30 D E T E R M I N A T I O N OF A T O M I C O R D E R I N G A R R A N G E M E N T S

The possibility of systematic displacement of the ordered atoms from their assigned sites, which would modify and perhaps improve the predicted intensity distributions, has not been taken into account here. The related study of Jamieson, Abrahams & Bern- stein (1969), however, suggests that such displacement effects in Nd2(MoO4)3 are small.

It is a pleasure to thank Dr W. D. Robertson of Yale University for use of his optical diffraction equipment and for helpful discussions.

BOWEN, M. 111, 1.

References

G. & GAY, P. (1958). Z. Kristallogr.

FONTAINE, D. DE (1966). In Local Atomic Arrangements Studied by X-ray Diffraction. Ed. J. B. COHEN & J. E. HILUARD. p. 51. New York-London-Paris: Gordon & Breach.

GAY, P. (1956). Miner. Mag. 31, 21. GUINIER, A. (1963). X-ray Diffraction bl Crystals, Imperfect

Crystals and Amorphous Bodies (English trans.) p. 254. San Francisco-London: Freeman.

JAGODZINSKI, H. (1964). In Advanced Methods of Co,stallo- graphy, Ed. G. N. RAMACHANDRAN, p. 181. London- New York: Academic Press.

JAMIESON, P. B. (1967). Nature, Lond. 214, 794. JAMIESON, P. B., ABRAHAMS, S. C. & BERNSTEIN, J. L. (1969).

J. Chem. Phys. 50, 86. MEGAW, H. D. (1960). Proc. Roy. Soc. A259, 59, 159, 184. TAYLOR, C. A. • LIPSON, H. (1964). Optical Transforms.

Ch. 5. Ithaca, New York: Cornell Univ. Press.

J. Appl. Cryst. (1969). 2, 30

The Crystal Structure of Arsenic at 4" 2, 78 and 299 °K

BY D. SCHIFERL* AND C. S. BARRETT

The James Franck Institute, The Unirersity of Chicago, Chicago, Illinois 60637, U.S.A.

(Received 10 June 1968 and in revised form 26 November 1968)

The lattice constants and the atomic position parameter, z, of a high purity, low strain, single crystal of arsenic have been determined. Low extinction reflections of filtered Mo K~ and Ag K~ radiation were used for the determination of z. The Bond precision technique with Mo K~ radiation was used for the determination of the unit-cell dimensions a and c.

Average values for a and c in ,~, and for z are:

4"2°K z=0"22764 a=3"7597 c=10.4412; 78°K z=0"22754 a=3"7595 c=10"4573;

299___3°K z=0"22707 a=3"7598 c= 10-5475.

The estimated standard deviation of z is + 0.00004 at 4-2°K, + 0.00002 at 78 °K and + 0.00005 at 299°K. The precision of a is estimated to be +0-0001 A and that for c to be +0.0002 A. The results for As are compared with those reported earlier for Sb and Bi.

Introduction

The present redetermination of the arsenic crystal struc- ture completes a series of experiments at this laboratory to redetermine with precision the structures of the Group V semi-metals As, Sb, Bi (Barrett, 1960; Cucka & Barrett, 1962; Barrett, Cucka & Haefner, 1963). Bradley (1924) first determined that As was isomor- phous with Sb and Bi, having a rhombohedra l struc- ture with space group R-3m and six atoms per unit hexagonal cell. The lattice constants a and c (hexagonal cell) of As have since been measured with powder methods a number of times, at room temperature and at elevated temperatures (Olshausen, 1925; Jung, 1926; Willott & Evans, 1934; Hagg & Hybinette, 1935; Stohr, 1939; Swanson & Fuyat, 1954). The most recent of

* Graduate student, Department of Physics.

these measurements were made in the temperature range 295°K-677°K by Taylor, Bennett & Heyding (1965). The atomic position parameter z was last meas- ured by Trzebiatowski & Bryjak (1938) with the powder method at room temperature.

Band energy calculations have recently been made for arsenic (Falicov & Golin, 1965; Golin, 1965; Lin & Falicov, 1966), for ant imony (Falicov & Lin, 1966) and for bismuth (Ferreira, 1967; Golin, 1968), and the elec- trical properties as well as the stability of the rhombo- hedral structure of these elements have been considered qualitatively in terms of z and the rhombohedra l angle e. t

Cohen, Falicov & Golin (1964) have shown that if c~ = 60 ° and z =0.250, corresponding to a simple cubic

t The relation between ~ and the axial ratio c/a for the equivalent hexagonal unit cell is sin (~/2)=1.5/[3+(c/a)211/2.

D. S C H I F E R L AND C. S. B A R R E T T 31

structure, then these elements would have metallic elec- trical properties. These authors have concluded that these elements would be stable in a structure with semi- metallic properties provided z-¢ 0.250 and c~¢ 60 °. However, no functional relationship between c~ and z has been suggested in these papers.

The theoretical interest in these structures justifies making very careful and precise measurements on them, and making intercomparisons of the results for As, Sb, and Bi.

The present measurements of As are the first to be made at cryogenic temperatures, the first with single crystals and the first by use of the Bond (1960) method of determining the lattice constants. Single-crystal determinations of z for As are preferable to powder determinations for three reasons:

(1) There are very few lines of the powder pattern that do not overlap.

(2) Samples of ground As powder consist of flat chips which tend to have a preferred orientation. Sev- eral methods for randomizing the orientation were tried, such as mixing the As powder with flour particles of the, same size, but without success.

(3) Higher intensities and greater accuracies in a and c determinations are obtained with single crystals.

Samples

For these measurements, a single crystal of As was furnished by Dr J. B. Ketterson of the Argonne National Laboratory. It had an impurity content reported to be less than 1 p.p.m. (Ketterson, 1965) and gave reflections 7 minutes wide in 0 at half-maximum. A good (00.1) surface was prepared by cleavage. A (11.0) surface was prepared by spark-cutting and chemical polishing with a solution of 5 parts hydrogen peroxide, 1 part water and 30 parts ammonium hydroxide. The cutting was surprisingly fast and somewhat ragged; the surface was made smooth by first swabbing it with the etchant and finally immersing it briefly in the etchant.

During most of the experiment, the crystal was kept in high vacuum or in a desiccator. On several occasions when it was unavoidably exposed to air, the crystal was found to acquire a layer of oxide on the surface; but this layer disappeared, leaving no apparent damage to the crystal, when the crystal was placed in vacuum for several hours.

Determination of z

The parameter z was determined at 4.2, 78 and 299 + 3°K from the integrated intensities of the 00.l (hex- agonal indices) reflections.

The reflections were observed on a low temperature diffraction unit consisting of a cryostat, precision dif- fractometer, scintillation counter, pulse height ana- lyzer and chart recorder. The unit was similar to the equipment used in the previous studies of Sb and Bi (Barrett, 1960; Cucka & Barrett, 1962; Barrett, Cucka & Haefner, 1963). The observations were made with

the moving counter, moving crystal method. The radi- ations used were Mo Koc and Ag K~ because each requires only small dispersion corrections and pro- vides many reflections with resolved K~I and K~2 com- ponents.

Good thermal conductivity between the sample and the coolant was assured by several pieces of copper wire which were soldered from the sample to its mount. These wires were soldered to the back of the crystal at one end with a low melting-point solder in order to avoid straining or oxidizing the cleavage surface.

The cleavage surface was aligned parallel to the vertical axis of the diffractometer; alignment guides used were light reflections observed with a telescope and the observation that several orders of X-ray re- flections showed negligible vertical displacement from the horizontal plane. The crystal was adjusted in position so that the irradiated spot was compared to that of the exit slit of the X-ray collimating tube by observations with a level telescope.

The usable area of the crystal, approximately 3 mm by 3 mm, was located close to the middle of the cleaved surface, which was about 1 cm high by 3 cm long. The effects of unavoidable variations in the crystal surface were reduced by irradiating different spots of the usable area, varying the beam width between 0.5 mm and 1.5 mm and the beam height between 0.5 mm and 2 mm in different runs.

The incident beam was defined by a collimating tube 21 cm long which contained narrow slits at each end, and a wider slit in the middle to reduce the amount of radiation scattered from the inner walls of the tube. The counter window was wide enough to accept the entire peak in the moving counter, moving crystal method.

When Mo K~ radiation was used a 0.004 inch Zr-foil Kfl filter was placed over the counter window. No attempt was made to measure the 00.3, 00.6 and 00.9 reflections because their Ks components were not well resolved and because calculations based on the other reflections indicated the possibility of appreciable error from extinction if these were used. The 00.12 and 00.21 reflections were too weak to be measured accurately and, therefore, were not used in the final calculations of z. The calculations were based on the resolved Mo K~ and Mo K~2 reflections with indices 00.15, 00.18, 00.24, 00.27.

The measurements were repeated with Ag Ks radi- ation filtered with either 0.003-inch Rh foil or 0.002- inch Pd foil placed over the counter window. The 00.3, 00.6, 00.9, 00.12, 00.15 reflections were not used. The 00.21 and 00.36 reflections at all temperatures and, at room temperature, the 00.30 reflections were too weak to be measured accurately and were not used in the final calculations of z. The calculations with Ag K~I and Ag K0~2 radiation were based on reflections with the indices 00.18, 00.24, 00.27 and 00.33 at room temperature and 00.18, 00.24, 00.27, 00.30 and 00.33 at low temperatures.

32 T H E C R Y S T A L S T R U C T U R E OF A R S E N I C AT 4.2, 78 A N D 299°K

The integrated intensity Io of each reflection was taken as I0 = I t - Q , where It is the total integrated in- tensity and Q is the background count, computed from measurements of background on both sides of the reflection, for the same angular range of integration. Each observed intensity Io was corrected by the appro- priate Lorentz and polarization factors for each line of the resolved Mo K7 or Ag Kc~ doublets.

No correction was made for absorption. For a thick crystal, larger than the beam, with reflecting planes parallel to the surface, such. as was used, the correction A for absorpt ion is A =½/z, where It is the linear ab- sorption coefficient; this correction is the same for all orders, independent of 0, and does not change the relatire intensities.

The parameter z, the temperature factor B, and the reliability factor R given by

R = [ X wr(FZobs-FeZat)2]l/2/[ X ~,,r(Fobs)' 221,'21. , r i"

where Fobs and Feal are the observed and calculated structure amplitudes and Wr is the weight given to the

r th reflection, were calculated for the data at 4.2°K, 78 °K and 299 + 3 °K. Each intensity was given a weight I/[It+Q]. For each temperature, z, B, and R were calculated from (1) the total data obtained at that temperature and (2) the data obtained with each line of the resolved K~ doublet on each run. Calculation (2) was done to test the possibility of systematic differences between results obtained with different wavelengths and to test the internal consistency of the results from each run. Refinement was carried out with the Busing, Mart in & Levy (1962) least-squares computer program.

The Freeman (1959) scattering factors, computed with Hartree self-consistent field electron densities without exchange, were used with anomalous disper- sion corrections for the K, L and M electrons (Inter- national Tables for X-ray Crystallography, 1962; Coo- per, 1963). As a check on the influence of the scattering factors used, refinement was also carried out by use of the Freeman scattering factors without any dispersion corrections. Individual results for z were then changed by no more than one unfit in the fifth decimal place. In these refinements the temperature factor exponent,

Table 1. Atomic position parameters, z, temperature factors, B, and reliability factors, R, for arsenic

Temperature Radiation (°K) z B R

Mo Kcq 4.2 0.22757 + 0.00006 0.048 + 0.014 0.027 Mo Kcq 4.2 0-22763 + 0.00002 0-229 + 0-003 0-006

Mo Kg2 4-2 0-22758+0.00009 0.047+0.019 0.037 Mo Kct 2 4.2 0-22762 + 0.00002 0-279 -+ 0-004 0-007

Ag Kctt (Rh filter) 4.2 0.22761 +0-00006 0.224+0.010 0.034 Ag Kct2 (Rh filter) 4.2 0.22772+0.00003 0"240+0"006 0.019

Mo K~tl 78 0.22751 _+0"00004 0"395+0"008 0.013 Mo Kal 78 0"22750 + 0-00003 0-320 + 0"006 0.019 Mo K~tl 78 0"22749 + 0.00009 0.321 + 0.022 0.035

Mo K~x2 78 0.22741 _+0"00004 0-391 +0.008 0.012 Mo Kcx 2 78 0"22764 + 0"00004 0-302 + 0"008 0"014 Mo Kc~2 78 0"22748 + 0"00011 0"262 _+ 0"028 0"040

Ag K~tl (Rh filter) 78 0-22757+0.00001 0.334+0.003 0"005 Ag Kcq (Rh filter) 78 0"22752+0-00007 0-294+0.013 0.036 Ag Kcq (Pd filter) 78 0-22754+0.00008 0-358+0.014 0"040 Ag K~l (Pd filter) 78 0.22760+0.00004 0.360+0.007 0.020

Ag K~x2 (Rh filter) 78 0.22758+0.00008 0.331 +0.017 0-042 Ag Kct2 (Rh filter) 78 0-22759+0.00002 0.345+0.004 0.012 Ag K~t2 (Pd filter) 78 0.22748+0-00009 0-347+0.007 0"019 Ag Ka2 (Pd filter) 78 0.22757 + 0"00005 0.358 + 0"009 0.025

Mo K~tl 301 0"22696 + 0"00002 0-933 + 0"007 0"003 Mo K~tt 300 0-22721 +0"00002 1"000+0-006 0"003 Mo KCtl 298 0-22713+0-00003 0"835+0"007 0-007

Mo Kg2 301 0"22699+0'00004 0"906+0"012 0"006 Mo Kcx2 300 0"22700 + 0-00007 0"875 + 0"020 0"012 Mo K~x2 298 0.22721 + 0.00001 0.858 + 0-003 0-002

Ag K~tt (Pd filter) 298 0.22725 + 0.00003 0.923 + 0-005 0"005 Ag K~q (Pd filter) 299 0.22698 + 0-00004 0-886 + 0.009 0-008 Ag K~q (Pd filter) 298 0-22724+0-00004 1.100+0.008 0.007

Ag K~t2 (Pd filter) 298 0.22750+0.00004 0.954+0.008 0-014 Ag K~t2 (Pd filter) 299 0.22730+0.00008 0-927+0.016 0.014 Ag Kct2 (Pd filter) 298 0.22733+0.00014 1.036+0.015 0.014

D. S C H I F E R L AND C. S. B A R R E T T 33

B, was smaller than in the other refinements by amounts in the range 0.01-0.03.

Table 1 lists values of z, B and R according to the wavelength used. The errors given for z and B are the estimated standard deviations Sz and sB (as opposed to the actual standard deviations az and a~) where the estimated standard deviations sp~ for the parameter pi is found from

2 =bu[ ~" Wr(F2bs-F2al)2l/(m-n), (1) Sp i r = l

where bu is the ith diagonal element of the inverse matrix, m is the number of observations, and n the number of parameters varied (Stout & Jensen, 1968).

The values ofz and B obtained with Mo Kel, Mo Ke2 and Ag K0q radiations are in close agreement* at all

* The apparent ly aber ran t values for B obta ined in the first M o K~ run are retained in Table 1 because (i) no reason for it, such as t runcat ion of reflection profiles, a spur ious burs t o f counts during the reflection-scanning process or heat leak in the cryos ta t could be identified and (ii) the z value obta ined f rom these reflections was reasonable and, therefore, indicated no reason to discount the whole run.

¢d ,/

1.004

1.002

1.000-

a 4.2 -~1 t " "

_ .

Bi i

Sb" As I I

I00 2 0 0 :500

T (°K) Fig. 1. Thermal expansion along alaex for As, Sb, Bi. (In Figs.

1-5 absence of any error bar indicates that it is too small to be indicated on the plots ; dashed or solid lines connect ing the da ta points are meant only as guides to the reader.)

1.010

1.008

1.006

1.004

1.002

1.000-

I I i

/

A~ /

/ -

/ /

/ /

/ / Sb - - / / / / / /

/ / / /

I IO0

T(°K)

I I 0 2 0 0 3 0 0

Fig. 2. Thermal expans ion along c for As, Sb, Bi.

temperatures; the differences in the values of z from refinement of the data of individual runs obtained with each wavelength lie within 0.00006 at 4.2°K, within 0.00019 at 78°K and within 0.00029 at 299 + 3°K. The results obtained with Ag K~2 radiation differ consider- ably from the others at room temperature. The dis- crepancy seems to be primarily due to the fact that the reflections obtained with the Ag K0~2 line had the lowest intensities and peak-to-background ratios. The results obtained from the use of Ag radiation with a Pd filter and with a Rh filter were not significantly different.

Final values of z, B and R were obtained by refine- ment of the combined data from all four wavelengths at each temperature. The advantages of this method are that" (a) the values of these parameters are based on more different 00.l intensities (five at room tem- perature and six at low temperatures) and (b) the quantity (m-n) of equation (1) is maximized. This refinement yielded the following values of z, B and R"

4.2 °K z = 0.22764 + 0.00004, B=0.233 + 0.007, R = 0.039 ;

78 °K z = 0.22754 + 0.00002, B=0.340+0.004, R--0.030 ;

299 °K z = 0.22707 + 0.00005, B = 0-923 + 0.015, R = 0.045 ;

where the errors refer to the estimated standard devi- ation sz. The previously accepted room temperature value, z--0.226, was obtained from powder samples only (Trzebiatowski & Bryjak, 1938; Wyckoff, 1960).

L a t t i c e c o n s t a n t s

The lattice constants a and c (hexagonal axes) were determined at 4.2 °K, 78 °K and 300 + 1 °K with Mo Kel radiation and the Bond (1960) method. The 00.24 (0~53 °) and 00.27 (0=66 °) reflections were used to determine c, and the 55.0 (0_ ~ 79 o) reflection to deter- mine a.

The apparatus mentioned in the preceding section was used, but with a more highly collimated beam. To check the beam divergence and alignment, a piece of X-ray film mounted on the cryostat at a distance R from the spectrometer axis was rotated about the axis through 180 °. The film was exposed to the direct beam twice, once when near the exit slit of the X-ray tube and once when at a distance 2R farther away. The angle A1 between the beam and the normal to the spec- trometer axis, which was vertical, was found from the relation sin AI =y/2R, where y is the vertical displace- ment between the centers of the two images on the film as measured on a comparator. The correction in a and c corresponding to Ax was of order 1 part in 106. The vertical divergence angle A0 of the beam was found from the relation sin A0 = (Wfar - Wnear)/2R where Wnear and Wfar are the vertical widths of the images produced when the film was near and far from the exit slit, respectively. The correction in a and c made for the beam divergence was of the order of 1 part in 105.

JAC 2 - 3

34 THE CRYSTAL S T R U C T U R E OF A R S E N I C AT 4"2, 78 AND 299°K

Before each run, after the crystal was cooled to the desired temperature, the tilt angle A2 between the re- flecting plane and the spectrometer axis was found by exposing to the appropriate reflection a piece of X-ray film mounted on the cryostat at distance R from the axis, then rotating the cryostat so as to sweep the film through the direct beam. The vertical displacement D of the center of the reflection image from the center of the direct beam image is related to A2 by sin A2 = D/R. The correction in a and e made for the crystal tilt was of the order of 1 part in 10 s.

Reflection positions were determined by point count- ing in steps of one minute of arc. For five minutes after the crystal was rotated through a large angle, as when going to a new reflection position, no readings were taken. If this precaution was not taken, the peak posi- tions were found to shift measurably, presumably be- cause of the movement of the lubricating fluid in the spectrometer.

The midpoint of each peak was found at intensity levels of-~- maximum and 32- maximum over background so that for each run two values of 0 were obtained. The mean, 0, of these two was used; their difference was always much smaller than the spread of measurements in the different runs with a given wavelength. No cor- rections were made for refraction or Lorentz and polarization effects.

Values of a and c were computed for each run with the wavelength value 221o K~t = 0-709300 A, from Bear- den (1964). The averages for each temperature are re- ported in Table 2. The errors listed are estimated from the spread of measurements of the several runs; these refer to precision. The room temperature results agree with the lattice parameters obtained by Taylor etal. (1965) within their limits of error. Figs. 1 and 2 respec- tively show the relative expansions a(T)/a(4.2 °K) and e(T)/e(4.2°K) plotted against temperature for arsenic, and for antimony and bismuth as well. The same scale is used in both graphs so that the thermal expansion in the a and c directions can be more easily compared.

It is interesting to note that the thermal expansion of arsenic in the a direction is extremely small; the present data plus that of Taylor etal. (1965) indicate that the relative change Aa is less than one part in 104 in the temperature range 0 to 700°K.

Table 2. Lattice constants of arsenic

Tempera ture a (A) c (A)

4.2 OK 3-7597 _+ 0-0001 10-4412 ___ 0.0002 78 °K 3.7595 +_ 0"0001 10.4573 _ 0-0002

300_+ 1 OK 3.7598 _+ 0.0001 10.5475 _+ 0.0002

Summary of results for As, Sb, Bi

Table 3 is a summary of the crystallographic data obtained in this laboratory for the Group V semi-metals. Listed for each element as a function of temperature are:

ahex, c, c/a, lattice constants referred to hexagonal axes; arhomb, Ct, lattice constants referred to rhombohed- ral axes; z, atomic position parameter in units of c.

All angles are in degrees, and all wavelengths in 5.ngstroms. The radiations and wavelengths used for determining the lattice constants are:

For As, Mo K~I with 2 = 0-709300 A (Bearden, 1964): For Sb, Cu Kfl with 2= 1.39217 A. (converted from Cauchois & Hulubei, 1947). For Bi, Fe K~ with 2 = 1.93728 A, (converted from Cauchois & Hulubei, 1947).

The lattice constant of primary interest here, c~, is not affected by the fact that different scales of wave- lengths were used. The error listed for each lattice constant is the estimated precision. The estimated standard deviation Sz is entered in the Table as the error in z for As and Bi. For Sb the errors in z at 4.2 and 78°K given in the Barrett, Cucka & Haefner (1963) paper are judged to have been underestimated

Temperature (°K)

As 4.2

78

299 + 3

Sb 4.2

78

298

Bi 4.2

78

298 _+ 3

Table 3. Summary of results for As, Sb, Bi

ahex C c/a arhomb O~ Z

3"7597 10"4412 2"7770 4-1018 54-554 0"22764 __ 0"0001 _+ 0"0002 _+ 0"0003 _+ 0"0001 + 0-001 _+ 0"00004

3" 7595 10"4573 2"7816 4" 1063 54"486 0"22754 _+ 0"0001 _+ 0"0002 + 0"0003 + 0"0001 + 0"001 + 0"00002

3"7598 10"5475 2-8053 4-1320 54" 126 0"22707 + 0"0001 + 0-0002 + 0"0003 + 0"0001 + 0"001 + 0"00005

4"3007 11 "2220 2"6093 4"4898 57-233 0"23362 + 0"0002 + 0-0005 + 0"0002 + 0-0002 + 0"002 + 0"00004

4-3012 11"2320 2"6114 4"4927 57-199 0"23364 + 0"0002 + 0"0005 + 0"0002 + 0-0002 + 0"002 + 0"00004

4"3084 11"2740 2"6167 4"5067 57"110 0"23349 + 0"0002 + 0"0005 __+ 0"0002 __+ 0"0002 + 0"002 + 0"00031

4"5330 11"797 2"6025 4"7236 57"35 0"23407 + 0-0005 + 0"001 + 0"0004 + 0"0005 + 0"01 + 0"00004

4"5350 11"814 2"6051 4"7273 57-28 0"23400 + 0"0005. + 0-001 + 0"0004 + 0"0005 + 0"01 + 0"00008

4"5460 11"862 2"6093 4"7458 57"23 0"23389 + 0"0005 + 0"001 + 0"0004 + 0"0005 + 0"01 + 0"00004

D. S C H I F E R L AND C. S. B A R R E T T 35

58 °

57*

OC 56 °

5 5 " - -

54*

I I I

~!~Sb.

~7299°K

% %

% %

• 4.2°K ~ ~ A s

• 78°K • 2990K "~~~\

3.0 4.0 5.0

( 0 . 2 5 - z ) 2x I04 Fig.3. Rhombohedral angle ~ v s . (0"25-z) 2 for As, Sb, Bi.

Dashed lines connect data points of the same temperature; solid lines connect data points of the same element.

o . z 2 7 8 - i I i I I _~

t - 4.2OK -

78"K 0.2276-- ~ . ~ --

-

Z

0.2274

- -

0 .2272~

o 2 z 7 o - - / ± I , I , ~ 4 . o o o ~ . z o o 5 4 . 4 0 ° s 4 . 6 o o

(£

Pig.4. Rhombohedral angle ~ v s . atomic position parameter z for As.

and are here increased to + 0-00004; this is a rather arbitrary value but is reasonable in comparison with both the As and Bi values and is within the scatter of the 6 determinations of z in Sb.

Discussion

The question of possible relationships that may exist between z and a is of physical interest. However, no such relationship has yet been derived from first prin- ciples. We consider here two simple algebraic relations which have been suggested, in informal discussions, as possible ways to correlate a with z. They may also be useful in estimating whether one parameter would approach its cubic value gradually or abruptly as the other gradually approached the cubic value in, say, a high pressure experiment.

A quadratic relation of the form

~ = K , (O .250- z)2 + K2 , (2)

where K1 and K2 are appropriate constants can be used in two different ways to correlate e and z. Considering the three elements at a given temperature, we see from the data points connected by broken lines in Fig. 3 that the experimental results for 4.2°K and 299 °K can be fitted well by a quadratic of type (2) with appropriate constants, and also that the data for 78 °K can be fitted fairly well. Extrapolation by relation (2) does not yield the value z = 0.250 for e equal to the cubic value of 60 °, suggesting that perhaps there would be an abrupt transition of z to 0.250 if e were caused to approach 60 ° slowly.

The data for As alone at the three temperatures can also be fitted very well by a relation of the form (2). The data of Sb alone at the three temperatures as well as that of Bi can be fitted within the assigned error bars by relation (2). The trends as temperature is varied are indicated by the solid lines for each element in Fig. 3. (The original plot of these data had a much expanded scale.) Extrapolation by relation (2) for each element does not yield z = 0.250 at ~ = 60 o for any of the three.

It is possible to fit some of the data with a linear relation of the form

(z = g324- K4, (3)

where/£3 and K4 are appropriate constants. The data for As at the three temperatures can be fitted within the assigned error limits by relation (3) as can be seen from Fig.4. The data for Bi is plotted similarly in Fig.5. Linear extrapolation does not yield z=0.250 for equal to the cubic value of 60 o. The data for Sb lead to no definite conclusion regarding relation (3) as can be seen from Fig. 5. A linear relationship between e and z for the three elements at a given temperature is not supported by the data, because neither the data for the three elements at 4.2 °K nor that of the three elements at 78 °K can be fitted within the assigned error limits by relation (3).

J A C 2 - 3 *

36 T H E C R Y S T A L S T R U C T U R E O F A R S E N I C AT 4.2, 78 A N D 299°K

All of the relations discussed above suggest that perhaps there would be an abrupt transit ion of z to 0-250 if e were made to approach 60 ° slowly. It is not clear at this time, however, whether any fundamenta l significance should be attached to any of these relations.

Discussions with L.M. Falicov, M.H. Cohen, S. Gol in and the late K. Haefner were very helpful. This work was supported by the U.S. Army Research Office Grant D A - A R O DAHCO4-67-C-0050 by the U.S. Office of Naval Research Contract No. Nonr 2121(11), and facilities provided by the Advanced Research Projects Agency.

References

BARREtt, C. S. (1960). Aust. J. Phys. 13, 209. BAR~Tr, C. S., CUCKA, P. & HAEFNER, K. (1963). Acta

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0.2542

0.2540

0.2538

0.2556

f 0.2334

t 0.2532 57.00*

/ l i

rn 4.2"K

0 7 8 *K

A 299"K

I i I 57.20* 57.40*

a

Fig.5. Rhombohedral angle ct rs. atomic position parameter z for Sb and Bi. (Trends are considered to be too uncertain to draw a line through the Sb points.)

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