Short Notes K95

phys. stat. sol. (a) 5 K95 (1971)

Subject classification: 12; 22.5

Department of Natural Philosophy, Aberdeen University

The Elastic Constants of Cadmium Fluoride

BY

S. HART

Cadmium fluoride crystallises in the fluorite structure common to the alkaline

earth halides, CaF2, SrFZ, BaF2. The elastic constants of these halides show a

smooth progression according as the cation mass and unit cell size increases.

Cadmium fluoride has a cation mass almost three times as big as calcium fluoride

yet the unit cell dimension is virtually the same.

It is instructive to look at the outer electronic structure of the metal atoms. In

Table 1 are set out the structures for Ca, Sr , Ba and for Zn, Cd, Hg. Also shown

are the structures for the alkali metals and for the noble metals Cu, Ag, Au.

Table 1

Outer electronic structure of some metals

6

6 2 lib 4p65s 55Cs 5p 6s

56Ba 5p 6s

6 6 lgK 3p 4s 37 3Li 2s l lNa 2p 3s

38Sr 4p 5s

Cu 3d 4s Ag4d 5s 7 9 A ~ 5d106s2

lo Hg5d 6 s Zn 3dl04s2 48Cd4d 5s

6 2

10 10 10 20Ca 3p 4s

29 47

30 80 I I

It will be seen that Zn, Cd, and Hg bear the same relation to the alkaline earths

in their electronic structure as do the noble metals to the alkali metals. Of the

noble metal halides only silver forms cubic symmetry crystals of the NaCl type

and only silver chloride, has received attention regarding its elastic constants (1,

2). In this material there is a large degree of failure in the Cauchy relation, c

= c

Hg only CdF forms crystals of the fluorite structure, the other halides of cadmium

being hexagonal. It is therefore interesting to compare the elastic properties of

CdF with those of the alkaline earth fluorides to see whether the same gross

changes will be observed between say CdF2 and S rF a s are observed between AgCl

and RbC1.

= 1 2

compared with the a l h l i halides. Of the halides of the metals Zn, Cd, and 44'

2

2

2

7 physica (a)

K 96 physica status solidi (a) 5

Although CdF has been known as an interesting optical material for some con- 2 siderable time and it has been available in single crystal form for almost 20 years (3)

no elastic properties have been reported. Large crystals of the material have been

grown in this DBpartment by Jones and Jones (4) and smaller oriented crystals in the

form of short thin cylinders have also been produced. These latter crystals were in

a particularly suitable form for this investigation.

The experimental method has been reported recently (5) and involves measuring

the resonant longitudinal (f ) and torsional (f ) frequencies of cylindrical crystals. The

crystals, 5 mm in diameter and a few centimetres long, were oriented by back re-

flection Laue photographs. Spectrographic analysis revealed the following impurities

in parts per million: Al: 30, Ca: 20, Co: 40, Mg: 30, and Ni: 50. The crystal density,

e , was measured by weighing in a i r and water and checked well with the value of

6.382 g/cm reported by Jones and Jones (4), this value being used in the calculations.

1 t

3

If the crystal has length, 1, the effective Young's and rigidity moduli, E and G

respectively are given by

E = 4 e 12f2 and G = 4 Q 1 2 2 ft , 1

where the Rayleigh correction factor may be neglected for the crystal dimensions

used in the investigation. The elastic compliances, s.., are obtained from these

moduli by the usual equations, Brown (6) 1J

1 - = 9 - 2SF, E 11

1 - i e = S + 4 S F , G 44

where 2 2 2 2 2

E = 2s (F - 4 F + 3& fi )E,

s = Sll - S12 - s44/2 , 2 2 2 2

F = a 2 $ + p r + y " s

and a, P , axes .

a r e the direction cosines relating the cylinder axis to the cubic crystal

In Table 2 are given the experimental results of the 9 crystals used.

Short Notes K 97

Table 2

2 Experimental results for CdF

~ ~~

crystal E G

P r No. (10 11 dyncm-2) (10'' dyncmd2) OL.

1 7.15 3.38 0.410 0.410 0.814

2 1 1 . 2 3 2.65 0.039 0.277 0.960

3 14.35 2.22 0.055 0.078 0.995

4 6.96 3.29 0.167 0,724 0.669

5 6.54 3.53 0.332 0.624 0.707

6 6.24 3.68 0.446 0.571 0.688

7 9.22 2.91 0.042 0.404 0.913

8 7.09 3.25 0.155 0.624 0.766

9 6.65 3.55 0.344 0.550 0.760

The data was reduced by least squares fitting to give the following results:

= 0.67 + 0.007, s s

10 cmdyn . On inversion these gave c = 18.6 + - 0 .5 , c = 2.17 + 0.2,and -2

c .

= 4.62 + 0.04, and s = -0.18 + 0 .02 in units of - 44 - 12 - l112 -1

11 44 - = 6.8 + 0 . 5 in units of 10l1 dyncm

When this data i s compared with the data of Gerlich (7) for SrF2 then the dis- 12

crepancies observed between AgCl and RbCl a r e generally repeated. The constant

c

is 3 times bigger than c 12

is 9 t imes bigger than c for CdF compared with 4 times for S rF2 and c 11 44 2 compared with 1 .5 t imes for SrF 44 2' So again the condition

fails drastically. c12 = c44 The data presented here cannot be compared with any other elastic data, how-

ever, it may be compared with the dispersion curves recently reported by Denham

et al. (8). The dispersion curves of phonon frequency vs. wavevector were calculated

for the [loo] , [ l l O ] , and [lll] directions for various fluorite structure ionic

crystals. At the long wavelength limit, (wavevector equal to zero) the slope of the

dispersion curves should be determined by the ultrasonic wave velocities. These

velocities may be determined from the elastic constants using the well known results

K98 physica status solidi (a) 5

2 Fig. 1. Dispersion curves for CdF

2 2 [loo] direction gv = c l l s Qvtl = evt2 = c 44 - '

[ l lO] direction

[lll] direction

2 1 2 2 1 ev, =;(Cl1 t. 2c12 + 4 ~ ~ ~ ) ' evtl = ev t2 = - (c 3 11 - c 1 2 + c44) ;

where V1, Vtl, and V are the longitudinal and two transverse velocities. t2

In Fig. 1 the acoustic branches of the dispersion curves are drawn for the three

relevant directions. Also shown are straight lines with slopes determined by the

ultrasonic velocities calculated from the present elastic constant results. It will

be seen 'that good agreement has been obtained. Similar agreement was observed for

the slopes at zero wavevector of the dispersion curves for the other fluorite structure

crystals investigated by Denham et al. , e . g. CaF2, PbFZ when literature values of

the elastic constants were used.

Short Notes K99

R e f e r e n c e s

(1) D.L. ARENBERG, J. appl. Phys. 2, 941 (1950).

(2) W. HIDSHAW, J.T. LEWIS, and C.V. BRISCOE, Phys. Rev. 163, 876 (1967).

(3) D.A. JONES, R.V. JONES, and R. W.H. STEVENSON, Proc. Phys. SOC. Bl 906 (1952).

(4) D.A. JOrJES and R. V. JONES, Proc. Phys. SOC. K9, 351 (1962).

(5) S. HART, phys. stat. sol. (a) 3, K187 (1970).

(6) W. F. BROWN, Phys. Rev. 3, 998 (1940).

(7) D. GERLICH, Phys. Rev. 136, 1366 (1964).

(8) P. DENHAM, G.R. FIELD, P . L . R . MORSE, and G.R. WILKINSON, Proc.

Roy. SOC. A317, 55 (1970).

(Received March 11, 1971)