IL NUOVO CIlVIENTO VOL. 21 B, N. 1 11 lViaggio 1974

Temperature Variation of the Electrical Resistivity of Alkali Metals.

~ATYA PAL

Physics Depa?'tment, Allahabad Unive?'sity - Allahabad

(ricevuto 1'11 Maggio 1973)

Summary. - - The t empera tu re var ia t ion of the electr ical resis t ivi ty of alkali metals has been s tudied within the free-electron approximat ion on the basis of the la t t i ce dynaznical model of Cheveau. The var ia t ional solution of the Boltzmann t r anspor t equat ion as developed by Zimau has been incorpora ted in the present study. The exper imenta l values of the elastic constants of alkali metals have been used ~or the calcula- t ion of the force constants appear ing in the secular equation for the la t t ice vibrations. The phonon spect rum has been computed with the help of the modified Houston's spherical s ix-term procedure as e labora ted by Betts e~ al. A dist inct ion between the normal processes and the Umklapp processes contr ibut ion to the to ta l e lectr ical resis t ivi ty has been made in a more real is t ic manner in the s tudy. The theoret ieMly c~lculated values of the electr ical resis t ivi ty of the a lkal i metals, except for l i thium, are in reasonable agreement with the exper imental data.

1 . - I n t r o d u c t i o n .

One of t h e b a s i c t a s k s of t h e s o l i d - s t a t e t h e o r y has b e e n to a c c o u n t for t h e

t r a n s p o r t p r o p e r t i e s of m e t a l s . T h e p r o b l e m h~s b e e n t h e s u b j e c t of con-

s ider~b l6 i n v e s t i g a t i o n s ince 1928 w h e n BLOCH (1) f i rs t t h e o r e t i c a l l y a c c o u n t e d

for t h e q u a l i t a t i v e f e a t u r e s of t h e r e s i s t i v i t y of me tMs . The m o s t i m p o r t a n t

l a n d m a r k in t h e s t u d y of t h e t r a n s p o r t t h e o r y is t h e c]assicM c a l c u l a t i o n s of

BARDEEI~ (2) w h i c h c o n v i n c i n g l y d e m o n s t r a t e d t h e v a l i d i t y o~ t h e n o w con-

v e n t i o n a l B l o c h - W i l s o n t h e o r y of e l ec t r i cM c o n d u c t i v i t y of me tMs . The s t u d y

(1) F. BLOGH: ZviSs. Phys. , 52, 555 (1928). (2) J . BARDEEN: Phys. Rvv., 52, 688 (1937).

195

19~ SATYA PAL

of the t r anspor t proper t ies of solids r emained obsenred for a considerable t ime due to inadequa te knowledge abou t the an iso t ropy of the lat t ice spec t rum and the electronic band s t ructure . I t was a lmost three decades la ter t h a t BAILYN (~) took into account the detai led iohonon spec t rum for calculat ing the electrical res is t iv i ty of alkali metals . B y this t ime sufficient informat ion abou t the phonon spec t rum and the electronic b a n d s t ruc ture had ga thered and this paved the way for a renewed ~ t taek on the p rob lem of the t r anspor t phenomenon in solids. Considering the e leetron-phonon in teract ion explicitly, COL~INS and Z~MA~ (4) calculated the t r anspor t coefficients of alkal i metals a t low tempera tu res . However , t hey did not take account of the detai led phonon spec t rum and ins tead resor ted to the simple Debye model for the la t t ice vibrat ions. Bi~oss and HoLz (~) and I-IASEGAWA (~) carr ied ont the calculat ions of electrical res is t iv i ty of alkal i meta ls b y tak ing into account bo th the elec- tronic band s t ruc ture and the an iso t ropy of the lat t ice spect rum. D ~ B Y and MA~CK (7) employed the exper imenta l phonon dispersion curves of sodium for the calculat ion of its resis t ivi ty . BAu (8) developed a m e t h o d to calculate the res is t iv i ty wi th the help of an exper imenta l ly measured dynamica l s t ruc ture factor. The me tho d enables one to include all mul t iphonon processes, Debye- Waller factors and the U m k l a p p processes in the electron-lat t ice in teract ion wi thout hav ing to use a phonon descript ion of lat t ice vibrat ions. However , because of the inadequa te knowledge of the s t ruc ture factor BAY~ conld not car ry out the detai led calculat ions for a n y meta l . GREENE and KO~N (9) have made an in teres t ing s tudy of the electrical res is t iv i ty of sodium. Al though the calculation is bel ieved to incorporate the m a n y - b o d y effects, the Umklapp - processes, t ime-dependen t effects and the dynamics of ions ~rom the nen t ron spec t romet ry da ta , the ag reement wi th exper iment is disappoint ing. The a d v e n t of the reasonable pse~dopotent ia ls and the s~ructure factors s t imula ted a fresh interes t in the s tudy of the t r anspo r t proper t ies of metals . An interes t ing a t t e m p t to ealenlate the t r anspo r t coefficients of alkal i metals has been made b y I~o~I~SO~ and D o w (~0). They ex t rac t the dynamics of ions f rom a phenom- enological dynamica l m a t r i x f i t ted to elastic constants and the electron-ion in terac t ion f rom the Animalu, Hei~te and Aba renkov pseudopotent ia l . Although the calculations allow for a detai led assessment of the anisotropy, the agree- men t be tween theoret ica l and exper imenta l resnlts is d is turbingly poor. DYNES

(3) iY[. BAILYN: Phys. t~ev., 112, 1587 (1958); 120, 381 (1960). (a) J. G. COLLINS and J. ~r ZI~AN: P~'oc. Roy. Soc., &264, 60 (]961). (5) H. BR0SS and A. HOLZ: Phys. Star. golidi, 3, 1141 (1963). (6) A. HASEGAWA: Jour Phys. Soc. Japan, 19, 504 (1964). (v) J. K. DAI~Bu and N. H. I~IAa~eH: Proc. Phys. Soc., 8~, 591 (1964). (s) G. BAu Phys. Bey., 135, h 1691 (1964). (9) ~r P. G ~ E mud W. Ko~s: Phys. Rev., 137, h 513 (1965). (10) J. E. I:~OBINSON and J. D. Dow: Phys. t~ev., 171, 815 (1968).

T E 1 H P E R A T U R E V A R I A T I O N O F TIt:E E L E C T R I C A L R E S I S T I V I T Y O F A L K A L I ~ E T A L S 197

and CAI~BOTTE (11) have ex tended the work of BAY~ (s) by using the dynamics of ions f rom the neut ron spec t rometry data. Their calculated electrical resistivities are in good agreement with the exper imental da ta in the low-temperature region. The calculations differ f rom those of GREENE and KOHN (9) in the t r e a tme n t of the electron-ion par t only. While GREENE and KOH~ use a phase- shift analysis with the Friede] sum rule to obtain the electron-ion scattering cross-section, DYNES and CA~ROTTE rely on the pseudopotent ia l theory. Re- cent ly HAYMAN and CARB0~TE (~) have extended and improved the work of DYNES and CAlC]~0TTE (~1) by including the effect of volume changes through thermal expansion. Their calculations of the electrical resis t ivi ty of the alkali metals sodium~ potassium and rubid ium compare well with the exper imental da ta over a wide t empera tu re range. :Recently I~ICE and SHA~ (la) have em- ployed differen~ pseudopotentials and the phonon frequencies obtained f r o m the neu t ron inelastic-scattering studies for the calculation of the low-tem- pera ture resist ivi ty of potassium. I~ICE and SHA~ obtain the resist ivi ty ex- pression in terms of the re laxat ion t imes f rom the Landau t ranspor t equat ion by /(ohler~s (1~) var ia t ional method. The relaxat ion t imes and the resist ivi ty expression as obta ined by RICE and SHA~ are more or less the same as the ones usual ly obta ined f rom the Bol tzmann equation wi thout the quasi-particle interact ion. Following the t r e a tmen t by RIcE and SEA~ (13), EKI~ (~5), T~O- I~I1VIENI(OI~I~' and EKIN (is) and EKIN and MAXFIELD (1~) have also calculated the low-temperature resis t ivi ty of potassium. They have calculated the elec- tr ical resis t ivi ty using several different pseudopotentials and have explicit ly separated the to ta l resis t ivi ty into the normal and Umklapp components. They obta in the resis t ivi ty expression f rom the Kohler first-order var ia t ional solution of the Bol tzmann t ranspor t equat ion with the s tandard tr ial fnnct ion (~s). I{AVE~ and WISE~ (1~) have also repor ted a theoret ica l evaluat ion of the low- tempera tm'e electrical resis t ivi ty of potassium. K av eh and Wiser 's calculations are based on the weak-coupling theory for describing the electron-phonon in teract ion and the Bol tzmann equation for describing the t ranspor t theory.

l%ecently Ct{]~VEAU (:0) has proposed a model for the latt ice dynamics of metals which satisfies the symmet ry requirements of a cubic latt ice. During

(11) ~R. C. DYNES and J. P. CAI~BOTTE: Phys. Roy., 175, 913 (1968). (12) B. HAYMAN and J. P. C~BOTTE: Canad. Journ. Phys., 49, 1952 (1971). (is) T. :~/I. RICE and L. J. SHA~: Phys. Roy. B, 1, 4546 (1970); 4, 674 (1971). (1~) M. KOHLE~: Zeits. Phys., 124, 772 (1948); 125, 679 (1949). (15) j . W. EKIN: Phys. Rev. Lett., 26, 1550 (1971). (16) T. ~. TROFIlW~ENKOFF and J. W. EKIN: Phys. Rev. B, 1, 2392 (1971). (17) j . W. EKIN ~nd B. W. ~r Phys. /~eq/. B, 4, 4215 (1971). (18) j . M. Zn~AI~: Electrons and Phonons, Chap. IX (Oxford, 1960). (19) M. KAVE~I and N. WISER: Phys. t~ev. Lett., 26, 635 (1971). (20) L. CH]~VEAU: Phys. Rev., 169, 496 (1968).

1 9 8 S ~ A VA~

the last two decades, a l though ~ n u m b e r of models (2~-.~) ~or the s tudy o~ la t t ice dynamics o~ cubic meta ls have been proposed, none could s tand a detai led crit icism, as t h e y nll suffered f rom one d rawback or ~nother. The Ch6veau model has been successfully employed for the e~lculations of the phonon dis- pers ion curves (20), the Debye-Wal]er factors (~) and the Griineisen p~rame- ters (2s) of a n u m b e r o~ cubic meta ls . I n v iew of the success o~ the model , i t w~s t h o u g h t of in teres t to ex t end it to the calculations of some other lat t ice dynamica l proper t ies of solids. I n the presen t s tudy, the au thor h~s incorpo- ra ted ~he Ch6veau model (sg) to the s tudy of the t e m p e r a t u r e dependence of the electrical res is t iv i ty of ~]k~li metals . As such, the results of calculations of the t e m p e r a t u r e va r ia t ion of the electrical res is t iv i ty of l i thium, sodium, potass ium, rub id ium ~nd cesium wi th in the free-electron approx imat ion on the b~sis of the l~ttice dynamica l model of C~]~VEAV are presented herein.

2 . - T h e o r y .

I n recent years , var ia t iona l calculat ions of the t r anspor t proper t ies of meta ls have been in vogue because of the ~pparen t s implici ty and physical directness. The var ia t iona l solution of the Bol tzm~nn equat ion has been ob- ta ined by Kon-LEt~ (1~) and SO~DI~EIMEI~ (29) and is clearly set for th b y ZI~[A~ (~s) f rom whom the general formula t ion has been borrowed. For a la t t ice of cubic s y m m e t r y the var ia t iona l expression for the electrical res is t iv i ty can be writ- ten a s (18)

(1) ~L : (1/k~ T)]~[~be - - ~b~,]2 P(kq, k') dk dk' dq

er ~1~ 2 v~b~ ~E~ dk )

where e is the electronic charge, k B the Bo l t zmann constant , k and k ~ are the wave vectors of the ini t ial and final electron states, P(kq, k ~) is the sca t te r ing probabi l i ty t h a t an electron in s ta te k is sca t te red to a s tate k ' wi th absorpt ion of a phonon of wave vector q, vk is the veloci ty and Ek the energy of the pe r tu rbed dis t r ibut ion of the fo rm

/~ = / o _ ~ ~ "

(21) j . ])~ LAU~AY: Journ. Chem. Phys., 21, 1975 (1953). (28) A. B, BHATIA: Phys. Rev., 97, 363 (1955). (28) A. B. BHATIA and G. K. HO~TO~: Phys. t~ev., 98, 1715 (1955). (2r p. K. SHAI~)~A and S. K. ffosHI: Journ. Chem. Phys., 39, 2633 (1963); 40, 662 (1964). (25) K. K~]~]3s: Phys. Rev., 138, A 143 (1965). (2o) p. K. SHAR~A and N. SING~: Phys. t~ev. B, 4, 4636 (1971). (27) p. K. SEAn~A and N. SI~GH: Phys. ~ev. B, 3, 1141 (1971). (2s) p. K. SHAR~A a~d N. SI~GH: Phys. Rev. B, 1, 4635 (1970). (29) E. H. S 0 ~ D H ~ I ~ : P~'oc. Roy. Soc., &203, 75 (1950).

T:E~I)]~I~ATI~I~E VARIATION 01 ~ THE ELECTI~ICAL RESISTIVITY" O:F ALKALI ~:ETALS l ~

The probabi l i ty of occupancy a t the equi l ibr ium is given by the Fe rmi funct ion

0 _ [ e x p - ~ ) / k ~ T ] ]i~- [(J~ -[- lJ -1.

Now for a la t t ice of cubic s y m m e t r y , the use of the t r ia l funct ion +~ = k. u, where u is a un i t vector in the direct ion of the applied electric field~ leads to the first-order var ia t iona l solution of the Bo l t zmann equat ion to be wr i t t en as the four-dimensional in tegra l

(2) 3zd~ ~ C~ K~(K.eq.~)~C~(K)dSFdSr

e~ = 2 k,~zd~2~N~S~ ~ JJ~'[1 - exp [ - ~,o,~/k~ ~]] [e~p [~o~o,~,lk~ T] - - 1 ] '

where k F is the Fe rmi radius, M the mass ol the ion, N the n u m b e r of ions per un i t volume, T the absolute t empe ra tu r e , S F the area of the Fe rmi surface, o%~ the angula r f requency of phonon of wave vector q and mode of v ib ra t ion p, eq.~ the polar izat ion vector of the q-p la t t ice mode, v and v' are the velocities of the electrons in the ini t ial s ta te k and final s ta te k ' respect ively, and K - - - - - k ' - - k is the sca t te r ing vector . The two surface integrals are t a k e n over the Fe rmi surface and the summat ion is over the th ree polar izat ion branches p for each wave vector q. C(K) is the Bardeen m a t r i x e lement and for a free-electron model is g iven b y (~)

[V(r) - - E J K 2 + W ( K ) q~ G(qr) , r = q~ + K2

where [ E - - V(r)] is the kinet ic energy of an electron in the lowest s ta te a t the surface of a tomic polyhedron of radius r and q~ is the screening pa rame te r def ined by

4y~n6 2

q ~ - W ( K ) '

n being the electron densi ty wi th

~E[1 4ki--K ~ 2kF+K]-I W ( K ) : ~ FLU-I- S k F ~ l n 2 k F - - K

a n d the Bardeen interference factor is given b y

G(x) = 3(sin x - - x cos x)/x ~ .

The evaluat ion of the double surface integrals in eq. (2) becomes difficult because of the dependence of the phonon frequencies e%~ and the factor (K. e~.~)

2 0 0 SATYA PAL

on the direction of the scat ter ing vector K. To overcome this difficulty

Bai lyn 's (3) averaging procedure has been used to compute the double average. The r ight -hand side of eq. (2) can be wri t ten as

lff ((F(K)))F.s. = ~ F(K) dSF dS~, s~

where

K 2 ( K �9 er ~ C~(K) F ( K )

2 [exp [~oo,,/k~ T] - - ~] [1 - - exp [-- ~o , , /k~ ~]]

Now to effect the average first all the vectors k and k ' t ha t have the same scattering vector K---- k ' - - k are separated and an average over all such vectors is taken. The advantage in performing a prel iminary average over all such com-

binations with a given K is t h a t the phonon parameters do not change during it. Nex~ an average over all K for a given K-magni tude is taken. Final ly an average

over all K-magni tudes is t aken which leads to the required average over the Fermi surface as

(3) S~ (K)dSdS '= fdKfd~F(K)s (k ) SdKSd~8(k) '

where s(k)= (dk~--KS) �89 and /~ (K) indicates the average over all k, k' with

the same K. Now for a spherical Fermi surface

(~) f d K f df2 s(k) = 4~r 3 k~ .

This gives the required average over the Fermi surface as

(5) :,:~~ J

where u = K/2k F. Thus making use of this relation the final expression for the electrical re-

sistivity due to phonon scattering can be wri t ten as

(6)

.f K2(1 - - u~)�89 �9 e~.~) 2 C2(K) d K .

T:EM~PEI~ATU~E VAt~IATIO~ OF THE :ELECTRICAL :RESISTIVITY OF ALKALI X~I:ETALS 20 ] -

3. - N u m e r i c a l computat ions .

The phonon frequencies o9~.~ and the polar iza t ion vectors e~.~ are ob ta ined

f rom the solutions of the secular equa t ion set up by CH~VEAU (~o). The exper-

imen t a l values of the e las t ic cons tan ts have been used to compute the force

cons tants appea r ing in the secular equat ion. The numer ica l values of the e las t ic

constants of the a lka l i me ta l s and the o ther p a r a m e t e r s used in the calcula t ion

are l i s ted in Table I . The e lec t r ica l res is t iv i t ies of a lka l i meta l s have been cal-

TABLE I. -- Physical constants ]or alkali metals used in the calculation.

Elastic constants Temperature Lat- Density Fer- V(r)--E 10 lo dyn cm -2 at which rice (gem -~) mi (eV) C11 C1 ~ Ca ~ elastic con- con- en-

st a~ts are stunt ergy measured (A) (eV) (~

Lithium 13.42 11.30 8.89 298 (~) 3.483 0.5326 4.72 --0.8

Sodium 7.69 6.468 4.31 300 (b) 4.291 0,966 3.16 0.08

Potassium 3.715 3.153 1.88 295 (c) 5.344 0.851 2.06 --0.02

Rubidium 2.96 2.50 1.71 170 (a) 5.585 1.58 1.86 0.03

Cesium 2.465 2.055 1.48 78 (~) 6.067 1.98 1.61 0.00

(a) T. SLOTOW~qSKI and J . TI~IVISO~NO: Journ . P h y s . Chem. Sol ids , 30, 1276 (1969). (b) R. t l . )/[ARTI~SOI'~: P h y s . Rev . , 178, 902 (1969). (e) 1 ~ A. SMITH and C. S. SMITI:I: Journ . P h y s . Chem. Sol ids , 26, 278 (1965). (d) E . J . GUTMAX and J . TRIVISO~NO: Journ . P h y s . Chem. Sol ids , 28, 805 (1967). (e) F. 5. ]~OLLAI~ITS an d J . TRIVISONNO: Journ . Phys . Chem. Sol ids , 29, 2133 (1968).

cula ted wi th the help of the modif ied Hous ton ' s m e t h o d for the ca lcu la t ion of

the phonon spect rum. The in teg ra t ion over K has been pe r fo rmed numer ica l ly ,

while the in tegra t ion over the solid angle z9 has been car r ied out wi th the help

of the modif ied Hous ton ' s spher ical s ix - t e rm procedure as e l abora t ed b y BETTS

et al. (30). Hous ton ' s m e t h o d gives proper s t a t i s t i ca l weights to each rec iprocal

la t t ice poin t and is i nhe ren t l y p re fe rab le to the sampl ing technique which relies

mere ly on t a k i n g a ve ry large n u m b e r of points in the reciprocal space. F o r

the eva lua t ion of the in tegra l

J = f l ( O , ~5)dY2

the i n t e g r a n d which has cubic s y m m e t r y is f i t ted in each of the six direc-

t ions to a s ix - te rm expansion in Kubic harmonics and the series averaged

(30) D. D. BETTS, A. B. BHATIA and M. WYMAn: Phys. Rev., 104, 37 (1956).

2 0 2 SATYA ~AL

analy t ica l ly over the complete solid angle. Thus the expansion of the in tegrand in Kub ic harmonics re ta in ing only six t e rms leads to the following expression:

4~ J--_

1081080 [117 603Ia -~- 76 544IB~- 17 4 9 6 I e +

+ 381 250I~)-[- 311 0 4 0 I ~ + 177 147I~],

where the subscripts A~ B, C, D, E and F denote the values of the in tegrand I(0, qb) along the six directions [100], [110], [111], [210], [211] and [221], re- spect ively.

While in tegra t ing over K a dist inct ion has been made between the no rma l processes and the U m k l a p p processes. The fac t t h a t the U m k l a p p processes cont r ibute a subs tan t ia l p a r t of the h igh- tempera tu re res is t ivi ty of monova len t me ta l s has been known for a long t ime (~). I t has been only recent ly t h a t i t had been recognized t h a t such processes are ve ry i m p o r t a n t a t low t empera tu re s as well (a,31.~). BA~Y~ (a) and I-IASEGAWA (8) assume t h a t the normal pro- cesses opera te in the range of the var iab le of in tegra t ion u f rom 0 to 0.63 in the first Bril lonin zone, while the range 0.63 to 1 corresponds to the U m k l a p p processes. This defined separa t ion be tween the range of the normal and the U m k l a p p processes seems ra the r artificial. Very recently, following the me thod of DY~ES and CARBOTCE (~0), BLACK (~a) has s tudied the cont r ibut ion of the U m k l a p p processes exclusively to the electrical res is t iv i ty of ~lkali metals l i thium, sodium~ po tass ium and rubid ium. B]SACK neglects the effects of mult i- phonon processes and depar tu res of the Fe rmi surface f rom a sphere in his res is t iv i ty calculations. The i m p o r t a n t role p layed b$ the U m k l a p p processes in the low- tempera ture electrical res is t iv i ty of the monova len t metals has also been corrobora ted by the recent work of EKI~ (~5), T~oPI~n~xoFF and EKIN (~s) and EKI~ and MAXI~IELD (~). I n the present calculations the separat ion be- tween the normal and Um kl app processes has been effected in a more realistic manner.

The no rma l processes occur subjec t to the conservat ion law K---- ]d-- ]e ~-- e/ where the phonon vector q is res t r ic ted to be in the first Bri]louin zone. To take account of the elastic anisot ropy, the in tegra t ion over K has been carried out numer ica l ly by res t r ic t ing the phonon wave vector to points within the first Bril louin zone. The l imi t ing values of K vectors can be ob ta ined b y finding the points of in tersect ion of the planes of the Brillouin zone bounda ry with the line having the direct ion of K.

I n ~n Umklapp process the sca t te r ing vector is given b y the selection rule K = k ' - - k = q -~ g, where g is a reciprocal la t t ice vector . I n this case K goes

(al) ~ . BAILu and H. B~OOKS: Bull. Amer. Phys. Soc., 1, 300 (1956). (a2) j . M. ZIMAN: Proc. Roy. Soc., A226, 436 (1954). (an) j . E. BLACK: Canad. Jouvn. Phys., 50, 2355 (1972).

T E M P E R A T U R E V A I ~ I A T I O B [ O F T H E E L E C T R I C A L R E S I S T I V I T Y O F A L K A L I M E T A L S 203

beyond the boundary of the first Bri l louin zone but q is constrained to lie w i th in it. The m i n i m u m value o2 K at which the Umklapp processes crop up can be easi ly determined from the g e o m e t r y of the reciprocal latt ice o2 a body- centred cubic structure. In the U m k l a p p region the phonon wave vectors which are needed for the average are obta ined wi th the help o2 the selection rule:

4 . - Results and discussion.

The low-temperature electrical res is t iv i ty values o2 l i thium, potass ium, rubidium and ces ium have been measured by NIAcDo~ALD et al. (34), whi le BEI~BIAN and MAcDonALD (35) h a v e reported the m e a s u r e m e n t s for sodium alone, l~ecent]y ]~KIN and MAXFIEL])(~7) h a v e reported the low-temperature

L~

~L

'7_

10

10 - -

1 0 - - -

10 .2 _

1 0 - 3 o 10

o

o

o o

o

o o

o o

o

o

o o

):x o

)-(

x / -<

20 50 I00 200 500 ?:empemature (~

~ i g . 1. - The electrical resistivity of l i thium as a function of temperature: retical curv~, x experimental values o f DUGDAL]] a n d GUGA~,

theo- o ~r et al.

(ad) D. K. C. ~r G. K. WHITE azl4 S. B. WOODS: Prov. Roy. Soc., A 235, 358 (1956). (~5) R. B]~H~AN and D. K. C. 1V[AcDoNALD: P~'OC. Roy. Soy., A209 , 368 {1951).

2 0 4 SATYA PAL

exper imenta l electr ical-resis t ivi ty values of potass ium. DUGDALE and GUGAI~ (as} h~ve also measured the exper imentu l res is t iv i ty values of l i th ium ( ( 8 0 . - -290)~ sodium ( (50- -295)~ and po tass ium ( ( 8 . 2 9 5 ) ~ whereas DU~DA~E and PIn .LIPS (37) h~ve de te rmined the exper imen ta l res is t iv i ty for rub id ium and cesium in the t e m p e r a t u r e range ( 2 . 3 0 0 ) ~ Very recen t ly COOK et aL (as) h~ve ~lso repor ted the res is t iv i ty measu remen t s of sodium f rom 40 to 360 ~ The theoret ica l ly calculated t e m p e r a t u r e var ia t ion of the elec- trieul res is t iv i ty of alkal i meta ls together wi th the exper imen ta l da ta have been p lo t ted in Fig. 1-5. I t is seen tha t , except for l i th ium, the calculated electrical- res i s t iv i ty values of the alkal i meta ls are in reasonable ~greement with the ex- pe r imen ta l results. However , for l i thium, rub id ium and cesium the calculated

10 I I I ~ I

/ lO o _

u

"<

10 -3 I I I I I 10 20 50 100 200 500

~empero~ture (~

Fig. 2. - The electrical resistivity of sodium as a function of temperature: retieal curve, o experimental values of DUGD~E and GvGA~, 1V[AcDoI~ALD, ~ COOK et al.

- theo- x B]~R~A~ and

(8r J .S . DUGDAL]~ and_ D. GgGA~: Proc. Roy. Sot., A 270, 186 (1962); A 254, 184 (1960). (3~) j . S. DVGDALE and D. PHILLIPS: Proc. Roy, Sot., A 287, 381 (1965). (ss) j . G. COOK, iYl. P. vA~ D~I~ l~]~:~:a and M. J. LAUBITZ: Canad. Journ. Phys., 50, 1386 (1972).

TEI~P]~RATUI~E VAlClATION OF T H E E L E C T R I C A L } r 1 6 5 OF A L K A L I M E T A L S 2 0 5

10 1 - -

"~10 ~ u

:d_

"&

~. 10 -1

o

~d 0J

10 -2 _

i i I I r I

5 I I i I I

10 20 50 100 200 500 ~emperoture (~

Fig. 3 . - The electrical rcsisHvity of potassium as a function of ~emperature: - theoretical curve, o expcrimen~sd values of DIJGDALE and GUGAI% x I ~ [ A o D o ~ A L D

et a~., ~ ]~KIN mad i~AXFIELI).

values of the electrical res is t iv i ty are lower t h a n the exper imen ta l ones through- out the t e m p e r a t u r e range s tudied and the d iscrepancy increases wi th the rise in t empera tu re . The d isagreement wi th exper imen t is more pronounced in the case of l i thium. The theore t ica l as well as the expe r imen ta l studies of l i th ium me ta l have always revealed someth ing special abou t its behaviour . Al though l i th ium is the s implest metal~ its complex na ture always singles it out f rom the res t of the alkal i metals .

I t has been found t h a t the dist inct ion be tween the no rma l processes and the U m k l a p p processes which was made in a more realist ic m a n n e r in the s tudy affected considerably the numer ica l values of the electrical resist ivit ies of the alkal i metals . A sys temat ic increase in the U m k l a p p contr ibut ion to the to ta l res is t iv i ty was observed wi th the rise in t empera tu re . I t was observed tha t wi th the rise in t e m p e r a t u r e the U m k l a p p cont r ibut ion s tar ts f rom an a lmost negligible value, increasing and increasing ti]l i t a t t a ins ~ m a x i m u m V~]llC a t

206 SATYA P A L

~ 10

=2

b

~" 10-

"~_

m-2

i i [ I i I J

).(o/

7"

m -3 i r I I I I I 2 5 10 20 50 100 200 500

temperczf;ure (~K)

Fig. 4. - The electrical resistivity of rubidium as a function of temperature: thee- retical curve, o experimental values of DUGDALE and PHILLIPS, • 1V~AcDoNALD et a~.

a par t icu lar t e m p e r a t u r e af ter which the fur ther increase in t empe ra tu r e causes the U m k l a p p cont r ibut ion to decrease slowly.

I t has been observed t h a t for l i th ium the U-process con t r i bu t i on a t t a ined the m a x i m u m value a round 50 ~ where it cont r ibuted abou t 60 ~o of the to ta l electrical resis t ivi ty. Beyond 50 ~ the U m k l a p p contr ibut ion s tar ts decreasing slowly and at 300 ~ it drops to abou t 50 ~ of the to ta l electrical resistivity. The U m k l a p p contr ibut ion to the electrical res is t iv i ty of sodium reaches its m a x i m u m value a round 15 ~ i t being abou t 45 ~ a t this t e m p e r a t u r e and near ly 35 ~o at 300 ~ For po tass ium the U m k l a p p contr ibut ion is m a x i m u m a r o u n d 10 ~ i t being roughly 60 % of the to ta l electrical resist ivi ty. At 300 ~ the U-process contr ibut ion drops down to abou t 50 ~o. I n the case of rub id ium the m a x i m u m Umklapp cont r ibut ion of abou t 55 % is reached a round 5 ~ and this decreases to abou t 40 % at 300 ~ However , for cesium, the m a x i m u m U-pro- cess contr ibut ion of abou t 35 ~/o has been observed a round 30 ~ and beyond

TElYII~]~RATUR]~ VAI~IATIOI~ OF T H E E L E C T ~ I C A ~ R E S I S T I V I T Y OF A L K A L I :~ETALS 207

I 1 1 l F I I /

/

10 1 __ ooo ~ 1 7 6

oooooo " J "10~ o o o

o 0

~10 -1

10 -2 :< - - / ~0 ~3 I I l I I [ I

2 5 10 20 50 100 200 500 Lempercltut 'e (~ )

Fig. 5. : The electr ical resis t ivi ty of cesium as a function of tempera ture : - - theo- re t ica l curve, o exper imental values ot DUGDAL]~ and Pm]~LIPS, x MAcDonAlD et al.

t h i s t e m p e r a t u r e t h e c o n t r i b u t i o n was f o u n d to f l u c t u a t e b e t w e e n (30 + 3 5 ) %

of t h e t o t a l e l ec t r i cu l r e s i s t i v i t y r i g h t up to 300 ~

The p r o g r e s s i v e d e v i a t i o n of t h e t h e o r e t i c g l r e s i s t i v i t y v~lues f r o m t h e ex-

p e r i m e n t a l d a t a , as one goes f com s o d i u m to ces ium, is to be a s c r i b e d to a n

ine regs ingJy d i s t o r t e d F e r m i su r f ace in t h e s e m e t a l s . I t ha s b e e n o b s e r v e d t h a t

o n l y in s o d i u m ~nd p o t g s s i n m t h e F e r m i surEgces a r e e f f ec t ive ly s p h e r i c a l (~9),

in r u b i d i u m (as,as) n e a r l y s% whiJe in l i t h i u m (4~) a n d c e s i u m (4~,~3) t h e sur faces

(39) p. SHOEN~EnG and P. J . STILES: P~'OC. Roy. Sot., A281, 62 (1964). (4o) K. OK~U~A and I. )/[. T~rPLETON: Phil. Mug., 7, 1239 (1962). (~1) A. T. STEWA]~T, J. J. DOXAGtIY, J. H. Kvs~ISS and D. M. ROCK~OX]~: Bull. Amev. Phys. Soe., 9, 238 (1964). (r K. OKO~VR~ and I. M. TEM~LETON: Phil. Mug., 8, 889 (1963); P*'oc. Boy. Soe., A287, 89 (1965). (aa) j . TRIVISO~NO end J. A. M~mPHY: Phys. Rev. ~, 1, 3341 (1970).

2 0 8 SATY~ ~AL

are appreciably different f rom spheres. The effect of the anisotropy of the Fermi surfaces of rubidium, cesium and li thium is clearly borne out by the increasing

discrepancy between the theoretically calculated resistivity values and the experimental ones. Because of the anisotropy of the Fermi surface the scat- ter ing will va ry with the character of the electron wave function on the dif-

ferent parts of the Fermi surface leading thereby to a change in the resistivity value. As the t ranspor t properties are very sensitive to the details of the scat- ter ing mechanism at low temperatures so even a relatively small distortion of

the Fermi surface may great ly change the resistivity values. Al though the resistivity expression is quali tat ively correct~ it will not s tand

up to a detailed criticism. The Bardeen formula is not so firmly based as its derivation suggests, and contains implicitly some assumptions and approxi- mations about the physics of the solid which cannot be justified rigorously.

The discrepancy between theory and experiment may be a t t r ibuted to the an- isotropy of the Fermi surface, to the use of the Bardeen model of electron scat-

tering, to the approximate calculation of the electron-phonon interaction, to the neglect of the tempera ture variat ion of the elastic constants and to o ther

anharmonic effects. However, it is concluded tha t the present s tudy which incorporates the

Bardeen model of electron scattering together with the lattice dynamical model

of CKgVEAU provides, under certain simplifying but ra ther general assumptions, a satisfactory explanation of the temperature variat ion of the electrical resistivity

of alkali metals.

The author is grateful to Profs. VAC~ASP~tTI and KRISgaXAJ'[ for their inspi-

ra t ion and encouragement . The financial assistance from the Council of Scien- tific and Industr ia l l~esearch~ India, is also thankful ly acknowledged.

@ R ! A S S U N T O (*)

Si ~ studiata Is v~riazione con la temperatura della resistivi~ elettrica dei meta~li alealini nell'approssimazione di elettroni liberi sulla base del modello dinamieo del retieolo di Chgveau. In questo studio si ~ ineorporata la soluzione variazionale del- l'equazione hi trasporto di Boltzmamn nella maniera sviluppat~ d a Zeeman. Per fl ealcolo delle 'eostanti di forza che appaiono nell'equazione seeolare delle vibra~ioni di retieolo si sono usati i valori sperimentali delle eostanti elastiehe dei metalli alcalini. Si ~ calcolato lo spettro dei fononi con l'aiuto della procedura sferic~ a sei termini di I-Iouston, modificata nella maniera indicata da Betts et al. Nello studio si 6 ope;rata, pifi realisticamente, una distinzione f r a i l eontributo dei processi normali e quello dei processi di ribaltamento. Si trova un ragionevole aceordo fra i valori teoriei calcolati della resistivit5 elettrica dei metalli alealini, a eecezione del litio, e i dati sperimentali.

(*) Traduz ione a eura della Redazione.

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1 4 - I l N u o v o C i m e n t o B .