron numbers greater than 98.

NUCLEIDIC MASS FORMULAE 377"

5. Roughness parameters

The nuclear mass surfaces may be represented by the following isobaric parabolas

M(Z, A) = M(Za, A)"I'½BA(ZA--Z)2"I'PA, (2)

where B a, Za , and P,t are the average steepness, minimum, and separation distance of the two parabolas of a given mass number A, and M(Z, A) is the mass of the nucleides of a given A and Z. Then, the total beta decay energy will be

Ep(Z, A) = Ba(Z~-Z-O.5)+Pa(Z, A)--PA(Z+ 1, A). (3~

It may be shown that, from eqs. (1), (2) and (3),

R(Z, N) = (Ba+IZ4+x+B,~_xZA_ ~-2 BaZa)

--Z(Ba+I+BA_I--2BA)--O.5(BA+x--BA_I)+(PA+I+Pa_I--2Pa), (4)

where

Pa+l = Pa+I(Z, N+I ) -PA+I(Z+I , N),

Pa-1 = PA-x(Z-1 , N ) - P A - I ( Z , N - 1 ) ,

2Pa = P.4(Z- 1, N+ 1 ) - P a ( Z + 1, N - 1).

(5)

(6)

(7)

It is clear from eq. (4) that the value of R is small compared with the masses of the six nucleides involved, and that, if B a and Z a do not differ greatly from Ba± 1 and Za± 1, respectively, then R(Z, N) is a good measure of the residual pairing interac- tion of the three pairs of nucleides indicated in eqs. (5) to (7). One obtains the same conclusion on the basis of the single-particle model 27).

The experimental values Rex p of roughness parameters have been examined in ref. a). Figs. 10 and 11 show the frequency distributions of the values of Rexp and of the roughness parameters Rcalo obtained with the formula masses. Hillman's equation displays a broader distribution and the others (Garvey's formula assumes zero value of R and is not shown here) exhibit much narrower distributions than the experimental one.

We calculated the root mean square deviation a of the Ro,t¢ values from the Rex ~ values by the following formula

0 2 = ~ • 8 " 1 , (8) n - k

where 51 is the square root of the sum of the squares of the experimental errors as- sociated with the six nucleidic masses in the calculation of (R,xp)j, n is the total number of Rex p values and k is the number of adjustable parameters in the mass

378 J. WING

equation. Table 1 list the values of a, n, and k of the present four mass formulas and the five previously compared 3). The last entry, labelled "experimental", is the root mean square deviation of the R=x v values from zero, and is equivalent to Garvey's

1.0

~ r

3.0

Roughness Parameter (MeV) .2;o :0;3

5.0

2.0

0 0.5 1.0 1.5 2.0 3.0 I i i i i I [ i J I I i i I [ I i i ~ i J i [ [ i i ~ = ' ~ " -

Hillman

1964 Moss Table

2.0

1.0

Fig. I0. Frequency distributions of roughness parameters for the empirical masses and Hillman's equation,

300

200

I 00

o "

400 14.

3O0

2O0

Roughness Porometer (MeV) -0 .4 -0.2 0 0.2 0.4 -0.1 0 0.1 0.2

i I = i i I ~ t i I I i i I i i i I ' I I i I ~ I I i . J

Kummel

SeegerL~l~; (1968) !

i I i I ' T i ~ ~ ~ t I , I ' i i I i = I i l ' i

Zeldes Myers

-0.1 0 0.1 -0.2 0 i I i l i I l l i l l I L i i l : Seeger (1965)

I , i i J I i i i I I I I I

#ing Cameron

I00 i |

_ . I I . . . . j , . j

Fig. 11. Frequency distributions of roughness parameters for various formulas.

L

sen~ompiric~ nucleidic mass

NUCLEIDICMASSPORMULAE 379

tr value. The numbers of adjustable parameters in the mass equations of Hillman, Cameron, and Zeldes are actually less than those indicated in table 1, as their constants in one mass region may not be applied to other regions. On the other hand, the non- adjustable coefficients in Seeger's and Garvey's formulas run into several hundreds. The latter assumes zero value of roughness parameter for all nucleides, whereas most Rex p values are not zero. One surmises that all the ~ values in table 1 are perhaps comparable to one another. Thus, the comparison of the root mean square deviations of roughness parameters is not as useful as the other methods of comparison in this work, and it is included here only for reference.

TABLE 1

The values of n, k, and a

Mass equation No. of input No. of adjustable Root mean square data n parameters k deviation a

Hillman 695 479 26.04

Cameron 695 408 10.01

Wing 695 34 6.49

Seeger (1965) 689 18 6.64

Myers 695 8 6.42

Kummel 688 43 7.68

Seeger 675 10 5.66

Zeldes 693 182 6.34

Garvey 675 0 6.25

Experimental 695 0 6.30

6. Isobaric analysis of beta-decay energies

We determined the values of B~, Za , and PA defined in sect. 5 from the total beta decay energies, using Dewdney's method of least squares analysis 28). In figs. 12 to 17 are plotted the ratios of the formula B a to empirical B a values, the differences between the formula Za and empirical Za values, and the differences between the formula Pa and empirical Pa values, all for the corresponding mass numbers. In general, except in Seeger's Pa difference plots, the four curves in each of these figures exhibit structures similar to each other and comparable to those in the corresponding plots in ref. 3), although they differ in the magnitude of deviations. Note that the quantity B a is an average steepness of the two isobaric parabolas of a given mass number. A B a ratio of unity in figs. 12 and 13 does not necessarily indicate complete agreement between the mass equation and the empirical datum.

In figs. 12 and 13, the peaks and dips around the mass numbers 90, 120, 140, and 208 are likely associated with the shell effects which have not been adequately treated

380 J. V~IHG

1.8

1.6

1.4

1.2

• "~ 1.0 o

"~ o.8 e n

_~ o.6

1.2

1.0

0 .8

0 .6

50 60 70 80 90 I00 I10 120 1:30 140 150 160 Moss Number

Fig. 12. Ra t ios o f the fo rmula B A to exper imenta l B A values, versus mass n u m b e r f r om 44 to 160.

1.4

1.2

1.0

0.8

~ 0 . 6

o "6 1.4

m 1.2

1.0

0.8

0.6

- Seeger~f~ / - K u m m e l - / / / ; 2

i t l.J

r - G a r v e y ^ = 1

160 170 180 190 200 2 I0 220 230 240 250 Mass Number

Fig. 13. Ra t io s o f the fo rmula B A to exper imenta l B A values, versus mass n u m b e r f rom 150 to 254.

NUCLETDTC MASS FORMULA~ 381

0 . 4 ~ - -

0.2 Ii I ~ :

~ -0.2

I~ -0"4 , (1968) ~ _-

N 0 . 2 X i Zeldes - i

0

-0.2 i

-o.4 ,50 60 70 80 90 100 I I 0 120 130 140 150 160

Moss Number

Fig. 14. Dcviatious of the formula Z A from experiment Z A values, versus mass number from 44 to 160. The vertical scale is in units of nuclear charge.

I I I I I I I I I i i i i I I I i I I i I I i ~ i l l l i i i I t I l I [ I I l l l l i I l i i l I t t 1

0.417, Seeoe r? I I ~ . ~ ( ,9e~)1 i

Vl' li i i . , j ~ . . . . !AA, II I~ ! ~ ~ f l \ l I ~' ' ~ I ~

T Ii

" /

-o .2F V 1 / G a r v e y ~ ~ ~/ i

L Zeldes~i " IO~ 4 / [ i , , , l i , , , l , i ~ i l i l i l l l t l i J I l l i l l i [ , l , , i ] I I I I ] I I I I ] I I

160 170 180 190 200 210 220 230 240 250 Moss Number

Fig. 15. Deviations of the formula Z4 from experimental Z 4 values, versus mass numtmr from 150 to 254. The vertical scale is in units of nuclear charge.

382 ~. W,NO

1.2

0.8

0.4

=~ o

~. -0 .4

a -0.8

3 o.8

I l J l l J I I l i l l J Z l J l l l l I l J l l l [ J l ~ l l J l l l l l i t I I ~ Z l l l J l I l l | I l l

I

- - I | .~ 1

- LKumme l

• _ / - Z e l d e s , ~o~- /Oo~ve, ~ ~ ~ ' ~ ' L " - , ,

-o.,K,, .... , .... ,,:,,,,,,,':~,v,,,,: .... , , v V -~ I I F ~ l I I I I I I I I I I I l l I I

50 60 70 80 90 I00 I10 120 I~0 140 150 M o s s N u m b e r

Fig. 16. Deviations of the formula P~ from experimental P~ values, versus mass number from 44 to 160.

1.2

0.8

0.4

0

"~ - 0 . 4

oE -o .8 I

o 0.8

0.4

0

- 0 . 4

Kummel . t

Af,~A/~,ir,~j,AAA ,,,,,, ,,, , t l l l l I l l 1 ( ~ | l I ~ | ~ .

,v, vvv,vvt ~- S e e g e r (1968) - -

I Gorvey7 A " Zeldes '; /I

t / ~'l - ~l | I tv l~ I v 1 / I - - v I I _

, ~, . ,y , v v "v ";~W,T,~,'~ 'VVt~, I ,'~l/^Vf..,,,~l , ' ,VV~,. ,~d; l , , i V ' , j v v 'L

~ a l l l i l l l , t r J i l i i l l l r l J n l l e J r l l l i r l n l l l n t n l l n l l I n l

160 170 180 190 200 210 220 250 ;>40 250 Moss Number

Fig. 17. Deviations of the formula Pa from experimental PA values, versus mass number from 150 to 254.

NUCLEIDIC MASS FORMULAE 383

in the mass formulas. Many other outstanding systematic deviations common to all the mass formulas appear in various mass regions. Seeger's peak at mass 192 is related to the large deviations of alpha decay energies for the Au, Hg, and TI isotopes (fig. 3), and Kummel's peak at mass 50 is related to the outstanding deviations for Ti isotopes (fig. 4). These two peaks of BA ratios at masses 50 and 192 were also observed in the five mass equations previously compared 3).

In the plots of the Z,t differences, we see several common peaks and dips which are associated with the shell effects as in the B a ratio plots. The two sharp peaks at masses 102 and 200, also observed in ref. 3), are probably not related to shell effects. These two peaks appear again in the Pa differences (figs. 16 and 17, but not in Seeger's Pa at mass 200), and less pronouncedly in the BA ratios (figs. 12 and 13). Note the relatively large deviations of Z,t values in the mass region of 172 to 180, a region of deformed nuclei. Seeger's large deviations extend to mass 190 (fig. 15). Both Kummel's and Seeger's equations contain sophisticated descriptions of nuclear deformation. This comparison suggests the need for further improvements in their treatment of nuclear deformation.

It is interesting to observe that, of all the mass equations compared here and in ref. 3), only Seeger's equations shows a distinct feature in the PA difference plots. The rest display similar structures to each other. This may not be surprising because many of the large systematic differences among the mass equations are removed in the calculation of the pairing interaction P,t. Seeger's equation predicts, for all the even mass nucleides, much larger PA values than the empirical values in the region below mass 104, and much small PA values above mass 104. The saw-tooth structure of the PA difference curves indicates that we have not yet understood the nucleon pairing effects completely.

7. Delayed-nucleon emission

We calculated the energies T n and Tp available for the emission of delayed neutron and delayed proton from the precursors, respectively, by the following expressions

Tn(Z, A) = Ep(Z, A)-Bn(Z + 1, A), (9)

Tp(Z, A) = - Ep(Z- 1, A ) - Bp(Z- 1, A ) - Epo(Z- 1), (10)

where B n and Bp denote binding energies of the last neutron and proton, respectively, and Epo is the kinetic energy for which the mean lifetime of a proton in the nucleus is 10-1, sec. We used the Epo values given in ref. 29), which were computed from barrier penetrabilities of protons. The absolute values of T, and Tp are trivial in the present discussion, but their signs should be positive for the emission of delayed nucleons to be energetically possible. Table 2 lists the calculated negative values of T, and Tp for the reported precursors of delayed neutron 30) and delayed proton 31, 32) emis- sion. Positive values are omitted.

384 J. w r N o

Kummel's, Zeldes', and Seeger's (1965) formulas yield positive T. and Tp values for all these precursors. Hillman's and Wing's mass equations which have the relatively simple structure of isobaric parabolas 3), give several negative T, and Tp values. Five mass formulas yield negative Tn values for 210T1; four of these formulas give negative T, only for this precursor. When more precursors of delayed-nucleon emission are found, we can have additional interesting comparison among the mass formulas.

TABLE 2

The negative values of Tn and Tp, in MeV, for the reported precursors of delayed-nucleon emission

Precursor HiUman Cameron Wing Myers Seeger Garvey

SSAs S7Br SSBr a~Br 90Br eaKr 9aKr e4Kr *2Rb 9SRb 18sSb xa7 I

laa I 139 I

eX0T 1

XeeTe X0aTe

roTe

T~

Tp

--0.92

--1.86

- - 1 . 8 9

--0.07

. 0 . 8 6

--0.81 --0.20

--0.46

--0.06 --0.56 --2.21 --0.40

--0.17

8. Conclusion

We have examined several methods of comparison of nucleidic mass equations. Each of these comparative methods deals with a particular mass relationship of the nucleides not far from beta stability. The present comparison has revealed many large systematic and random deviations of four recently developed mass formulas from the empirical mass regularities. Many of the systematic deviations may be a result of inadequate treatment of the shell effects, nuclear deformation, or nucleon pairing interactions in these formulas. It is interesting to note the similarities and differences among these semi-empirical mass formulas which are based on various nuclear models. Whereas the number of adjustable parameters in any one mass equation has little to do with the goodness of fit with the empirical mass data, the mathematical structure in the equation is most relevant.

A nucleidic mass equation that is intended for extrapolation to the region far away from beta stability must at least be able to reproduce all the known mass systematics

NUCLEIDIC MASS FORMULAE 3 8 5

of the nucleides. None of the many semi-empirical formulas which the author has examined hitherto satisfies completely this basic requirement. For the purpose of estimating the unknown nucleidic masses, the author still prefers local extrapolations from the empirically observed mass systematics. A set of extrapolated masses con- sistent with many empirical regularities including those discussed here is under development.

The author is most grateful to H. Kummel, P. A. Seeger, N. Zeldes, and G. T. Garvey for receiving their tabulations of mass data prior to publication and for helpful discussions.

Appendix

The following is a list of corrections to be made in ref. 1 o).

(a) Page 132, table 1, in the expression for B(N, Z), the subscripts for 6 should be No and Zo, not NO and ZO. The 2nd and 6th headings should have small case of s in the subscripts instead of capital S.

(b) Page 137, in eq. (3.2), insert +½b~ after ~ and replace the comma at the end of this formula by a semicolon. The two lines below eq. (3.2) should read "assuming n~ >> 1, this can be written". In the first part of eq. (3.4) add +½c'~Np after ½b'~N i.

(c) Page 139, eq. (3.9), the last term within the brackets should be ¢o~zk£ k. The 7th line f rom bot tom should read " . . . just ni~izk£k ( n u m b e r . . . "

(d) Page 142, eq. (3.21), the last term within the brackets should be -~Wz2/A. (e) Page 144, the symbol u c in front of the last term in the first line of eq. (3.37)

should be deleted.

(f) Page 145, table 2a, the value for ~184 should be - 6 . 3 6 x 10 -1. The symbols (743, Cs4, U~ w and Uc ew should be replaced by c43, cs4, u~ w, and ~c w, respectively.

(g) Page 147, table 2b, first column, all the symbols 0 should be capitalized and the third entry from last should read ~23oo Add footnote at bot tom of the table: " F o r ~" 1 2 6 , k"

formulae I to V, inclusively, r/~o = r/5o . n ,,

References

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Press, 1967) 4) M. Hillman, Proc. 2nd Int. Conf. on nuclidic masses, Vienna, 1963 (Springer, Vienna, 1964);

U.S. Atomic Energy Commission Report BNL-846 (1964) unpublished 5) A. G. W. Cameron and R. M. Elkin, Can. J. Phys. 43 (1965) 1288 6) P. A. Seeger, U.S. Atomic Energy Commission Report LA-3380-MS (1965) unpublished 7) W. D. Myers and W. J. Swiatecki, Nucl. Phys. 81 (1966) 1 8) J. H. E. Mattauch, W. Thiele and A. H. Wapstra, Nucl. Phys. 67 (1965) I 9) A. H. Wapstra, Z. Naturforsch. 21a (1966) 68

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386 J. wlr~G

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A90 (1967) 23