Acta Physica Academiae Scientiarum Hungaricae, Tomus 30 (3), pp. 231--240 (1971)

PHONON FREQUENCY DISTRIBUTION FUNCTIONS OF COPPER, NICKEL AND VANADIUM

By

JAI PR A K A SH , B. S. SEMWAL and P. K. SHARMA

PHYSICS DEPARTMENT, UNIVERSITY OF ALLAHABAD, ALLAHABAD, INDIA

(Received 28. IX. 1970)

The frequency distribution funetions of phonons in copper, nickel and vanadium have been determined by root sampling technique using BHATIA and HORTON'S model of eleetron- ion interaction. The results ate compared with curves obtained from inelastie neutron scatter- ing experiments and other theoretieal ealculations. The calculated distribution for copper agrees in its broad features with the experimental spectrum of SVENSON et al. obtained by neutron seattering method. In the case of nickel, a reasonably satisfactory agreement is obtained wi th the frequency distr ibution measured by M o z r a et al. in incoherent neutron scattering experiments. For vanadium, the present as well as other theoretical ealculations show consi- derable discrepancies with experiment. The lattice specifie heats and the equivalent Debye temperatures obtained from the calculated frequency distributions agree reasonably well with the experimental data for copper and nickel, but not in the case of vanadium.

I. Introduction

The frequency distribution function G(co) of the normal modes of vibra- tion of crystals is a quanti ty of considerable importance in the study of their many thermal and transport properties. During the last few years there has been considerable interest in the experimental study of lattice vibrations in metals through the measurement of frequency - - wave rector dispersion rela- tions with the help of inelastic neutron scattering technique [1]. Analyses of these dispersion relations by means of the Born - r o n K•225 model have pro- vided information about interatomic force constants which in turn has been used to obtain the frequency distribution functions. For certain metals like nickel and vanadium, which scatter neutrons primarily incoherently, G@) has been directly measured by incoherent inelastic neutron scattering experiments. Ir would be interesting to consider the theoretical side of the problem in the light of a suitable lattice dynamical model.

h is now well established that conduction electrons in metals consider- ably modify their vibration frequencies and these are responsible for the failure of the Cauchy relation. BnATXA [2] and BaATIA and I-IoaTON [3] have propounded an elastic force model for studying the phonon frequencies of cubic metals by considering the ion -- electron interaction through the screen- ing of the long-range Coulomb forces between the ions. The ion--ion interaction is described by the first two terms in a Taylor expansion of the potential energy. The model has provided a plausible description of vibration spectra

1 Acta Physica Academiae Scientiarum Hungaricae 30, 1971

2 3 2 JAI PRAKASH et al.

and heat capacities of alkali metals [2, 4] (Li, Na, K) and noble metals [3, 5] (Ag, Au). I t has also been used to explain the temperature variation of thermal expansion [6], eleetrical and thermal resistivities [7], and the x-ray D e b y e - - Waller factor [8] of a number of cubic metals. Recently SANGAL and SHARMX [9] have studied the lattice vibrations in transition metals of body-centred cubic structure on the basis of this model by considering the second-neighbour interionic interactions.

In the present paper we r e p o r t a computation of the phonon frequency distribution functions of eopper, nickel and vanadium on the basis of the BHATIA and HOaTO~ [2, 3] model and compare the results with measured distributions and with those obtained by other models. Earlier work [10--12] on the frequency distribution of vanadium using force models does not show satisfactory agreement with experiment. Ir was thought worth while to examine this metal in the light of BHATIA'S model. From the computed distributions, the lattice specific heats of these metals ate evaluated and compared with available calorimetric data.

II. Secular determinant

The secular equation for the detcrmination of angular frcqueneies co of the normal modes of vibration in a cubic metal can be written as

lO(q) - - Mco 2 I] = 0, (1)

where M is the mass of the atom and I is the unir matrix of order three. In BHATIA and HORTO~'S model [2, 3], the elements of the dynamical matrix D(q) are given by

Di(q) = 40:1(3-~ C 1 C2- -C 2 C3- -C 3 C1) "~

-~40:2(2-C iCj C iCk) -~-

+ 2 K e f - 1 a 3 q2,

D;j(q) : - 40:2 S i S i �91 2 K e f -1 a 3 qi q j ,

( fcc) (2a) and

Dii(q ) = 8(0:1�91 (1 --C a C 2 C3) -~

+ 4 K e f -1 a 3 q~,

D;j(q) : - 80:2 S i S i C K � 9 1 -~ a 3 qi q j , (2b)

(bcc)

f = l A - K ~ q ~ (3) 4:r e ~ n 2

where

.Jlcta Physica Academiae Sci~ntiarum Hungaricas 30, 1971

PHOI~ON FREQUENCY DISTRIBUTIOtN FUNCTIOI~S 2 3 3

Here q, is the q Cartesian componen t of the phonon wave vec to r q; n is the numbe r of electrons per unir volume; a i s the semi-lattice pa ramete r ; ~: and ~2 ate the force constants for the f i rs t -neighbour ion- - ion in teract ion; and the pa rame te r ICe arises f rom e l e c t r o n - i o n interact ion. A comparison of the long-

- - th r curvr - - - - Jacobsen" $ curve f r o m x - r a y

cxpcr i mcnts . . . . Sinha's curvr f rom ~"-: nculron )

scat te r ing delta 60 - - - " - - S v e n s s o n e l al . curvr f r o m 4

ncu t ron sca t t c r i ng "~ i ex pr m r nts 'ii

�9 ii

�9 ~- , x.."-;l~ : ~

C ~~ H

0 i ~ f , L,

t 2 3 �91 xlO ~l f r c q u c n c y co ir) r a d / s r

F i g . 1. The phonoa frequency distribution function of copper

wave limit of Eqs. (2) with the elastic m a t r i x for a cubic crys ta l yields the following relat ions between elastic constants and force constants :

and

1.

~~ = ( a / 2 ) ( C : : - - C ~ 2 - - C ~ ~ ) ,

~ 2 = - - a ( C n - - C : 2 - - 2C44),

K~ =- (2C:: - - C,2 - - 3C,,) , ( f c c ) (4a)

~ : = ( a / 2 ) ( C ~ ~ - - C12),

zr z = - - ( a / 2 ) ( C : : - - C~2 - - 2C4s),

K ~ = C1~ - - C 44 . ( b c c ) (4b)

.Acta Physica Academiae Scientiarum Hungaricae 30, 197I

234 JAI PRAKASH et al.

II l . Numerica l computat ion

The computa t ion of the phonon f requency dis t r ibut ion functions for copper, nickel, and vanad ium has been made by root sampling technique for a discrete subdivision of wave r e c t o r space [13]. In order to get a fairly reason- able su rvey of frequencies, the reciprocal space w a s divided into minia ture eells with axes one-fort ie th of the length of the reciprocal lat t ice cell. Using

Table I

Values of constants used in the calculation

Metal

Copper Nickel Vanadium

C11

17.620 24.60 22.795

Elastic c o n s t a n t s (10 u dynes/cm i)

I C~ t t

12.494 15.00 11.870

C.

Lattice parr

8.177 3.603

12.38 3.524

4.255 3.028 I

Density (gro/cm')

9.018 8.91 6.022

s y m m e t r y considerations, v ibra t ion frequencies were determined from the roots of secular Eq. (1) at nonequiva len t points (1686 points for copper and nickel, and 1661 points for vanad ium) lying within the 1/48th irreducible p a r t of the f i rs t Brillouin zone. Each f requency was weighted according to the symmet r ica l ly equivalent points , which gave 192,000 frequencies in the whole Brillouin zone. T h e n u m b e r of frequencies falling into intervals 3co = 0.05 X 1013 rad/sec were counted and f rom these the his togram giving the f requency dis- t r ibu t ion was constructed. The calculated f requency distr ibut ions ate dis- p layed in Figs. 1 - -3 along wi th exper imenta l curves and other theoret ical calculations. The curves are drawn with a rb i t ra ry units for G(o~), b u t are nor- malized to the same atea. The numerical values of the elastic constants and other pa ramete r s used in the calculations ate given in Table I. The elastic constants of copper refer to 0~ and ate values ex t rapo la ted from the measure- ments of OVERTO• and GAFFNEY [14] at 4 .2 ~ For nickel and vanad ium the elastic eonstants refer to room tempera tu re and are t aken f rom the mea- surements of DE KLERK [15] and ALERS [16], respect ively.

Using the computed f r equency distr ibutions, the cons tan t volume speci- fic heat Cv per grato a tom was calculated in the usual manner b y numerical in tegrat ion. In Fig. 4 the calculated C~ ate compared with exper imenta l values. A more sensit ive comparison wi th calorimetric da t a is made in terms of the Debye t e mpe ra tu r e O. The O versus T curves for copper and nickel are dis- p layed in Figs. 5 and £ along wi th empirical data.

Acta Physiea Ar Scierttiarum Hungarieae 30, 1971

PHONON FREQUENCY DISTRIBUTION FUNCTIONS 235

IV. Discussion

A. Frequency distribution

For convenience we discuss separately the results for the three metals. (1) Copper. The frequency distribution of the normal modes of vibra-

tion in copper has been experimentally obtained both through diffuse seatter- r

t q Curve from Bhatia's model Chr et al. curvr frorn

75 neutron expcriments Mozr r al. curvr f r o m nr rorl scQttr da ta

[ ........ B i rger lcQu r 1 6 2 D / f rorn n r scattr ti

45 .! i

"9~ ." ["

x - ~~!>~..".L .4',<. i'.ir,

o I , . : f l vI - : l i l \ '~1- /::-.�91 l" l i ~ ',,

/ " / . "~

1 2 3 4 5 "6 xl013 frr ca in rod I sec

Fig. 2. The phonon frequency distribution function of nickel

ing of x-rays and slow neutron spectroscopy. JACOBSEN [17] has determined the frequency distribution of copper by fi t t ing the third-neighbour Born-- ron K•225 force constant model to the frequency versus wave vector dispersion relation obtained from bis diffuse x-ray scattering measurements at room temperature. Too much reliance cannot be bestowed on bis result, however, because of the inherent uncertainty in diffuse x-ray experiments. Str~nA [18] and SVENSSON et al. [19] have carried out reliable calculations of the frequency distribution of copper using sixth-neighbour general force eonstant models determined from the analysis of phonon dispersion relations measured at room temperature by neutron spectrometry. These distributions ate shown in Fig. 1. The neutron scattering results agree broadly in the location and intensity of major peaks, but differ appreciably from those obtained by JACOBSEN on the high frequency side. A perusal of Fig. 1 shows that our calculated spectrum

Acta Ph.ysir Academiae Scientiarum Hrtngaricae 30, 197J

2~• JAI PRAKASH et al.

is in satisfactory agreement with tha t of SVENSSON et al. The major peaks and the high frequeney end appear at the same position in both the spcctra.

(2) Nickel. Information about the frequency distribution of lattice vibrations in nickel has been obtained directly by CHERI~qOPLEKOV et al. [20] and MOZER et al. [21] using incoherent inelastic neutron scattering techniques. These data are plotted in Fig. 2 together with the distribution calculated by BIRGENEAU et al. [22] from the fourth-neighbour Born-- von K•225 model f i t ted to their frequency versus wave factor dispersion relations measured by neutron spectroscopic method. For normalization, the eut-off frequency of the experimental curves is taken as 6.84 • 1013 rad/sce. I t will be scen that our spectrum qualitatively resembles the experimental curve of MOZER et al. The major peaks in the ea]culated and experimental curves are found to be at nearly the same positions. The heights of the calculated peaks are greater than the experimental values, but their form is similar to tha t of the experimental curve. Our calculated spectrum shows striking simi]arity with that of BIRGE~EAV et al. with rcgard to the location and the intensity of peaks.

(3) Vanadium. EISENHAUER et al. [23] and MozEtt et al. [21] have obtained the frequency distributions of vanadium from incoherent inelastic neutron scattering experiments. Thcse distributions are shown in Fig. 3 along with the curves caleulated by I-IEiNDRICKS et al. [24] using a noneentral three- force-constant mode], and by CLARK ct al. [25] using a noncentral four- constant model. Becausc of uncertainty in the upper end of the experimenta] curves, we have arbitrarily cut them off at 5.65 X 1013 rad/sec for normalization.

A perusal of the various frequency distributions plottcd in Fig. 3 rcveals several striking fcatures. The calcu]ated and the experimental distributions differ with regard to the position and intensity of peaks. The calculated curves show two distinct widely spread maxima with a sharp dip between them, the higher frequency peak having ]ess intensity than the low frcquency one, while the experimental eurves have f]at maxima with a small dip between them and the high frequency peak is more intense than the low frequency peak. The maximum frequency of our calculated distribution is in fair agreement with experiment though the experimental value is not well defined. An overall eomparison of the various computed frequency distributions reveals that they all resemble each other in their broad features, but none of them agree with the experimental curves. Our calculated curve is, however, somewhat nearer to the experimental curve with regard to the location of the dip and the high frequency peak. I t appears from the present study tha t none of the models is a satisfactory description of vanadium.

B. Specific heat

The experimental data for the specific heat of eopper below 20 ~ have been taken from the work of Kox and KEESOM [26] while those for above

Acta Physica Acaderniae Scientiarttm Hungaricae 30, 1971

PHONON FREQUENCY DISTRIBUTION FUNCTIONS 237

20 ~ f rom papers by GIAUQUE and MEADS [27] and MARTIN [28]. The la t te r two measurements ate inconsistent and v a r y within wide limits. In the case of nickel, the exper imenta l hea t capacit ies have been obta ined f rom the measurements of EUCKEr~ and WEUTH [29] and BUSEY and GIXUQUE [30], those for vanad ium f rom the measurements of CLusIus et al. [31]. The specific heats of nickel repor ted by the different workers are fairly eonsis tent in the

75

- - p r c s r c a l c u l a t i o n - - - H c n d r i c k s r o. l .curvr - - - - - C l a r k ~t al. c u r w ........ e x p e r i m e n t a l c_urv~ d u e t o

E i s r r al. . . . . r 162 cu rv r dur t o

6C

r

4~

.o

�9 ~- 3(

3

~D

L 1 2 3" 4 5 xlO ~3

F r e q u c n c y �91 i n - r a d l sr

Fig. 3. The phonon frequency distribution function of vanadium

t empera tu re range studied. The exper imenta l specific heats have been p lo t ted af ter correction for the electronic cont r ibut ion . The values of the coefficient of the electronic specific heat , 7, used for this purpose, ate given in Table I I .

I t will be seen f rom Fig. 5 tha t the theoret ical and exper imenta l 0 - - T curves for copper are similar in shape, a l though the theoret ical values are sys temat ical ly higher. However , the discrepancy between t h e m is nowhere greater than 30/0 . Be tween 70 and 130 ~ the theoret ical O values lie ve ry close t o the measurements of MARTIN. Ir is a par t icular ly str iking feature of

Acta Physica Academiae Seientiarttm Hungaricas 30, 1971

238 JAI PRAKASH et. al;

Table II Eleetronie specifie heat eoeffieients for copper, nickel and vanadium

Metal 7 (~cal/mo]r deg z) Source

Copper

Nickel

Vanadium

172

1740

1550

R A y N E a)

CLUSIUS a n d SCHACHXN6En ~)

CLUSIUS e t a l . c)

a) J. RAYNE, Phys. Rev., 95, 1428, 1954. b) K. CLusIus and L. SCHACHIN6ER, Z. Naturforsch., 7a, 185, 1952. c) K. Cr.usius, P. FRANZOSINI and U. PIESBERG~N, Z. Naturforsch., 15a, 728,

1960.

E

~ / j ~4

t~

c~3

t~

> 2 r

1

0 ^ ' ~ ~ ~ J ' I I I I I

50 100 150 200 tCmpr ~

Fig. 4. Comparison of the ealculated and observed lattice speeifie heats of eopper, nickel and vanadium. Experimental data: �9 eopper; �9 niekel; A vanadium

the theore t ica l curve t h a t the m i n i m u m appea r s to h a v e abou t the r ight dep th and posi t ion.

As is seen in Fig. 6, the e x p e r i m e n t a l O values for nickel lie ve ry close to the expe r imen ta l values. The difference be tween the observed and theo~etical values nowhere exceeds 4~ For v a n a d i u m , the theore t ica l and expe r imen ta l O values revea l considerable discrepancies: the expe r imen ta l values being 30~o higher t h a n theore t ica l ones. These discrepancies m a y be a t t r i bu t ed to the neglect os the t e m p e r a t u r e dependence of elastie cons t an t s and o ther anhar - monic effects, due in pa r t to the a s sumpt ion of shor t - range interionic in ter- actions in the theory . Using n e u t r o n sca t ter ing exper iments , TURBERFIELD and

Acta Physica Academiae Scientiarum Hungaricae 30, 1971

PHONON FREQUENCY DISTRIBUTION FUNCTIONS 2 3 9

EGELSTAFF [32] h a v e o b s e r v e d t h a t t h e f r e q u e n c y s p e c t r u m o f v a n a d i u m con-

t a i n s a t a i l a t h i g h c r f r e q u e n c y . T h e y h a v e a lso f o u n d t h a t t h e s p e c t r u m does

n o t o b e y t h e u s u a l D e b y e r 2 l a w in t h e l o w e r f r e q u e n c y reg ion . T h e s e a b n o r m a l

b e h a v i o u r s i n d i c a t e t h a t t h e r e is s o m e t h i n g spec i a l a b o u t t h e l a t t i c e d y n a m i c s 0~o 420

o

400 + .~ o + o o 4- o + + + , ~ . + o ~ ~ +~ 4 " + +

o o o o

380 ' [ ~ I I I ' I 50 100 150 200

~r ~

Fig. 5. The Debye temperature of eopper a s a function of temperature. Solid line shows the present calculation. Experimental points: O KOK and KEESOM, X GIAUQUE and MEAI)S,

~- MARTIN

330 ~ ~

"~* ~ ~

o 320 ~ |

X 44+ X 310 ~ •

0 I I ' I f 1 ' I = 40 80 120 160

tcmpcra~urr ~ Fig. 6. The Debye temperature versus temperature curve for nickel. Solid line represents the present calcu]ation. Experimental points: �9 EUCKEN and W E R T H , -~ B U S E Y and GIA+JQOE

of v a n a d i u m . I t e m e r g e s f r o m t h e p r e s e n t s t u d y t h a t BHATIA a n d HORTON'S

e l e c t r o n gas m o d c ] g ives a r e a s o n a b l e r e p r e s e n t a t i o n o f t h e r m a l p r o p e r t i e s o f

c o p p e r a n d n icke] , b u t n o t t h o s e o f v a n a d i u m .

A c k n o w l e d g e m e n t s

The a. thors are thankful to the Council of Scientific and Industrial Research for finan- cial assistance. One of us (B. P.) is thankful to the University Grants Commission for the award of a Senior Researeh Fellowship. The Work was partly supported by the Department of Atomic Energy, Bombay.

.4cta Physica Academiae Scien|iarum Hur~garicae 30, 1971

2 4 0 JAI PRAKASH et al.

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Acta Physica Academiae Scientiarura Hungaricae 30, 1971