Ultrasonic parameters in the born model of the rubidium halides

J. Phys. Chem. Solids

Pergamon

Press 1970. Vol. 3 1, pp. 2397-2400.

Primed in Great

Britain

ULTRASONIC PARAMETERS IN THE BORN OF THE RUBIDIUM HALIDES R. W. ROBERTS Physics Department.

University

and CHARLES

of North

(Rweivrd

Carolinaat

9 Februury

MODEL

S. SMITH

Chapel

Hill. N. C. 275 14. U.S.A.

1970)

Abstract- Values of all dC/dP have been obtained at 295°K and are strikingly smooth when plotted against nearest neighbour distance. dw When combined with known temperature coefficients of the elastic constants. they lead to values of 5 = dlnBr/d7J. which are uniformly - I X IO-’ deg-’ for these crystals with low Debye temperature. The resulting Born model exponential parameters. d,,/p. are precisely linear with do. in contrast with ddp values where compression [ were used in the necessary correction for vibrational effects. Values of d&/dP are uniformly 5.6. a general magnitude well accounted for by the Born model which. however. predicts a significant systematic variation in the four halides. In all these features the Rb halides are similar to the Na and K halides.

1. INTRODUCTION

A PAPER of this title by Roberts and Smith [ 1I on the sodium and potassium halides appears in a recent issue of this journal. This note reports the experimental measurement of the pressure derivative of the single crystal elastic constants of the rubidium halide sequence. It extends to the rubidium sequence the analysis of the previous paper for the parameters in the Born model in the Hildebrand approximation. and compares these ultrasonically based parameters with preaccepted viously values derived from compression work. The note also extends to the rubidium halides the discussion of the pressure derivative of the isothermal bulk modulus. The experimental techniques used. data reduction. analysis. notation and conclusions are largely those of the previous paper; exceptions are noted in appropriate places. The note is also sectioned in a parallel fashion.

Rubidium fluoride proved to be especially difficult because it is extremely hygroscopic. by far the worst so of the group. This material was handled exclusively in a glove box, but even so, acoustic seals were difficult to make and to retain throughout a pressure run. For this reason. the important pressure effect on the longitudinal wave was verified on a[ 1001 crystal obtained especially for this purpose. 3. RESULTS

Table 1 shows the experimental dC/dP for the longitudinal[l lo] stiffness, Ci,. and the two shear stiffnesses C and C’. Also shown is the result for the adiabatic bulk modulus. B,, obtained directly by combining the measurements. and for the isothermal bulk modulus. LIT. which was derived from dB,ldP as described below. Each row in Table 1 plots smoothly. almost Tmhle I. Values

of

dC/dP clt 295°K

for the rubidium halides 2. EXPERIMENT

The experiment results were obtained on [ 1101 single crystals procured from SemiElements. The crystals that were used passed our stringent X-ray tests and the sterling cooperation of Mr. Joseph Barrett, formerly of that company, is gratefully acknowledged.

RbF

RbCl

RbBr

Rbl

---..=;I C’ C,r B. Br 2397

6.51

6.88

6.91

7.02

4.93 -0.70 5.57 5.69

S.88 -0.56 548 5.62

6.03 -0.55 5.45 5.59

6.26 -O.Sl 5-44 560

2398

R. W. ROBERTS

linearly. against nearest neighbour distance. do. just as was observed previously for the sodium and potassium halides. It should be noticed further that dB,ldP and dBrldP are each essentially invariant in the sequence. 4. DERIVED

QUANTITIES

Table 2 gives the auxiliary thermodynamic data adopted for use in this section and the next. Sources were as in]11 with two exceptions.

and C. S. SMITH

judicious extrapolation

of the B’s of the other

three.

Other thermodynamic input data are the dlnC/dT), of Haussiihl[4] which have been combined to give the dlnB,/dT),B shown in Fig. 1. Our dB,ldP. together with Table I, then give dlnB,/d& in the manner the figure shows. From dlnB,/dT), we obtain .$ = dlnB,/ dT& by using equation (4) of[ I]. The derived ultrasonic 5 shown in Fig. 2 are remarkably

Table 2. Basic data at 295°K for the rubidium halides

d” (IO-B cm) RbF RbCl RbBr Rbl

The

2.826 3.291 3445 3.671

c’p

f&l (I O.-Z’cm? B, molecule-‘) (kbar) 45.13 71.29 81.77 98.94

important

277 162 138 III

(IO lBergdeg-’ molecule-‘) 8.421 8.567 8.59.5 8.630

nearest-neighbor distance. on a specimen of our crystal by the National Bureau of Standards X-ray diffraction group under the direction of Mr. Howard E. Swanson; the value shown will appear in a forthcoming section of NBS Monograph 25. This do is 0.4 per cent larger than the old measurement used on Born model analysis[2] and, because it enters the analysis to the fourth power. the change is significant. It has in fact smoothed our Born model parameters and we are most grateful for the willing, and speedy, cooperation of NBS. The volume thermal expansion coefficients. B. given for RbCI. RbBr. and RbI are the - 40°C measurements of Hengleinl31. increased by a common 0.04 deg-l to bring them to 295°K. The temperature adjustment was determined by a comparison of the available data for these and other alkali halides as a function of Debye temperature. The common value of the adjustment reflects the low Debye temperature of these materials. The B value for RbF was arrived at by

P (IO-‘deg.

‘)

y I.40 I.39 I .42 I.56

(0.94) I .03 I.08 I.23

I

VY (IO-*)

& (kbar)

3.87 4.23 4.53 5.67

26-i I56 I32 IO5

I

I

do. for RbF was measured

Rb

30

2.5 Nearest

Neighbor

Halides

40

35 D~stonce.

d,

(A)

Fig. I. The temperature coefficients of the adiabatic bulk modulus at 295°K vs. nearest neighbor distance. 4,.

invariant at - 1 x 10-l deg-I. again in contrast with values based on temperature and pressure derivatives obtained in compression experiments[2]. The ultrasonic 5 for the Rb sequence are smoother than even those for the Na and K sequences. The Rb halides are essentially classical solids at 295°K; 5 is

ULTRASONIC

PARAMETERS

IN THE

BORN

$&,,l=

-325

Nearest

3.5 Nmghbor

Dirfonce,

4.0 d,

(A)

Fig. 2. Derived ultrasonic 6 = dlnBdd77, at 295°K. and derived Born model repulsive parameter dJp = n + 1 vs. nearest neighbor distance, d,,. Previously accepted values based on compression data are also shown.

uniformly small under these circumstances as Swenson[S] has pointed out for other cubic solids, but it is clearly not zero here. The second derived quantity, dBJdP. was obtained using equation (5) of[l]. The correction involved in going from the adiabatic to the isothermal modulus was again larger than the first approximation suggested by Swensonl51. 5. BORN

MODEL

The repulsive interaction that appears in the Born model may be written in a power law version

or in an exponential

version

WR=Dexp{-$[(E)“‘-I]}.

(2)

Use of the Hildebrand vibrational approximation, the equilibrium condition and the definition of the bulk modulus yields the result.

2399

MODEL

4+3x(1 -5T) [l-T@]

_2 ’

(3)

together with corresponding expressions for An and D(dJp),[l]. The quantity A is 3&B,/ (ae2/d,,) and the curly and square brackets are the vibrational correction terms arising in the Hildebrand approximation which is surely appropriate for the classical Rb halides. It may be pointed out that either do/p or n + 1 is physically the ratio - r*W”/rWk, where r is any crystal distance and the primes signify differentiation with respect to that distance. Values of this quantity are given in Fig. 2. The ‘ultrasonic’ values employ the ultrasonic 5. of the lower part of the figure, in the curly bracket vibrational correction; the ‘compression’ values are those quoted by Tosi[2J who employed the compression 5. Conclusions similar to those for the Na and K halides[l] apply here, with the additional observation that in the Rb sequence ddp = n+ 1 is precisely fineur with d,,. It was in this connection that the new value of the RbF lattice constant was important. Linearity was nearly, but not precisely. found in the previous two sequences. The quantity dBr/dP vs. the ultrasonic Hildebrand repulsive parameter is shown in Fig. 3 for all three halide sequences. The different halides are not distinguished because they read uniformly from left to right. F-Cl-Br-I. in all three sequences (which would not have been the case had compression based parameters been used). It is seen that the twelve data are practically invariant at a common value of 5.5, a remarkable result. Close inspection shows a small slope in each sequence which. further. decreases systematically in the order Na. K. Rb. The Born model in the power law version leads to the simple result. dBr/dP = (n +7)/3. In deriving this expression vibrational terms have been omitted. but it can be shown that, if included. they largely cancel. leaving a small residue of around + 3 per cent. Figure 3 shows that this expression gives the general

2400

K. W.

ROBERTS

and C.

S. SMITH

magnitude of dB JdP quite satisfactorily. but that the slope is too large. The exponential version of this expression yields the same slope for practical purposes. but magnitudes approximately 10 per cent lower and hence not in quite as good agreement with the facts as the power law version. 0 Na

Acknocldedgemenf

AK . Rb

OL

I

a

9 Rspulslve

IO Parameter.

AEC. I

12 :

= nf

13

I

I. 2.

Fig. 3. Ultrasonic dBr/dP at 295°K for the sodium. potassium and rubidium halides vs. repulsive parameter &/p = n+ I. The members of each sequence occur monotonically from left to right in the order F, Cl. Br. I. The solid line is the preidction of the Born model in the repulsive power law version.

-The

research

was provided

was supported

hy the

by ARPA.

REFERENCES

I

I

II

Equipment

3. 4.

5.

ROBERTS R. W. and SMITH C. S.. J. Plays. C‘izrm. So/i& 31. 6 19 ( 1970). TOSS M. P.. Solid Srate Ph.vsics (Edited by F. Seitl and D. Turnbull). Vol. 16. p. I. Academic Press. New York (I 964). HENGLEIN F. A.. Z. Elecrroclrcm. 31.424 (I 925). HAUSSUHL H..Z. Phys. 159.223 (1960). SWENSON C. A.. J. Ph.vs. Chem. Solids 29. 1337 (I 968).