438

J. AppL Cryst. (1993). 26, 438-447

The Neutron Transmission of Single-Crystal Sapphire Filters

BY D. F. R. MILDNER, M. ARIF AND C. A. STONE* National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

AND R. K. CRAWFORD

Intense Pulsed Neutron Source, Argonne National Laboratory, Argonne, Illinois 60439, USA

(Received 29 July 1992; accepted 12 January 1993)

Abstract The wavelength dependence of the transmission probability of a beam of neutrons through super optical quality single-crystal sapphire at room tem- perature has been measured. The measurements extend over the neutron wavelength range 0.05- 1.2 nm. Sharp dips in the transmission caused by Bragg reflection are not found. The data below the dip around 0.2 nm in the cross section have been fitted to a function that accounts for multiphonon scattering. Measurements performed on single crystals with increased lattice distortion (or mosaic spread) show an increase in the cross section at long wavelengths. The attenuation of room-temperature A1203 is not significantly different from that for liquid-nitrogen-cooled MgO.

Introduction Large perfect single crystals are often used as filters to produce a thermal-neutron beam relatively free of fast-neutron background (Brockhouse, 1959). In general, a useful neutron filter material must have wavelength-dependent cross sections such that the total cross section is low at the thermal energies of interest but large at epithermal and higher energies. The efficiency of the filter is dependent on the magnitudes of various cross sections: (1) the absorption cross section O'abs, which is usually linearly dependent on the neutron wavelength and always independent of temperature; (2) the coherent Bragg- scattering cross section o-e~, which is dependent on the neutron wavelength, temperature and crystal orienta- tion and perfection (this can be reduced by suitable orientations and by using highly perfect crystals); (3) the incoherent elastic cross section crin c, which is usually small and independent of wavelength; and (4) the inelastic or phonon-scattering cross section

* Present address: Department of Chemistry, San Jose State University, 1 Washington Square, San Jose, California 95192, USA.

O'inei, which also varies with neutron wavelength and is dependent on the crystal temperature (the value of O'inel can be reduced significantly by cooling to low temperatures because the phonon population is thereby reduced). For a material to be an effective filter, the sum of these cross sections at thermal energies must be small compared to the total cross section O-to t for neutrons at the higher energies, which need to be filtered out of the beam.

Sapphire (A1203) is an effective fast-neutron filter because its transmission for neutrons of wavelengths less than 0.04 nm (500 meV) is less than 3% for a 100 mm thickness. It is also an effective filter of thermal neutrons with wavelengths less than about 0.1 nm, since there is a great density of high-order reflections available to scatter the incident beam. Nieman, Tennant & Dolling (1980) have measured the wavelength dependence of the total neutron cross section for single-crystal sapphire. They have shown that high-quality single-crystal sapphire at room temperature is a significantly more efficient thermal- neutron filter, with higher transmissions for wave- lengths longer than 0.1 nm, than either single-crystal silicon or quartz (SIO2), even when the latter materials are cooled to liquid-nitrogen temperature (77 K). They also conclude that cooling the A1203 filter to 77 K to increase the transmission is scarcely justified by the complication and expense. For example, Tennant (1988) states that the transmission of 0.12 nm neutrons through 101.6mm sapphire increases by only 18% upon cooling from 300 to 77 K. Cooling the sapphire filter has only a small effect on its neutron transmission because of its high Debye temperature (0o = 1040 K).

Freund (1983) has reviewed a variety of single- crystal filter materials and calculated the total neutron cross sections from transmission measurements. He uses a simple model to determine the single-phonon, multiple-phonon and absorption cross sections as functions of the neutron wavelength and fits the data to a general formula with two adjustable parameters. There is good agreement between the calculated

0021-8898/93/030438-10506.00 © 1993 International Union of Crystallography

D. F. R. MILDNER, M. ARIF, C. A. STONE AND R. K. C R A W F O R D 439

neutron cross sections and the experimentally determined cross sections for several materials, e.g. single-crystal silicon at both 300 and 77 K. There are, however, substantial differences between his results for sapphire and the experimental results of Nieman et al. (1980), especially at neutron energies greater than about 100 meV. Freund suggests that improvements in the quality of sapphire single crystals will reduce these differences and thereby decrease the problems associated with elastic scattering of the higher-energy thermal neutrons.

Born, Hohlwein, Schneider & Kakurai (1987) have examined the intensity transmitted by a single crystal of A1203 over a range of orientation angles within _+ 2 :~ and have observed no significant changes. They have found that a 90 mm-long sapphire-crystal filter has a transmission of about 0.8 for wavelengths in the range 0.12-0.24 nm and 0.07 for epithermal neutrons. These transmission results are in good agreement with those obtained by Nieman et al. (1980). However, unlike Nieman et al., who suggest that the effective attenuation coefficient can be minimized by fine-tuning the crystal orientation, Born et al. have found that there is no need to tune the crystals for every wavelength. They find that the beam intensity transmitted by their crystals does not depend on the crystal orientation. Their 7-ray diffraction measurements indicate the crystalline perfection and homogeneity of the high-quality sapphire crystals that are now available. The mosaic spreads (5-15") are very much lower than typical angular divergences for thermal-neutron beam tubes. Hence, it is unnecessary to fine-tune the filter crystal orientation to eliminate certain Bragg reflections in order to increase transmission.

The results of Born et al. (1987) suggest that commercially available single-crystal sapphire has a sufficiently high degree of crystal perfection for its quality to be of no great concern in neutron-filter applications. Their measurements do not show how the total cross section varies as a function of the neutron wavelength. This wavelength dependence is important for comparison of the performance of sapphire relative to other neutron filter materials over a specific range of wavelengths. We report here a measurement of the wavelength dependence of the total neutron cross section for high-quality single- crystal sapphire. We also report measurements of neutron transmission as a function of the grade of crystal, in particular of the degree of lattice distortion that affects the crystal mosaic.

Table 1. Grades o f single-crystal sapphire

Scatter of light (number of voids)

1 2 3 4

Lattice distortion (crystal mosaic) A B C D

Hemex Hemlux Hemlite Hemlux Hemlite Hemcor Hemlite Hemcor Hemcor

Hemcor

Recycle

studied crystals obtained from Crystal Systems Inc.* of Salem, Massachusetts, grown using the heat- exchange method (HEM). The A1 premium 'hemex' grade is the highest available grade of sapphire and is almost free of lattice distortion. We have made measurements of the transmission probability of neutrons as a function of wavelength for three pieces of this high-grade single-crystal sapphire filter at ambient temperature. The size of each of the crystals is 25.4 x 25.4 x 27.0mm, with the c axis parallel to the 27.0 mm edge. The faces perpendicular to the c axis are optically polished.

Since the premium-grade sapphire is only available in small sizes, it is necessary to place a large number of small pieces together in order to make a large filter. This introduces a broadening of the effective crystal-lattice mosaic of the filter as a whole. It is reasonable to question whether a lower grade of crystal with a larger intrinsic lattice distortion, which is available in larger sizes, is adequate as a neutron filter. The grading of single-crystal sapphire is performed according to two specifications. The first is the amount of lattice distortion, or crystal mosaic, which is measured by crossed polarizers using monochromatic light. The second is the amount of light scatter, which is a measure of the number of voids in the crystal. There are four gradings of each specification (see Table 1). Because the wavelengths of neutrons transmitted by the filter are so much shorter (<1 nm) than those of visible light (> 100 nm), the scatter of neutrons occurs at very small angles. Unless these voids are extremely small, in which case they give rise to lattice distortion, little difference would be expected between filters as a function of the number of voids. On the other hand, when the crystal mosaic increases, differences in the transmission properties might be expected, though this depends on the collimation of the primary neutron beam.

We have also made neutron transmission measure- ments through two crystals of 42.8 mm-diameter sapphire rod, 53.0 mm long, cored within 2 ~ of the

Filter material A sapphire filter is usually composed of a number of super optical quality crystals with the [001] axis parallel to the incoming neutron beam. We have

* The filter materials identified in this paper do not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that these materials are necessarily the best available for the purpose. The grades hemex, hemlux, hemlite and hemcor are trade names of Crystal Systems Inc.

440 NEUTRON TRANSMISSION OF SINGLE-CRYSTAL SAPPHIRE FILTERS

[001] direction. The crystals are of grades B1 (superior grade 'hemlux') and C1 (standard optical grade 'hemlite'), with increasing lattice distortion but no increased light scatter relative to the A1 hemex grade. This enables any questions regarding the number of voids and any small-angle scattering of neutrons to be eliminated. It makes possible a comparison of the transmission properties of the various grades of crystal as a function of lattice distortion, independent of the second specification. The crystals are not polished because neutrons are only sensitive to the form of the surface when they are incident at a glancing angle.

Neutron measurements

The neutron transmission measurements were ob- tained on the C3 beam line, which views a 2 0 K liquid-hydrogen moderator at the Intense Pulsed Neutron Source (IPNS) at Argonne National Laboratory. The source operates at 30 Hz with 14 laA proton current at 450 MeV on the enriched-uranium target, which produces a delayed neutron fraction of 0.0283 (Epperson, Carpenter, Thiyagarajan & Heuser, 1990). This results in a time-independent background that degrades the quality of the results at the longest wavelengths, for which there are few neutrons in the pulsed beam. The spectrum peaks around 0.25 nm, indicating an effective moderator temperature of 60 K [2ma x = (2/5)1/22T ] . There was no filter in the neutron beam and collimation was provided by crossed converging SoUer collimators (Crawford, Epperson & Thiyagarajan, 1988), giving a collimation of 3.38 mrad.

The sapphire crystals were placed in the sample position of the small-angle neutron diffractometer (SAND). The measurements were recorded as a function of time of flight on two flat ( 'pancake') detectors, low-efficiency BF 3 ionization chambers, placed before the collimation and after the sample at distances of 5.9 and 9.0m, respectively, from the liquid-hydrogen moderator. An additional He 3

detector was placed immediately after the second pancake detector. A schematic diagram of the experimental arrangement is shown in Fig. 1. The first detector before the sample acts as a beam monitor against which all measurements are normalized. Differences in intensity in the beam monitor are caused by time-averaged variations (< 1% about the mean) in proton-beam current, but not by variation in moderator characteristics, which would influence the thermal peak. Hence, differences in neutron spectra are negligible and differences in beam currents delivered in the various runs can be ignored.

Each data set was taken for 3 h, though measurements with no sample were taken for twice this duration and have been normalized to the same running time. The data from all three detectors were recorded with channel widths At = 10 Its and the data in each successive set of nine channels have been summed to reduce the statistical variation, giving a channel width of At = 90 gs, which corresponds to A2 = 0.004 nm for the downstream detectors. Large statistical fluctuations in the data make it difficult to discern trends at long wavelengths. Because the cross section is slowly varying in this region, the data have been further summed, so that A2 = 0.008 nm for 0 . 4 < 2 < 0 . 8 n m and A 2 = 0 . 0 1 6 n m for 0 . 8 < 2 < 1.2nm. This does not spoil the resolution significantly.

The data in the two pancake detectors give a crude wavelength scale. However, in practice, a more detailed calibration is performed using the Bragg dips in the transmitted beam corresponding to the largest plane spacings of polycrystalline aluminium caused by various aluminium windows within the beam. These dips are attributed to A1 (111), (200) and (220) diffraction, corresponding to 2 = 0.468, 0.405 and 0.286 nm, respectively. No other aluminium dips are observed. In theory, there are also dips in the time-of-flight transmission spectra corresponding to Bragg diffraction from A120 3, but these are not pronounced. In addition, there are two other calibration points corresponding to diffraction from

5.9m ~----~ 3.1 m -.91---305 mm ~IP'-

filter sample moderator BF detector BF detector He 3 detector source 3 3 ~ ,,

Fig. 1. A schematic diagram of the experimental arrangement for the neutron-filter transmission measurements. Each BF3 detector is masked with B4C 'crispy mix' with an opening of 15 mm diameter. In addition, a 12.7 mm diameter 10 atm (1.01 MPa) He 3 detector, masked with cadmium with a 3.2 mm diameter pinhole centered on the beam, is placed behind the second detector.

D. F. R. M I L D N E R , M. ARIF, C. A. STONE AND R. K. C R A W F O R D 441

pyrolytic graphite crystals at specific Bragg angles for which measurements have been performed with the same experimental arrangement.

The data in the two downstream detectors have been normalized for each data set by the integral of the data in the upstream detector over the thermal range (0.15-1.2 nm). We use the data taken in the He 3 detector for determining the transmission T(2), attenuation Z'(,;t) and total cross section a(2) of the filter as functions of wavelength 2. If the neutron count rates for a given 2 with the filter of thickness t in and out of the beam are 1,(2) and Io(2), respectively, then

I,().) = Io(2)T(2 ) = lo(2 ) exp [ - - N a ( 2 ) t ] , (1)

where N ( = 2.35 x 1019 mm -3) is the number density of AI20 3 units. The transmission T(2) for a given run is obtained from the ratio of the counts with and without the sample, modified by background subtrac- tion for each case, that is

T().) = [ I , ( 2 ) - B].~amplJ[l , (2)- BJempty, (2)

from which the microscopic cross section a(2) is obtained. (Because this is a ratio measurement, differences in detector efficiency as a function of 2 can be ignored.) This measurement has been performed for (a) single hemex blocks; (b) multiple hemex blocks; and (c) both the hemlux and hemlite rods•

The greatest difficulty has been in estimating the appropriate value of the background B for each measurement because the data in the downstream detector never reach a constant background. The value of B greatly affects the statistical fluctuation, as well as the values, of the results at the longest wavelengths where the data are not significantly different from the background. Hence, we need a careful approach to the estimation of the background (see Appendix A).

Earlier spectra taken as a function of time of flight on the C1 beam line, which also has converging collimation but over a wider angle (~ 8.7 mrad), show for the downstream detectors evidence of many sharp dips at specific wavelengths. Some of these reflections correspond to the Bragg condition for single-crystal sapphire. However, they are not as sharp as those found for the MgO filter studied previously (Carpenter, Mildner, Cudrnak & Hilleke, 1989) because of the much narrower mosaic of the single-crystal sapphire. The calculation of the positions of these Bragg dips and the dependence of the cross section on wavelength is given in Appen- dix B.

R e s u l t s a n d d i s c u s s i o n

The data in the downstream detectors that cover the wavelength range 2 = 0-1.2 nm appear to be similar for all data sets, although the data suffer from poor

statistics for wavelengths above 0.8 nm. The level of statistical noise masks dips in the transmission data that correspond to scattering at the Bragg condition, even for the poorer-grade material. The transmission as a function of wavelength for different numbers, n, of hemex crystals is shown in Fig. 2. It is given by T(t) = e x p ( - n Z t ) , where t is the thickness of an individual crystal and Z" = Na is the macroscopic cross section or attenuation coefficient.

Values of S().) derived from data sets with n = 1, 2 and 3 are shown in Fig. 3. The data are the same within statistics, which indicates that they are

transmission for multiple hemex blocks lt~--,t_ , , ~ - ~ r ~ ~ - ' " ~ - ' I ' , , I ' ' ' T ~ - ' ~ r = ~

. . . , : , o , o , ° ° ° ° . . ° ° 08 k ~ . . ~ . . . . . . • • a ' ~ eqPQ~ ID - * ' ~ ' * . ~ ' * * *" ÷no ° * o , oO + ° *1

= F "-" " " ~"t. '"..:": "..'" :" "00 = O / °t" '~'d, .~ ". "0" " "'" .- i . .~: .%: . . . "-. • . . . . . . ¶ 0-67.. • ; . . . . . . . . . . -

. - - ~ : # _~ E 2,'."

0 . 4 ~ , n = l - - e-. -~" • n = 2

t _ .,. ;o "" " 3 • n = 3 0.2 ,

0 ~ ~ , I , , , I ~ , , I , , , I i ,~_1 , , , 0 .2 0 . 4 0 .6 0 .8 1.0 1.2

wavelength ~. (nm) Fig. 2. The transmission probabili ty as a function of wavelength

for different numbers n of AI20 3 hemex crystals of thickness 27.0 mm at ambient temperature. The data shown are for n = !, 2 and 3. Channel widths are A2 = 0.004 for ). < 0.4, A2 = 0.008 for 0.4 < 2 < 0.8 and A). = 0.016 for 0.8 < ). < 1.2 nm, to reduce statistical fluctuations at long wavelengths.

attenuation for multiple hemex blocks 0.02[_~,, ! , : , 1-T-T~T~-T-T-r, ~-~-T, , ,

i , , ]

* n = l -~ 0 .015 • n = 2 1

• n = 3 g ~1 . . . . . . a b s o r p t i o n I,

o : L C [_ °

o 0 . 0 1 ~ °° *°*: o 4 . m =

1- *': . 0 : " '° i t- .°0 . L g ° , ." • =

=

0 .2 0 .4 0 .6 0 .8 1.0 1.2 w a v e l e n g t h )~ ( r i m )

Fig. 3. The at tenuat ion as a function of neutron wavelength for premium-grade single-crystal AI20 3 derived from the transmis- sion data for multiple hemex blocks. The data include the three data sets shown in Fig. 2. The line is a fit of the data for n = 3 to a function of the form Z = A , i + C ( I - e x p l - [ ( B / ) . 2)+ (D/)#)]}). Also shown for comparison is the absorption cross section for AlzO3.

442 N E U T R O N TRANSMISSION O F SINGLE-CRYSTAL SAPPHIRE FILTERS

qualitatively consistent and vindicates the method of accounting for the background from the booster target (see Appendix A). The data can be fitted to a function given by Freund (1983), which accounts for multiphonon scattering. This can be further gen- eralized for anharmonic effects, and the data for n = 3 have been fitted to a function of the form

Z" = A2 + C(1 - exp {-[(B/22) + (D/24)]}), (3)

with the following parameters:

A = 5.96 ~tm- ~ nm- 1 (corresponding to o/2 = 2.537 barns nm- 1),

C = 0.0252 m m - ~ (corresponding to o" = 10.72 barns), B = 1.17 x 10 - 3 n m 2,

D = 3.42 x 10 - 6 nm 4.

The inclusion of the anharmonic term significantly improves the fit around the minimum in the attenuation. The relatively low value of C (the free-atom cross section for AlzO3 is 14.04 barns) is probably caused by the very few data points at small 2. Also shown in Fig. 3 is the theoretical absorption cross section, aa = 2.575 barns nm-~, or 0.463 barns at 2200 m s- 1, calculated from the known absorption cross sections for aluminium and oxygen (Sears, 1986).

The transmissions of the hemlux and hemlite rods appear to be identical, with the hemlite having a slightly lower transmission than the hemlux, although the data suffer from poor statistics for 2 > 0.8 nm. Similarly, therefore, the attenuation as a function of 2 appears to be identical (see Fig. 4), with the hemlite cross section slightly higher than for hemlux. In the region of the greatest transmission (2 = 0.2-0.3 rim),

Table 2. Fittin9 parameters to equation (3)

Z = A2 + C(1 -- exp { - [(B/22) + (D/24)]}).

Parameter Units Hemex Hemlux Hemlite

A lam -1 nm -x 5.96 (10) 8.72 (15) 8.61 (15) C ~tm-x 25.2 (3) 25.3 (6) 25.8 (19) B 10 -3 nm 2 1.17(10) 2.34 (14) 2.71 (18) D 10 -6 nm 4 3.42 (43) 5.77 (499) --3.73 (555)

the attenuation of the hemlite is 15% greater than that of the hemlux, though for 2 > 0.4 nm there is very little difference between these two grades of crystal. Over the range of wavelengths measured, these poorer grades of sapphire have a contribution to the attenuation of approximately 0.002mm -1 (o-~ 8.5 barns), independent of wavelength, in addition to the absorption cross section. This is attributed to the lattice distortion in these crystals. We have fitted the data to the function given by (3) and give in Table 2 the fitting parameters for all three grades of sapphire. The cross sections for the poorer grades in the thermal range are much greater than those found for the hemex-grade crystals (with the hemlux data giving A = 8.72 ~tm-~ nm-1, corresponding to o-/2 = 3.71 barns n m - ~, of which 1.14 barns n m - 1 is attributable to single-phonon scattering, and C = 0.0253 m m - t , corresponding to a = 10.76 barns). Note that the har- monic term for the poorer grades is much greater than for the hemex grade and the anharmonic term is negligible. Hence, there remains a compromise between transmission and cost for the lower-quality sapphire crystals.

We show in Fig. 5 our time-of-flight attenuation data for ambient temperature, super optical quality

attenuation for hemlux and hemlite 0.02~c-- ' I ' ' ' I ' ' ' I ' ' ' I ' ' ; ' ' .. 1 , :

• hemlux -

' i l::: 0"O15E- t o hemli te ,-~

E ~ . . . . . . a b s o r p t i o n

" 0.01 _ . ~ 0 g o o .

0.005 ~ _ ' - ~- t~

1.2 wavelength X (nm)

Fig. 4. The attenuation as a function of neutron wavelength for superior-grade and standard-optical-grade single-crystal AIzO 3 derived from the transmission data for hemlux and hemlite cylindrical rods. The line is a fit of the hemlux data to a function of the form 2: = A2 + C(1 - exp {-[(B/22) + (D/).4)]}). Also shown for comparison is the absorption cross section for AIzO 3.

0"02 f '

• Cassels formula _ _~o o Tennant 300K ~× x Tennant 80K

. . . . . . ab so rp t ion

ko * . ~ . . - ' ~

0.1 0.2 0.3 0.4 0.5 0.6 wavelength X (nm)

Fig. 5. A comparison of the time-of-flight transmission data for the hemex grade (from Fig. 3) with that of Tennant (1988) taken with a crystal spectrometer at 300 and at 80 K. Our data have been fitted to an anharmonic form of the function given by Cassels (1950), 2: = A2 + C[1 - (22/2B)(1 - exp {-[(2B/22) + (D/).4)]})]. Also shown for comparison is the absorption cross section for AlzO3.

D. F. R. MILDNER, M. ARIF, C. A. STONE AND R. K. C R A W F O R D 443

Table 3. Fitt in9 parameters to equation (4)

Z" = A2 + C [1 - ( 2 Z / 2 B ) ( l - exp { - [(2B/J?) + (D/,,I.4)]})].

Parameter Units Hemex Hemlux Hemlite

A I,tm -1 nm -1 6.03 (10) 8.75 (16) 8.61 (16) C t, tm-1 28.4 (4) 27.7 (7) 27.6 (8)

2B 10 -3 nm/ 5.78 (21) 5.97 (48) 5.84 (67) D 10 -6 nm 4 13.14(163) 6.14 (260) 2.35 (275)

and single-crystal sapphire fitted to the function given by Cassels (1950), and see also the work of Scharenberg (1972), which gives a more satisfying account of the multiphonon cross section. The form, also with an anharmonic term in the Debye-Waller function, is

S = A2 + C[1 - ( 2 Z / 2 B ) ( 1 - e x p { - - [ ( 2 B / J L 2)

+ (D/24)]})], (4)

with the following parameters:

A = 6.03 I-tm- l nm - (corresponding to a/). = 2.565 barns nm- ~),

C = 0.0284 m m ~ (corresponding to a = 12.09 barns),

2B=5 .78 x 10 -3nm 2,

D = 1.31 x 10 ...5 nm 4.

This function gives a better fit than that of(3) (see Fig. 3). The fit is also closer to the free-atom cross section for AIzO 3 at small 2. Although there is considerable uncertainty in the data at long wavelengths, this function matches the data well in the long-wavelength region, which is a result of both absorption and single-phonon scattering. We give in Table 3 the fitting parameters of (4) for all three grades of sapphire. Here, the values of the harmonic term are comparable for all three grades and the coefficients of the linear term in 2 are the same as for (3).

Also shown in Fig. 5 is a comparison to the data of Tennant (1988) taken at specific wavelengths at both 300 and 80 K. The present data show a slightly lower attenuation than the 300 K data of Tennant, which is perhaps indicative of the degree of perfection of the crystals. The transmission data of Nieman et al. (1980) were probably taken using single-crystal sapphire with a much larger mosaic and lattice distortion than is found in the presently available crystals that are used for filter material. Our results show that the elastic Bragg peaks are drastically reduced for high-quality sapphire. It is clear that there is a small improvement in the transmission for sapphire when the crystals are cooled, and the data approach the theoretical absorption limit at long wavelengths. However, the improvement is small and for many applications this may be insufficient to warrant the expense and inconvenience of cryogenics.

The data for the cross section taken from the transmission measurement for the hemex-grade crystals may be compared with the calculations of Freund (1983). Evidently, the Cassels function [(4)] describes the data as a function of neutron energy (see Fig. 6) better than does the Freund calculation. In the low-energy region, the Freund calculation gives an overestimate of the single-phonon scattering. In the short-wavelength region where the cross section is a result of multiphonon scattering and increases towards the free-atom cross section, the calculation underestimates the experimental data; this was also found for the data of Nieman et al. (1980). We obtain a better fit to the data in the short-wavelength region if anharmonic effects in the Debye-Waller factor are taken into account.

We compare the data for ambient-temperature AI20 3 with the results for single-crystal MgO at 77 K given by Carpenter et al. (1989). Certainly the Bragg dips are not so deep or so pronounced; the scattering length of aluminium is less than that of magnesium (see Table 4). However, the density of possible reflections is much greater. The most important difference is that the MgO filter consists of about 100 aligned blocks and the effective mosaic spread is probably much larger than the mosaic of the sapphire crystals. Hence, on the one hand, the MgO filter removes many more neutrons for a given Bragg reflection than does sapphire. On the other hand, the absorption cross section of aluminium is greater than that of magnesium, so that the transmission window should not be so deep. In addition, the free-atom cross sections are such that the attenuation for fast neutrons is 0.033 m m - 1 for AIEO 3 and 0.038 mm i for MgO.

We show in Fig. 7 a comparison of the attenuation taken from the time-of-flight measurements for A120 3

e -

_- Freun° / / ! 0 = ~ calculation , ' f i r / i

i ~ ~ . ' ~ ~Cassels ! ~" i "-" ~ formula i

~ F

= o 1 i5.001 0.01 0.1 1

energy (eV) Fig. 6. The measured neutron cross section for room-temperature

hemex-grade sapphire as a function of neutron energy, together with the Cassels function [(4)], compared to the calculations of Freund (1983).

444 N E U T R O N TRANSMISSION OF SINGLE-CRYSTAL SAPPHIRE FILTERS

Element Mean scattering length ( x 10-15 m)

A1 3.449 O 5.803

Mg 5.375 S i 4.149 C 6.646

Table 4. Cross sections for some elements

Bound coherent scattering Incoherent scattering Absorption cross Free-atom cross section (barns) cross section (barns) section* (barns) cross section (barns)

1.495 0.0092 0231 1.398 4.232 0 0.00019 3.749 3.631 0.077 0.063 3.421 2.163 0.015 0.171 2.030 5.551 0 0.0035 4.730

* For 2200 m s-~ neutrons.

at ambient temperature and that taken from measurements for MgO at 77 K, using the reported fits of the data to (3). Evidently, there is not a great difference between the two. In theory, the cooled MgO filter should have a lower attenuation (by a factor of three) than room-temperature AlzO3 for long- wavelength neutrons. However, there clearly remains plenty of inelastic (single-phonon) scattering at these wavelengths for MgO, which can be eliminated by the use of much lower temperatures than 77 K.

Summary and concluding remarks We have measured the wavelength dependence of the transmission probability of a beam of neutrons through single-crystal sapphire at room temperature. The measurements extend over the neutron wave- length range 0.05-1.2 nm. Such a filter is useful for wavelengths greater than about 0.1 nm, with the maximum transmission at about 0.2 nm. We do not find sharp dips in the transmission at specific wavelengths corresponding to Bragg reflections in sapphire. We find that no great improvement in the transmission data is obtained by fine-tuning the orientation of the sapphire filter relative to the neutron beam. We also find that the transmission for

0.01

'E E

0.005

c

~ -'~, I I I I I

A ugo K) , , , I , , , I , , , I , , , I , , , , , ~

0.2 0.4 0.6 0.8 1 1.2 wavelength ~ (nm)

Fig. 7. A comparison of the attenuation as a function of wavelength using the fit of [(3)] to the time-of-flight data taken for AI2Oa at room temperature and MgO at liquid-nitrogen temperature (Carpenter et al., 1989).

the premium grade of sapphire (though not for the superior and standard optical grades) is better than that obtained through the equivalent thickness of crystal studied by Born et al. (1987)

Room-temperature sapphire is approximately as effective a filter as the liquid-nitrogen-temperature MgO filter of the same length. This statement would change if high-grade large MgO crystals were available. A problem inherent with both filters is that oxygen has a fast-neutron window in the total cross section at 2.35 MeV. This means that, if the beam port looks directly at the core with its fission spectrum, there will be some increased background from this source that would be absent for other filters such as single-crystal silicon. This is less of a concern if there is some other additional filter in the beam or if the beam port is tangential. However, we note that the transmission at 2.35 MeV for sapphire is still less than that for silicon of the same length.

We acknowledge the assistance of Mark Felt of Crystals Systems Inc. and discussions with Jack Carpenter of the Intense Pulsed Neutron Source at Argonne National Laboratory. We also acknowledge the assistance of Ernest Epperson on the SAD instrument (Epperson et al., 1993) on the C1 beam line, on which some earlier preliminary measurements have been made. We thank John Copley for his comments on the paper. This work has benefited from the use of the Intense Pulsed Neutron Source at Argonne National Laboratory; this facility is funded by the US Department of Energy, BES-Materials Science, under contract W-31-109-Eng-38.

APPENDIX A Data correction

Much of the background comes from the delayed neutron fraction in the moderated beam from the enriched uranium target of the pulsed-spallation neutron source. These delayed neutrons, which represent a fraction 0.0283 of the total for 450 MeV protons (Epperson et al., 1990), are assumed to be moderated with the same spectral distribution as the prompt pulses. However, they produce a background in a time-of-flight instrument that is independent of

D. F. R. M I L D N E R , M. ARIF, C. A. STONE A N D R. K. C R A W F O R D 445

time but is dependent on the particular sample. These delayed neutrons are important to the correction of the measured data that are most sensitive to the background, especially at long wavelengths.

The data for the downstream detectors cover the wavelength range 2 = 0-1.2 nm (see Fig. 8), but it is evident that the data in the monitor have not approached the true background until at least 2 nm. In order to estimate the background in the downstream detectors at 2 = 1.2 nm, we assume that the ratio of the intensity at the thermal peak, with the background subtracted, to that at a wavelength of 1.2 nm is the same for both the detectors, though the data for the downstream detectors are modified by attenuation through the sample. That is,

[I ~(peak) - B~] exp ( - Napt)EIz(peak) - B] - = [Ii(1.2 n m ) - BL] exp ( - N a l . z t ) × [I2(1.2 n m ) - B] -1 (A1)

The values of 11 and Iz at the thermal peak and at 1.2 nm are obtained from each data set and the value of B~ from the intensity in the first pancake detector at 2nm. The cross sections trp and tr~.z at the wavelength of the thermal peak and at 1.2nm, respectively, can be estimated from the known cross sections for aluminium and oxygen. Because both detectors are similar pancake detectors, we may ignore the detector efficiencies in (A1). Hence, the value of the background B in the downstream pancake detector can be estimated for each data set. This is not strictly valid for the He 3 detector, whose efficiency as a function of 2 is different.

We find that the values of the background B obtained by this method obey an exponential decay function,

B = B o exp (-2"t), (A2)

O 16000 o o o

"o 1 2 0 0 0 E

m 8000 c

0 "o = 4000

o m

c o q o

[ - ~ - 1 ~ - ' ' I ' ' ' I ' ' ' I ' '-¢-Vc~-T~ ~/~ A 220

~ \ aluminium _: - ~aV~ 2 0 0 {

\\'~., ,111

sapphire - ' ~ . _ ~ ~ - - ~ -~ I", ' ,

0.2 0.4 0.6 0.8 1.0 1.2 wavelength ~ (nm)

Fig. 8. Typical data sets taken on the downstream detector as a function of wavelength for different numbers n of AI20 3 hemex crystals of thickness 27.0 mm at ambient temperature. The data shown are for n = 0 (empty), 1, 2 and 3. Also shown are the positions of the highest a luminium and sapphire Bragg dips.

dependent on the thickness t of the sapphire sample. This result is shown in Fig. 9, where the results for the hemex data have been fitted to an exponential with a constant ~, = 0.0358 mm -~, which may be con- sidered an 'average' cross section. This corresponds to # = 15.26 barns, which is close to the free-atom cross section (14.04 barns) for AI20 3. If the time- independent background is caused only by the differential flux q)o(2) of delayed neutrons, the background is given by

B = c i e (2)tpo(2)exp [-L ' (2) t ]d2, (A3) o

where e (2) is the detector efficiency and c is a constant. Differentials with respect to t for both (A2) and (A3) give the average cross section,

~_, = S e (2)rpo(2).S(2) exp [-- Z(2)t] d2 0

x e (2)0o(,5,) exp [-- Z(2)t] d2 (A4)

The value of ~ = 0.0358 mm - t indicates that the weight of the cross section is to emphasize the short wavelengths, and many of the moderated delayed neutrons from the liquid-hydrogen moderator that contribute to the background have a wavelength below 0.05 nm. The fact that the results using (A1) for the estimate of the background obey a simple exponential function is vindication both of the method for estimating the background and of the assumption that the time-independent background originates from the moderated delayed-neutron fraction.

The data in the downstream detectors are also contaminated by frame-overlap neutrons; that is, by

¢'t °' ~ ' 1 ' ' ' I ' ' ' I ' ' ' [ ' ' '

"o ~ s t a e- 7 Is O -I, ! - - , _

~e o 6 - ¢~

~ -

o a ~ 5 - -__

i i lar ,"x I i i 1 1 i J I I ~0 2 4 6 8 10

. t h i c k n e s s ( m m ) Fig. 9. The estimated background for data sets for different numbers

n of AlzO 3 hemex crystals of thickness 27 .0mm at ambient temperature and for hemlux and hemlite crystals of thickness 53.0mm. The results for the hemex data are fitted to B = B o exp ( - 2"t), where Z" = 0.0358 m m - 1.

446 N E U T R O N TRANSMISSION O F SINGLE-CRYSTAL SAPPHIRE FILTERS

longer-wavelength neutrons from the previous moderated pulse. For the 30 Hz source and the distance, from the moderator to the detector, this occurs at a wavelength of 1.465 nm. Frame overlap introduces a systematic error into the measured intensities that is estimated to be of order 0.5% by comparison with the spectra from the ups t r eam detector, for which frame overlap occurs for wavelengths g rea te r than 2.235nm. This has a negligible effect on the measured transmission for intensities well above the background, but introduces a small error of less than 0.3% when the attenuated intensity is only twice that of background, for an attenuation constant of 0.005 mm-1. All data presented have been corrected for frame overlap.

APPENDIX B "' Cross sections

We expect dips in the data corresponding to wave vectors that satisfy the Bragg condition

2k. r = I rhktl 2, (B1)

where "Chk I i s the reciprocal-lattice vector of the crystal corresponding to the crystal plane with Miller indices (hkl). The neutron wavelength 2 is related to the wave vector k by 2 = 2rt/lk[. With the Ewald construction, wave vectors k that are available for Bragg reflection are given by

k = (2rt/2)k o = (27z/2)(~x + fly + ~z), (B2)

where ko is the unit vector in the direction of the neutron beam and ~, fl and y are the direction cosines of k o with respect to the principal axes x, y and z of the A120 3 crystal. Sapphire has rhombohedral symmetry (space group R3m), so a reflection labeled hkl has an associated reciprocal-lattice vector

qkl = (2~/ao)( h + k)x

+ (2~/31/2ao)(h - k)y + (27Z/co)lz , (B3)

where a o and c o are the lengths of the hexagonal unit-cell edges (ao=0.4759 and Co= 1.2991nm) (Newnham & de Haan, 1962). Herice, the possible wavelengths corresponding to the Bragg dips are given by

2 = {(2/ao)[ct(h- k) + fl(h + k)/3 I/2] + (2/Co)7l}/[(4/3ag)(h 2 - hk + k 2) + 12/cg].

(B4)

For each measurement, the crystals are aligned so that the collimated white neutron beam is incident parallel to the c axis, viz in the [0011 direction. Hence, Bragg dips should be expected for reflection planes (hkl) at wavelengths

2 = 2col/[l 2 + (4c2/3a2)(h 2 + hk + k2)], (B5)

Table 5. The planes (hkl) and their corresponding wavelengths for the Bragg re f l ec t ionsexpec ted for A120 3 when a collimated white neutron beam is incident

in the [0011 direction

The atom positions in the hexagonal unit cell are +__ (0 0 z) and + ( 0 0 z + 1/2) for aluminium, where z=0 .352 , and __+(x 1/4), _+(0 x 1/4) and _ ( x x - 1 / 4 ) for oxygen, where x = 0.306. Hence, the symmetry gives reflections only for - h + k + l = 3n, the AI atoms have no contribution to the structure and the O atoms have no contributionwhen h

factor when l is odd = k = 0 and I is odd.

Reflection hkl 2 = 2dsinOhkt(nm ) [Fhkt[ 2 OUZ " Multiplicity

006 0.4331 5314 0.5n 1 104 0.4006 3441 0.288n 3 012 0.3727 149 0.180~ 3 018 0.2811 54 0.381n 3 116 0.2368 17595 0.265~ 3

1,0,10 0.2363 291 0.403~ 3 0,0,12 0.2165 12362 0.5n 1

119 0.2110 7727 0.326n 3 113 0.2009 7727 0.160n 3 208 0.2003 1326 0.288n 3 024 0.1863 7701 0.180n 3

0,2,10 0.1859 149 0.320~ 3 1,1,12 0.1794 8904 0.364n 3 0,1,14 0.1766 3748 0.430g 3

with angles of inclination Ohk i with respect to the incident beam given by

tan Ohkt = (31/2/2)(ao/co)l/(h2 + hk + k2) 1/2. (B6)

We have generated in Table 5 the set of reflection planes (hkl), together with their corresponding wavelengths and multiplicities, for the Bragg dips that should be found in the sapphire transmission data, assuming that the white neutron beam is incident in the [0011 direction. For (hkO) reflections, the angle of inclination Ohk 0 = 0 and these do not contribute to the scattering cross section. We have found those planes in sapphire that have a nonzero structure factor for this experimental arrangement and have identified those having large structure factors with dips that might be found in the transmission data. They are not so pronounced as in the MgO neutron transmission data (Carpenter et al., 1989).

The elastic coherent scattering cross section o-el is given by

~el = (22Nc/2N) ~ (mhk, IFhk,12dhkl), (BT) hk~

where Nc is the number of unit cells per unit volume and N is the number of atoms per unit cell. The sum is taken over all reflections hkl, where ml,kZ is the multiplicity, Fhk I is the structure factor and dhk z is the lattice spacing for the reflection hkl. The structure factor is given by

Fhkl = E b) exp { - B~[sin (Ohkl)/,)o] 2 } J x exp [27ti(hx~ + kyi + lzi)], (B8)

D. F. R. MILDNER, M. ARIF, C. A. STONE AND R. K. CRAWFORD 447

where Ohk I is the Bragg angle for the reflection hAl given by (B6). The sum is taken over all a tom positions j within the unit cell, where x j, yj and zj give the position, bj is the scattering length and Bj is the Debye-Wal ler factor of the j th atom. The Debye- Waller factor is given by Weinstock (1944) as

B = (6hZ/mAkBOD)Cb(T/OD), (B9)

where h is Planck's constant, m A is the cross-section- weighted harmonic mean atomic mass (A = 17.35 for A1203), kB is the Boltzmann constant, T is the absolute temperature and OD is the Debye tempera- ture of the crystal. The function ~(T/Oo) is given by

OD/T

• (T/OD) = 1/4 + (T/OD) 2 ~ ~d~/(e ¢ - 1). (B10) o

Using the static incoherent approximat ion, Cassels (1950) has estimated the short-wavelength elastic cross section which is extinct for perfect single crystals. Hence, the mul t iphonon scattering cross section is given [see also Scharenberg (1972)] as

O'mp h : O'free{1 - (22 /2B)[1 -- exp( -2B/ ) .2 ) ]} , (Bl l )

where 2B is the Debye-Wal le r constant. By expanding this function, Freund (1983) has derived his simplified form

O'mp h = O'free[1 -- exp (-- B/,,;.2)]. (B12)

Note that, for O'mp h << O'free, both forms reduce to the same O'mp h : O'freeB/). 2.

References

BORN, R., HOHLWEIN, D., SCHNEIDER, J. R. & KAKURAI, K. (1987). Nucl. Instrum. Methods, A262, 359-365.

BROCKHOUSE, B. N. (1959). Rev. Sci. Instrum. 30, 136-137. CARPENTER, J. M., MILDNER, D. F. R., CUDRNAK, S. S.

HILLEKE, R. O. (1989). Nucl. Instrum. Methods, A278, 397-401.

CASSELS, J. M. (1950). Prog. Nucl. Phys. l, 185-215. CRAWFORD, R. K., EPPERSON, J. E. & THIYAGARAJAN, P. (1988).

Advanced Neutron Sources, edited by D. K. HEYER, pp. 419-426. Proc. lOth Meeting on the International Collaboration on Advance Neutron Sources, Los Alamos, October 1988. London: Institute of Physics.

EPPERSON, J. E., CARPENTER, J. M., CRAWFORD, R. K., THIYAGARAJAN, P., KLIPPERT, T. E. & WOZNIAK, D. G. (1993). J. Appl. Cryst. Submitted.

EPPERSON, J. E., CARPENTER, J. M., THIYAGARAJAN, P. & HEUSER, B. (1990). Nucl. Instrum. Methods, A289, 30-34.

FREUND, A. W. (1983). Nucl. Instrum. Methods, 213, 495-501. NEWNHAM, R. E. & DE HAAN, Y. M. (1962). Z. Kristallogr.

117, 235-237. NIEMAN, H. F., TENNANT, D. C. & DOLLING, G. (1980). Rev.

Sci. Instrum. 51, 1299-1303. SCHARENBERG, W. (1972). Kristallfilter fiir Kalte und

Thermische Neutronenstrahlen, Report Ji.il-875-RX. Kern- forschungsanlage Jfilich, D-5170 Jfilich, Germany.

SEARS, V. F. (1986). Neutron Scattering, Part A, edited by K. SKOLD t~ D. L. PRICE, pp. 521-550. Methods of Experimental Physics, Vol. 23. New York: Academic Press.

TENNANT, D. C. (1988). Rev. Sci. Instrum. 59, 380-381. WEINSTOCK, R. (1944). Phys. Rev. 65, 1-20.