TEMPERATURE DEPENDENCE OF ELASTIC CONSTANTS AND THE DEBYE TEMPERATURE OF NACL AND KCL SINGLE CRYSTALS

\

NASA TECHNICAL TRANSLATION

NASA TT P-l4,932

TEMPERATURE DEPENDENCE OP ELASTIC CONSTANTS AND
THE DEBYE TEMPERATURE OF NaCl AND KCl SINGLE CRYSTALS

A. V. Sharko, A.

Botakl

Translation of: "Temperaturnaya
Zavlslmost' Upruglkh Postoyannykh
1 Temperatura Debaya Monokrlstallov
NaCl i KCl," Izvestiya Vysshlkh
Uchebnykh Zavedenll, Flzlka; vol. 13,

no,

6, 1970, pp 22-28

OF ELASTIC CONSTANTS AND THE DEBYE
TEHPERATTJRE OF MaCl AND KCl SINGLE
(Linguistic Systems, Inc., Cambridge,
mass.) 13 p HC $3.00 CSCL 2nL

G3/25

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON, D.C. 20546 JUNE 1973

NASA TT F-l4,932

TEMPERATURE DEPENDENCE OF ELASTIC CONSTANTS AND /22*

THE DEBYE TEMPERATURE OF NaCl AND KCl SINGLE CRYSTALS

A. V. Sharko, A. A. Botakl

Measurement of the temperature changes of elasticity consjtants
are promising in the regard that they provide the possibility of
determining many hard to measure physical characteristics of solids.
Thus, for example, the Debye temperature and energy of a crystal
lattice may be found according to the temperature changes of the
elasticity constant of single crystals of the hali^-s ' of alkali
metals.

The temperature changes of the elastic properties of single
crystals of NaCl and KCl are determined in the present study and
the Debye temperature of these compounds is found. The measure-
ments were made by the ultrasonic pulse method with the use of the
DUK-6V flaw detecter. Using the method .of relative measurement,
we determined C,^ — ^the longitudinal wave propagation velocity in
an infinite medium, C-, — ^the longitudinal wave propagation velocity
in a thin rod, the cross section of which is much less than the
length of elastic waves in the sample and C is the transverse
wave propagation velocity. The essence of the method of relative
measurements is the following:' the apparent length 1" of the test 1
specimen was determined according to the scale of the depth gage
of a DUK-6V ultrasonic pulse flaw detecter, adjusted according to
a standard sample with a known ultrasonic propagation velocity C^. ;
its true length 1 was measured with the use of a micrometer.
Having designated the elastic wave propagation velocities, res-
pectively C and Cq. in the investigat-ed ■.ahd^^tarid-ard. .§a:5pI^-§-land

* Numbers in righthand margin indicate pagination of foreign
text.

n — ^th-e number -i of th.e multiple echo pulse reflection, a desired
value of a velocity: C OBay- be deterjnined from tJie relationship

/'-«^-^o. I (1)

The samples for the measurements have the form of rectangular
bars with faces coinciding with the "cleavage planes of the single
crystal. In order to remove internal stresses and the consequences
of mechanical treatment, the samples were subjected to a kneeling
at a temperature of 600°C with subsequent slow cooling. Particular
attention was given to the surface quality of the working faces of
the samples and the orientation relative to the planes {lOO). ^ In /23
recording the temperature dependences of the propagation velocities
of the ultrasonic oscillations the specimen was glued to the flaw
detecter sensor with a mixture consisting of 88^ epoxy resin and 12^
polyethylene polyamlne, and placed in a cryostat. The values of
the elastic constant C^, and Young's modulus E^p,„ were determined

from the well-known formulas

C^-Y^; C,^Y^,

(2)

where p is the density of the substance, determined for different
temperatures according to the formulas from [1]:

P(0 = 2.1680(1- 1,12-10-'^ — 5-10-8^-1 fori NaCi,
■ p(0 = 1*9920(1- 10.5- 10-5^-0,4.10-^^21 forlKCl,

(3)
(.4)

here t is the temperature in °C. Variations in the length of the
specimen l(t) in relation to temperature was calculated according

to the formula

/(0 = /o[l + J°«(0^f] I C5)

where a(t) is the coefficient of linear expansion, which, according
to Henglein [2] may be determined according to the formula

where p(t) and p are the densities respectively at t and 0°C.
In order to find the shear modulus, pulses of transverse waves at
a frequency of 2.5 mHz were sent into the specimen, using prismatic
search, heads with an angle of inlet of the ultrasonic oscillations
of 50°, which made it possible to transform longitudinal waves into
shear waves. In the general case, with the incidence of an ultra-
sonic wave onto a plane parallel plate at a certain angle to the
normal, both longitudinal and transverse waves arise in the Aatter.
At the chosen angle of incidence of the ultrasonic wave, complete
internal reflection of longitudinal wave takes place, and only the
transverse wave, oriented at a certain angle to the crystallogra-
phic axes, is propagated in the specimen. Taking account of this
angle made it possible to determine the orientation factor P(y5
[3]

where y are the direction cosines of the angles, formed by the
transverse wave propagation direction with the crystallographic
axes .

According to the measured values of C , we find the values of
the shear modulus G^^ in the direction of propagation of the shear

/

waves according to th.e formula

-=/?•

(8)

Calculation of the remaining elasticity constants was performed
with the use of the following equations:

" f.oo' " (S„-5„)(5„+2S,,)'

(9)

/24

C-»|2

s,,

(5„-S,,)(5„ +-25,,)

;

■-(1:1

c^ V •

2-2f-^V

• ''x = 3(S„-i- 25„).

Here a Is Polsson's ratio and k is the uniform compression co-
efficient .

Ex-crapolation of the experimental values of the low tempera-
ture' dependences of the elastic constants to 0°K gives the following
values :

For NaCl Caa

Si 2

11

11

5.625 • 10- dyne/cm ; Ci 2 - 0...973 v .10. . dvne/cra ^
1.310 • 10 dyne/cm ; Sii = 0.1784 • 10" cmVdyne^
— 0.0290 • 10- cmVdyne; S^i= 0.7368 • IQ-^cmVdyne;

For KCl Ci 1 = 4.950
C^n = 0.658
S12 =-0.020

1 1 2

10 dyne/cm ; Ci2= O.6IO
1 1 2

10 dyne/cm ;

1 1 2
10- cm /dyne; 844= 1.527

Sii= 0.2073

1 1 2
10 dyne/cm ;

_i 1 2
' 10 cm /dyne;

_i 1 2

10 cm /dyne.

/25

TABLE 1

TABLE 2

Ultrasonic Propagation

Velocity in a NaCl
■'• Single Crystal

Ultrasonic Propagation
Velocity in a KCl
Single Crystal

T'K

300
290
280
270
260
250
240
230
220
210
200
190
180
170
ICO
150
140
130
120

P

g/cm3

2J613
2,1639
2,1663
2,1687
2,1711
2, 1732
2, 1759
2, 1782
2,1806
2. 1829
2,1851
2, 1874
2. 1890
2,1919
2. 1943
2.1902
2. 1984
2,2005
2,2025

Propagation 'velocity
of ultrasonic
vibrations, m/sec

'to

4697
4710
4724
4737
4759
4771
4790
4803
4813
4829
4848
4800
4873
4882
4895
4909
4921
4935
4951

441(?

442/

4447'

4471

4492

4512

4551

4550

4574

4594

4615

4035

4002

4073

4093

4714

4731

4751

4772

2426

2426

2427 '

2427 j

2427 !

2427 I

2-127 j

2427 i

2427 I

2427 I

2427

2428

2428

2428

2428

2428

2429

2429

2429

T°K

P '

■g/cmS

(

/Propagation
of ultrason:
vibrations.

C/oo

Ci

c±

300

1.9864

4481

4402

1774

290

.1,9884

4501

4412

1776

280

,1,9906

4515

4424

1777

270

'1,9926

4527

4452

1778

200

1.9948

4550

4463

1780

250

1,9908

4562

4475

1781

240

1.9988

4581

4483

1780

230

2.0008

45'JS

4500

1781

220

2.0030

4G1C

4519

1780

210

2.0049

4C3I

4539

1781

200

2,0069

4010

4553

1781

190

2,0087

4G0O

•1509

17ii2

|80

2,0107

4072

■lot; 1

1784

170

2.0127

4GS3

4501

1785

160

2.0147

4 COS

40 12

1785

150

2.0165

4719

4020

1780

140

2.0185

4733

4043

1787

130

2.0203

4741

4058

1787

120

2.0221

4756

4003

1789

110

2.0241

4700

4083

1789

100

2,0259

4785

4090

1788

90
80

2,0277
'2.0297

4799
4811

4710
4734

1789
1790

Measurement of the elasticity constants are given in
Tables 1, 2, 4 and 5.

TABLE 4 ,

Elasticity Constants and Elasticity
Coefficients of NaCl

rK

2'/)

270
200
250
210
230
220
210
200
190
180
170
160
150
140
130
120

Elasticity constaints Elasticity

II , / 2 vx inn ■^ / n

xio-" dyne/cm

c..

Cu

•1,760

1,314

4. SOI

1.30G

4,831

1,298

1,853

1.291

■'i.OGtt

1/282

4.948

!.27l

4.901

1.200

5.025

1.252

5.052

1,244

5.0'Dl

1,233.

5,134

1.218

5,178

1.202

5.200

1.194

5.224

1.185

5.258

1.169

5.294

1,155

5,323

1.145

5.359

1,128

5.399

1.111

1,273

1.274

1.276

1,277

1,279

1.280

1.282

1.283

1.285'

1.286

1.287

1.289

1.290

1,292

1,294

1,295

1.297

1,298

1.299

::oefflcients

x>0"cm /dyne.

Su

0,2380
0,2357
0,2334
0.2306
0.2284
0,2263
0.2238
0.2212
0,2192
0.2169
0.2148
0.2127
0.2105
0.2088
0.2063
0.2049
0.2032
0,2012
0. 1992

-5„

0,0514
0,0504
0,0491
0,0431
0.0473
0,0464
0.0452
0.0443
0.0433
0.0423
0„0414
0.0402
0.0393
0.0386
0.0376
0.0367
0,0360
0,0350
0,0340

5u

0.7855
0,7849
0.7837
0,7831
0.7819
0.7812
0.7800
0.7794
0.7782
0.7776
0.7770
0.7756
0.7752
0.7740
0,7726
0,7722
0.7710
0,7702
0,7698

Tables 3 and 6 contain values as a function of time of the aniso-^

trophy factor a= (5,, - 5,2) - — S44I and a - C- ^" ~ ^'- I the uniform

2 \ .2 I

compression coefficients k, Poisson's ratio a and cubic elasticity

^ ^ £n.'^ '- ;, the measure of the resistance to deformation, ex-
pressed by the shear stress, applied in the plane {110} in the
direction < 100 > C = " — li-i and the ratio C^^f■C^2\, characterizing

the deviation from the Cauchy relation Ci^ — Cu, Prom Tables 3

and 6 it is obvious that the elastic anisotropy factor ', C^^•. — ^i— — *— /

of KCl and NaCl monocrystals Increases with an Increase in temper-
ature. Extrapolation of the experimental data to the fusing
temperature shows that the elastic anisotropy factor up to the ""
fusing temperature remains less than unity, that is, there is no
point of elastic anisotropy in a potassium chloride single crystal
which agrees with the data of references [4, 5].

TABLE 3

Elastic Properties, the ratio Ci,it/Ci2
and Anisotropy of NaCl

/26

C'X

c .

KX

"""X o

fiooX

1 .. .

1°K

xio-"^

x'«-"S"

eA

xio"""^^

,,dyne ,, cm'=^

xio-"^2rxio"^ —

3

cm

cm^i

dyne

cm^

dyne

300

1.72S

0.969

2.466

0.737

0.1033

4.201

0,4056

0,3181

290

1.747

0.976

2.471

0.729

0.1063

4,242

0.4047

0.3194

280

1.768

0.983

2,476

0.722

0.1090

4.285

0,4033

0,3207

270

1,788 •

0,989

2.484

0,714

0.1126

4.335

0.4020

0.3221

2G0

• 1,813

0.997

2,491

0.706

0,1152

4. 380

0,4014

0.3242

250

1,838

1,007

2.497

0.696

0.1179

4,422

0,4005

0.3254

240

1.867

1.018

• 2.501

0.687

0,1210

4,407

0.3993

0,3273

230

1.886

1.024

2.510

0,680

0,1242

4.520

0.3934

0.3286

220

1,901

1.033

2.513

0.675

0, 1266

'■ 4.561

0,3975

0,3295

210

1,929

1.043

2.519

0,667

0.1290

4.609

0.3969

0.3311

200

1,958

1.057

2.523

0.657

0.1323 ■

4,655

0,3903

0,3327

190

1.988

1.073

2,527

0.648

0.1350/

4.700

0.3957

0.3330

180

2,003

1.081

2,529

0.644

0,1378.

4.750

0.3054

0,3348

170

2,020

1.091

. 2.531

0.640

0,1396 •

4.788

0,3951

0.3357

160

2,044

1.107

2.533

0.633

0,1420

4.835

0,3948

0,3369

150

2.069

1,121

2,535

0,626

0,1445

4.SS0

0,3945

0,3381

140

2.088

1,132

2.539

0.621

0,1463

4.922

0.3939

0.33S9

130

2,115

1,150

2,539

0.614

0.1499

4,970

0,3939

0.3401

120

2.144

1,169

2.541

0,606

0.1517

5.020

0.3930

0.3409

TABLE 5

Elasticity Constants and
Elasticity Coefficients of KCl

(Elastl

jlty Constants

; Elasticity Coefflcieni

7"°K

X 10-

. dvney

cm

.X 10 \^ cm -/dyne

c„

c.,

Cu

Six

~Sn

5,.

300

3.989

0.725

0.625

0.2507

0.0152

1.000

290

4.028

0.724

0.027

0,2581

0.0113

1.595

280

4,000

0.724

0.629

0.2501

O.Oi.'O

1.590

270

4.095 .

0,723

0.630

0.2532

0.042G

1.587

200

4,130

0.723

0.632

0.2510

0,0118

1.582.

250

4.155

0.722

0.033

0,2500

0^0412

1.5S0

240

4,195

0.722

0.633

0,2488

0,0404

1,5.S0'

230

4.230

0,721

0.635

0,2409

0.0395

1.575

220

4.208

0.720

0,635

0,2445

0,0390

1.575

210

4,300

0.719

0,636

0,2421

0.0380

1.572

200

4,330

0,717

0.637

0,2404

0,0373

• 1.570

190

4.300

0.715

0,638

0.2387

0,0305

1.507

180

4.300

0,712

0,640

0,2307

0,0355

1.502

170

4.415

0,710

0,041

0,2353

0.0350

1.5G0

100

4.450

0.705

0,642

0,2333.

0.0340

1,558

150

4,490

0.704

0,043

0,2315

0,0335

1.555

140

4.520

0,700

0,044

0,2299

0.0330

1.553

130

4.512

0.092

0.645

0.2282

0.0.320

1.550

120 .

4,575

0.090

0.047

0,2273

0.0310

I.54G

110

4.000

0.085

0.018

0,2252

0.0305

1.543

100

4.040

O.OSO

0.048

0.2237

0.0295

1,513

90

4.070^

0.075

0.019

0.2222

0,0290

1.541

80

4.700;

0,070

0.650

0,2190

0.0280

1.538

The point of elastic anisotropy of NaCl single crystals is found
near the temperature 680°Kj which also agrees with the analogous
temperature measurements of [6, 71.

The Debye temperature of KCl and NaCl single crystals was
determined according to values of Ca i , C12, and C^I^ extrapolated
to 0°K- The calculation was performed according to the formula:

9^\ f- Y f — y 19/(18 + V3)] fiS, t)\ ,
4t:V'A k j \ P I (10)

/27

e?,=:

where S =

C,.

Cn + C

44

C,2 — C44

'44

; C., , V and p are respectively

fixe elasticity constants, molar volume and crystal density

determined for 0°K; N, h and k are the double Avogadro number

(for the halogens of alkali metals) and the Planck's and Boltzmann's

constants. The values of the function f(s, t) were determined /28

according to special tables [8, 9] with the use of Stirling's

interpolation formula and were equal to 1.4068 for NaCl and 2.1129

for KCl. The density of the substance at 0°K was determined

according to formulas (3) and (4) and the following results were

3 3

obtained: p„ „, = 2.2260 g/cm and p„„-, = 2.0432 g/cm . After
substitution of the values found in formula (10) 0^ = 321. 4°K and
0p = 244. 1°K were found for NaCl and KCl single crystals respect-
ively, which is in good agreement with data from heat measurement
[10, 11].

TABLE 6 . , /27

Elastic Properties, Ratio C4i/Ci2 and
Anisotropy of KCl

T

C'X

xio-"^^

KX

xio-"^

?"

-a X

xio"fai-^

dym

£ino X

xio-"-^

''^ 2

}5<io'- ^"^

dyne

Pols

300

1.632

0.8G2

1,813

0,383

0.4949

3.850

0.5516

0.4070

200

i.r,,-.2

O.RGG

1.825

0.379

0.4951

3,875

0.5480

0.4073

2S0

l.f",(".8

0.8G9

1.83G

0.377

0.4953

3.900

0,5446

0,4032

270

i.68r,

0,871

1.8-17

0.374

0,4975

3.950

0,5415

0.4087

2G0

1.703

0.874

1.859

0.371

0,4977

3.975

0.5379

0.4096

250

1.710

^0.877

1 .8GG

0.369

0.4988

4,000

0.5358

0.4101

210

1.7^0

'0.878

1.880

0.3G7

0,5008

4.020

0.5319

0.4111

230

1.755

0.881

1,890

0.3G2

0,5011

4.050

0.5291

0.4116

220

1. 77'!

0.882

1,903

0,358

0.5040

4.090

0,5255

0,4126

210

1.791

0,885

1.913

0.355

0,5059

4.130

0.5228

0.4132

200

1.80G

0.889

1.921

0,353

0.5073

4,160

0.5205

0.4147

son' s
patio

TABLE 6 Ccontinued]

Elastic Properties, Ratio Cm/Ci2
Anlsotropy of KCl

and

T

C'X

xio-"^

KX

xio--'^^^

190

1.822

0.892

1.930

0.351

180

1,829

0.899

1.938

0.350

170

1.852

0;902

1.945

0.346

160

1.872

0,910

1.953

0.343

150

1,893

0,913

1,9G6

0,340

140

1.910

0.920

- 1.973

0,337

130

1.925

0.930

1.975

0,334

120

1.943

0.937

1.985

0.333

110

1.957

0.946

1.990

0.331

100

1.980

9.953

2.000

0.327

90

1.997

0.9GI

2.007

0.325

80

2,015

0.9G9

2.013

0,323

— ax

XIO"

cm
dyne

„^„_

.0.5083

4.190 .

0.5088

4.225

0,5097

4.250

0.5117

4.285

0.5125

4.320

0.5136

4.350

0.5146

4.382

0.5148

4.400

5.5158

4.440

0.5183

4.470

0.5193

4.500

0.5212

4.550

£,00 X I •/• X

xio-"<^^#xio"/^^'

d dyne

0.5181
0.5160
0.5142
0.5120
0.5086
0.5058
0.5063
0.5038
0,5025
0.5000
0.4983
0,.4963

Polsson' s
rat:

pio

0,4148
0.4143
0.4150
0.4157
0.4164
0.4169
0,4173
0.4176
0.4180
0.4188
0.4193
0,4197

10

REFERENCES

1. International Critical Tables (McGraw-Hill Company, N. Y.),
V. 3, p. 43, 1928.

2. Hengleln, Pr. A., Zelt . f. Physlk Chemle, vol. 115, p. 91,
1925.

3. Byuren, V.: Defekty v Krlstallakh [Defect In Crystals],
Moscow, IL, vol. 27, 1962.

4. Nlknarov, S. P. and Stepanov, A. V.: FTT, vol. 4, 9,
p. 2576, 1962.

5. Norwood, M. N. and Briscoe, C. V.: Phys . Rev., vol. 112,
p. 45, 1958.

6. Stepanov, A. V. and Eydus, I. M. : ZhETF, vol. 29, pp. 5, 669,
1955.

7. Overton, I. and Swim, R. T. : Phys. Rev., vol. 84, p. 758,
1951.

8. DeLaunay, J.: Chem. Phys., vol. 30, p. 91, 1959.

9. Botakl, A. A'., Popova, Yu. Ya., Suslova, V. N. and Sharko, A. V.
Izv. , TPI (In print) .

10. Kittel, Ch. : Vvedeniye v fiziku tverdogo tela [Introduction to
Solid State Physics], Izd. AN SSSR, Moscow, 1962.

11. Zeytts, F. : Sovremenaya teoriya tverdogo tela [Contemporary
Solid State Theory], GITTL, Moscow, 1949-

11

STANDARD TITLE PAGE

1. Report No.

NASA TT F- 14, 932

2. Government Accession No.

3. Reclpient;s Catalog No.

4. Title and Subtitle

TEMPERATURE DEPENDENCE OF ELASTIC CONSTANTS AND
THE DEBYE TEMPERATURE OF KaCl AND KCl SINGLE

&. Report Date

June 1973

6. Performing Organization Code

7. Author(si

A. V. Sharko, A. A. Botaki

8. Performing Organization Report No.

10. Work Unit No.

9. Performing Organization Name and Address

LINGUISTIC SYSTEMS, INC.

116 AUSTIN STREET

CAMBRIDGE, MASSACHUSETTS 02139

11. Contractor Grant No.

IiASW~2k82

13. Type of Report & Period Covered
TRANSLATION

12. Sponsoring Agency Name and Address

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON, D C. 20546

14. Sponsoring Agency Code

15. Supplementary Notes

Translation of: "Temperaturnaya Zavisimost ' Uprugikh Postoyannykh i
Temperatura Detaya Monokristallo-v NaCl i KCl," Izuestiya-Vysshikh
Uchebnykh Zavedenii, Fizika; vol. 13, no. 6, 1970, pp 22-28.

16. Abstract

The ultrasonic pulse method was used to measure the temperature

changes, elasticity moduli, the constants Cv, and S. , the anisotropy

Ik Ik

factor, the uniform compression coefficient arid Poisson's ratio in the

temperature range 300 - 120°K for NaCl and 300 - 80°K for KCl. According

to the values of the elastic constants, extrapolated to 0°K, the Debye

temperature of the compounds is found.

17. Key Words (Selected by Author(s))

18. Distribution Statement

UNCLASSIFIED ■ UNLIMITED

19. Security Classif. (of this report)
UNCLASSIFIED

20. Security Classif. (of this page)
UNCLASSIFIED

21. No. of Pages
11

22. Price

/