CONDUCTION ELECTRON SPIN RESONANCE IN PALLADIUM

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CONDUCTION ELECTRON SPIN RESONANCE IN PALLADIUM

P. Monod

To cite this version:

P. Monod. CONDUCTION ELECTRON SPIN RESONANCE IN PALLADIUM. Journal de Physique

Colloques, 1978, 39 (C6), pp.C6-1472-C6-1477. �10.1051/jphyscol:19786589�. �jpa-00218081�

JOURNAL DE PHYSIQUE

Colloque

C6,

suppliment au no

8, Tome 39,

aotit

1978,

page

~6-1472

CONDUCTION ELECTRON S P I N RESONANCE I N P A L L A D I U M

P. Monod

Laboratoire de Physique des SoZides, Uniuersite' Pmis-Sud, 91405 Orsay, France

Rdsum6.- L'observation rdcente de la rdsonance de spin des dlectrons de conduction du Palladium est ddcrite en ddtail. Cette observation qui est la premisre dtude d'un mdtal de transition par cette mdthode, permet de mesurer l'interaction dlectron-electron responsable de l'augmentation de la sus- ceptibilitd statique. Dans notre cas, cette interaction se traduit par l'apparition d'un signal de resonance dtroit et intense et dont l'asymdtrie est due 1 la presence d'ondes de spin paramagndti- ques. Appliquant les mgthodes classiques d'analyse des mdtaux simples au cas du Palladium nous en tirons le facteur g(Pd) = 2,25 2 0,03, le coefficient d'augmentation de la susceptibilitg

(m*/m) (1 + B )-I % 16 ainsi que la largeur de raie rdsiduelle B basse temp6rature inf6rieure h 100 gauss.

: D

nombreux problsmes restent poses par la dynamique interne des Electrons.

Abstract.- The recent observation of CESR in palladium is described in detail. This is the first transition metal to be studied by this method, which allows also a determination of the electron- electron interaction responsible of the susceptibility enhancement. Indeed, this interaction gives rise to a resonance signal which is relatively narrow at low temperature (< 100 gauss residual width) and intense, and whose asymmetryis presently assigned to paramagnetic standing spin waves. Applying

to the case of Palladium the classical analysis carried for simpler metals yields the

g

factor g(Pd) =

2.25 +

0.03 and the susceptibility enhancement factor (mS/m)

(1

+ B ~ ) - ~ % 16. However, a number of problems concerning the internal dynamics of electrons are left.

1. INTRODUCTION.- Palladium metal has been investi- gated in the last decade in an extensive manner for it's magnetic properties, band structure and trans- port coefficient. As we wish to contribute a new in- formation concerning the spin resonance properties of this metal we will first recall some of the main features of

Pd

together with the classical results of spin resonance in simple metals.

Palladium being the last transition metal of the second serie should have a complete 4d1° shell.

However due to s-d hybridization the Fermi sea con- tains 0.36 electron/atom and 0.36 hole/atom, as es- tablished by band structure / I / and D.H.V.A. measu- rements /2/. The Fermi surface is divided into a

r

centered electron volume, roughly spherical, and a three dimensional d hole "jungle gym" of intersec- ting cylinders at point

X

in the classical notation of the fcc Brillouin zone polyhedron / l / . Remaining smaller volumes of electrons at

X

and recently dis- covered d holes at

L

will not concern us here as the density of state is overwhelmingfy due to the d hole

"jungle gym". Indeed this part is responsible for the large specific heat 131, susceptibility / 4 / and scattering properties

151.

As Pd is a moderately heavy metal spin orbit interaction has to be taken into account

a

p r i o r i in,the band structure. As shown by a tight-binding calculation 161 the effect of this perturbation is large only at selected points

on the Fermi surface and this will play an essential role in the evaluation of g factors / 1 , 7 / . In con-

trast to spin orbit interaction, electron-electron and electron-phonon interactions are not included in the band structure calculation and their existence is mainly inferred from the difference between the calculated and measured quantities / 2 / . The electron- electron interaction leads to an enhancement of the spin susceptibility which has been described in the Landau Fermi liquid framework / 8 / or in the equiva- lent paramagnon theory 191. However this type of de- termination has yielded widely different values for these interactions so that a direct determination of it is important.

From the electron spin resonance point of view Palladium raises a number of experimental and theoretical problems as compared to the simpler me- tals studied by this technique so far (these are the alcalis and noble metals and Mg, Be and Al). The quantity which is measured is the transverse suscep- tibility of the conduction electron spins : in the case of a non-interacting electron gas Dyson /lo/

and Feher and Kip / I t / have shown that the resonance occurs as a single peak at the Larmor precession frequency of the electrons, the analysis of which gives the g factor of the electrons, the spin life time T2, the diffusion coefficient D of the electrons and a quantity proportional to the static pauli sus-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786589

c e p t i b i l i t y of t h e e l e c t r o n s . I n presence of Fermi l i q u i d i n t e r a c t i o n s 1121 a d e t a i l e d s t u d y of s l a b s of Na and K 1131 h a s r e v e a l e d t h e presence of a se- r i e of resonances i n t h e immediate v i c i n i t y of t h e o r d i n a r y one : t h e s e have been shown t o be parama- g n e t i c s p i n waves whose d i s p e r s i o n r e l a t i o n d i r e c t l y g i v e s a c c e s s t o t h e Fermi l i q u i d parameters. I n t h e c a s e of t h e a l c a l i m e t a l s , t h e s p i n enhancement i s modest ^(% 20 %) and t h e Fermi s u r f a c e i d e a l l y simple, allowing a v e r y d e t a i l e d comparison w i t h t h e theory.

We w i l l s e e t h a t although much more pronounced, the- s e e f f e c t s a r e l e s s s u i t a b l y analysed i n t h e c a s e o f Palladium.

P r i o r t o our measurement a number of s c a t t e - r e d i n f o r m a t i o n was a v a i l a b l e , p e r t a i n i n g t o t h e probable s p i n resonance p r o p e r t i e s of Pd. D i f f e r e n t e s t i m a t e of g f a c t o r s on t h e e l e c t r o n Fermi s u r f a c e around X and

r

e x i s t e d / 1 / , / 6 / b u t only r e c e n t l y was made a c a l c u l a t i o n f o r t h e d h o l e " j u n g l e gym" / 7 / , g i v i n g g(d h o l e s ) = 2.31, on average, w i t h v e r y l a r - ge a n i s o t r o p y f r o m g < 0.5 t o g > 3. Experimentally, s p i n s p l i t t i n g z e r o e s from DWA work / 2 / r e v e a l a l s o g v a l u e s a t s e l e c t e d p o i n t s b u t do n o t correspond t o t h e q u a n t i t y measured i n a dynamical experiment. A p a r t i c u l a r d e t e r m i n a t i o n of g i s a v a i l a b l e through t h e magneto-mechanical f a c t o r g ' measured by Huguenin and coworkers

1141

f i n d i n g g l ( P d ) = 1.77 a t room temperature, from which i t f o l l o w s from t h e r e l a t i o n g

-

2 = 2

-

g ' t h a t g ( g l ) = 2.23. The most encoura- g i n g p r e d i c t i o n came, i n f a c t , from t h e s t u d y of s p i n resonance of d i l u t e a l l o y s of Mn i n Pd b y A l q u i e and coworkers

/

151. They showed t h a t , a s t h e Mn s p i n was s t r o n g l y coupled t o t h e Pd e l e c t r o n s p i n , they could, by e x t r a p o l a t i n g t h e i r r e s u l t s t o v a n i s h i n g Mn c o n c e n t r a t i o n , p r e d i c t t h e dynamical p r o p e r t i e s of Pd t o be : g(Pd) = 2.25

+

0.01 a t Helium tempera- t u r e and, most i m p o r t a n t , a l i n e w i d t h of 250 gauss and a s i g n a l i n t e n s i t y i n agreement w i t h t h e s t a t i c s u s c e p t i b i l i t y . This prompted us f o r a d i r e c t s e a r c h on pure Palladium m e t a l .

2. EXPERIMENTAL TECHNIQUE AND RESULTS.- The t e c h n i - que we have been u s i n g i s t h a t of s p i n t r a n s m i s s i o n 1131. T h i s technique i s i d e a l l y s u i t e d f o r t h e s t u - dy of pure m e t a l s where t h e mean f r e e p a t h i s long and t h e conguction e l e c t r o n s p i n s i g n a l i n t e n s i t y low and e a s i l y confused w i t h t h e background. The c a s e of Pd appears t o be marginal i n t h a t t h e reso- nance s i g n a l i n t e n s i t y i s v e r y l a r g e and t h a t t h e s p i n d i f f u s i o n c o e f f i c i e n t seems v e r y much s m a l l e r

t h a n t h a t of monovalent m e t a l s . Indeed pure Pd re- sonance h a s been observed a s w e l l w i t h t h e morecom- mon r e f l e c t i o n t e c h n i q u e / 1 6 / . The t r a n s m i s s i o n

t e c h n i q u e i s however e s s e n t i a l f o r t h e s t u d y of t h e e x c i t a t i o n and propagation of paramagnetic s p i n wa- v e s . We simply r e c a l l t h a t t h e sample i s i n t h e form of a t h i n f o i l clamped t i g h t l y between two sym- m e t r i c TE 101 c a v i t i e s tuned a t 9.2 GHz. The exci- t i n g c a v i t y i s f e d by t h e r . f . g e n e r a t o r and indu- c e s s p i n resonance on one f a c e of t h e sample. Those e l e c t r o n s which w i l l d i f f u s e a c r o s s t h e sample t h i c k n e s s i n a time s h o r t e r t h a n t h e i r t r a n s v e r s e r e l a x a t i o n time T emit a coherent r a d i a t i o n i n t h e

2

o t h e r c a v i t y , which i s d e t e c t e d by i t ' s p r o j e c t i o n on a s u i t a b l e r e f e r e n c e phase, by a balanced mixer.

I n t h e c a s e of a v e r y t h i n and p u r e metal s l a b s t a n d i n g s p i n waves occur : i f a d i s p e r s i o n r e l a t i o n e x i s t s , w ( k ) , t h i s w i l l g i v e r i s e t o a s e p a r a t e r e - sonance f o r each k v a l u e s e l e c t e d by t h e boundary c o n d i t i o n s 181. The requirement f o r sample prepara- t i o n a r e : 1) high p u r i t y , h i g h r e s i s t a n c e r a t i o ; 2 ) v e r y t h i n . We have p u r i f i e d commercially a v a i l a - b l e Pd by prolonged oxydation i n a i r a t 1 2 0 0 ° ~ i n a q u a r t z tube. Three c o n s e c u t i v e s t e p s of r o l l i n g and a n n e a l i n g were t h e n n e c e s s a r y t o achieve a f i n a l t h i c k n e s s of 5 t o 15 microns. The f i n a l , h i g h l y po- l y c r i s t a l l i n e f o i l s , had a r e s i s t i v i t y r a t i o which was s i z e dependent a t a r a t e of Ap (size)= 5f l x l r 3 vilcm. em-', i n d i c a t i n g s u r f a c e s c a t t e r i n g , and e x t r a p o l a t i n g t o a bulk r e s i s t i v i t y r a t i o 2.000.

Great c a r e was taken t o avoid hydrogen contamination.

The t h i c k n e s s was estimated. ( t o about 3 %) by weigh- t i n g and s u r f a c e measurement.

Although g r e a t c a r e was used t o mount t h e samples t i g h t l y between t h e r e s o n a t i n g c a v i t i e s v e r y l a r g e r . f . power was t r a n s m i t t e d a t room tem- p e r a t u r e by t h e t h i n n e s t samples. T h i s was e a s i l y t r a c e d back t o be due t o t h e d i r e c t s k i n e f f e c t t r a n s m i s s i o n a s i t s t r o n g l y d e c r e a s e s upon c o o l i n g , t o g e t h e r w i t h a r o t a t i o n of t h e phase of 132O p e r decade amplitude change ( a s expected from t h e trans- m i s s i o n c o e f f i c i e n t exp - ( I

+

i)L/G(T)). 1t disap- peared below n i t r o g e n temperature.

A t temperatures below 20 K l a r g e spin resonance were v i s i b l e a s shown i n f i g u r e 1 and 2, superim- posed on a v e r y uniform background of o r b i t a l trans- m i s s i o n (unresolved c y c l o t r o n motion). However a s compared w i t h a normal m e t a l conduction e l e c t r o n resonance t h e Pd s i g n a l h a s many uncommon f e a t u r e s t h a t we have t o d e s c r i b e i n d e t a i l .

JOURNAL DE PHYSIQUE

1 ~ 1 ~ 1 1 ~ 1 ~ 1 1 ~ 1

Pd

R R R 1250

Fig. 1 : T y p i c a l t r a c e r e c o r d i n g of Pd s p i n r e s o - nance. The sample t h i c k n e s s i s 15 ym and t h e tempe- r a t u r e 4.0 K. The asynrmetryin t h e l i n e shape i s ma- d e a p p a r e n t by t h e d o t t e d l i n e which has been drawn t o b e symmetric t o t h e low f i e l d s i d e of t h e l i n e w i t h r e s p e c t t o t h e maximum p o i n t . This s i g n a l i s i s o t r o p i c w i t h t h e magnetic f i e l d d i r e c t i o n w i t h r e s p e c t t o t h e sample and corresponds t o a s i t u a t i o n where t h e t h i c k n e s s i s about e q u a l t o t h e s p i n mean f r e e path. The r e f e r e n c e phase used i s s h i f t e d by

110° from t h a t of normal metal s p i n resonance

b l l l l ~ l l l l ~ l l l l ~ l l l l ~ ~ ~ l d

2 0 2.5 3.0 3.5 4.0

H,

(K gauss)

Fig. 2 : Pd s p i n resonance f o r a t h i n sample where t h e t h i c k n e s s i s l e s s t h a n t h e s p i n mean f r e e p a t h . The r e f e r e n c e phase i s s e t a t t h e "normal" m e t a l phase. The h i g h f i e l d n e g a t i v e l o b e i s i s o t r o p i c and i s i n t e r p r e t e d a s a m a n i f e s t a t i o n of a s t a n d i n g s p i n wave. A s u b s t r a c t i o n of a s l o p i n g b a s e l i n e h a s been o p e r a t e d

It appears a t f i r s t s i g h t t h a t i n f i g u r e s 1 and 2 t h e resonance s i g n a l cannot b e made t o appear l i k e a symmetric L o r e n t z i a n l i n e . Such a s i t u a t i o n can occur i n t h e a l c a l i m e t a l s 1131 o r i n aluminium 1171 and i s a m a n i f e s t a t i o n of unresolved s p i n waves We have t r i e d t o g e t r i d of t h e a s m e t r y of the.

shape i n d i f f e r e n t ways 1) by i n c r e a s i n g t h e tempe- r a t u r e , t h e s c a t t e r i n g r a t e i s increased and inde- ed t h e asynrmetry i s decreased b u t s o i s t h e i n t e n -

s i t y of t h e s i g n a l which v a n i s h e s ( s e e f i g . 3 ) above 22 K without developing t h e c h a r a c t e r i s t i c shape 1181 (with A and B l o b e s ) t y p i c a l of t r a n s m i s s i o n of a normal m e t a l of t h i c k n e s s l a r g e r t h a n t h e s p i n mean f r e e p a t h / 1 9 / . 2) By changing t h e magnetic f i e l d a n g l e w i t h r e s p e c t t o t h e sample t h e s i g n a l shape and amplitude i s observed t o b e completely i s o t r o p i c , t o b e t t e r t h a n a few %, i n complete con- t r a d i c t i o n w i t h u s u a l s p i n waves s p e c t r a . 3) By changing t h e sample t h i c k n e s s t h e shape of theasym- metry i s modified from f i g u r e 1 t o f i g u r e 2 and a l s o

t h e r e f e r e n c e phase used. For o u r t h i n n e s t Pd sample t h e r e f e r e n c e phase i s t h a t used f o r a normal m e t a l 1181. I n a s much a s one can d e f i n e a g f a c t o r by t h e p o s i t i o n of t h e peak of t h e l e a s t asymmetric s i g n a l , t h i s parameter i s observed t o change s l i g h - t l y w i t h sample t h i c k n e s s and on t h e average of s i x samples i s g(Pd) = 2.25 ? 0.03. From f i g u r e 4 i t appears t o b e independent of temperature a t low tem- p e r a t u r e . The l i n e w i d t h i s a l s o t e n t a t i v e l y d e f i n e d a s t h e width a t h a l f h e i g h t . F i g u r e 5 shows how i t i n c r e a s e s w i t h i n c r e a s i n g t e m p e r a t u r e 1201. The r e - s i d u a l width ( e x t r a p o l a t e d a t 0 K) c o n t a i n s a con- t r i b u t i o n which i s s i z e dependent a t a r a t e of 0.1 gauss/cm-l i n d i c a t i n g an u n u s u a l l y l a r g e s u r f a c e s p i n s c a t t e r i n g term.

Fig. 3 : The i n t e n s i t y of t h e resonance f o r t h e s i - gnal o f f i g u r e 1

versus

temperature (0). The squares

IQ) r e f e r t o t h e same sample a f t e r s p o i l i n g t h e r e - s i s t i v i t y r a t i o . Although d e f i n i t e l y . i n t h e " t h i c k l i m i t " c a s e where t h e t h i c k n e s s i s l a r g e r t h a n t h e s p i n mean f r e e p a t h , t h e observed s i g n a l shape f a i l s t o develop A/B l o b e s a s t h e temperature i n c r e a s e s . See r e f e r e n c e s 1171 and 1181

Fig. 4

:

"g" factor as a function of temperature defined by the position of the resonance peak for the least asymmetric signal phase setting. The lar- ge variation observed between two samples is an il- lustration of the lineshape problem

o

426

R R R

o

1250

Fig. 5

:

The line width at midheight of the (asym- metric) signal as a function of-temperature. The data above 10 K should not be used without proper correction for lineshape. For comparison see refe- rence / 151

3 .

ANALYSIS AND DISCUSSION.- In the case of a nor-

mal metal of thickness less than the spin mean free path the transmitted signal is given by /19/

:

where ho is the exciting field,

6

the normal skin depth,

X

the free space wavelength of the radiation at frequency

w,

R the sample thickness, X the Pauli susceptibility of the metal and

wo =

9 ^{Ho is the}

Larmor frequency of the spins. We will discuss these different parameters in turn

:

Most important is the signal intensity defi-

ned by the product of the signal height by the line width

:

this quantity is proportionnal to (~1%) and is found to be constant for R

< 9

microns. We have compared this quantity to the same one determined with copper samples /21/. This allows to determine approximatively the ratio

of

the spin susceptibili- ties/cm3 of two metals as we use the same geometri- cal conditions. We find

('Icm3

₃₃

+

¹⁰

(spin resonance)

x/cm3 (cu)

where the large uncertainty reflects the undetermi- nacy of the anomalous skin depth in each case. By comparison it should be noted that the ratio of the measured static susceptibility of Pd at He tempera-

ture to the (calculated) Pauli susceptibility of copper gives

'Icm3

= 8 7

(static)

x / m 3

(Cu)

so that the large intensity inferred by resonance is only a fraction of the total susceptibility mea- sured

;

nevertheless this crude comparison establi- shes the fact we are indeed observing the d hole spin resonance of Palladium. This is in accordance with the ratio of susceptibilities of Mn impurities and Pd electrons inferred by Alquie 1151. Although the linewidth, and correspondingly T2, are poorly defined experimentally it is possible to define a low temperature region below 10 K where the widthis approximately constant

:

as this residual linewidth contains an important contribution from surface scattering, one determines the bulk linewidth by extrapolating our data to zero inverse thickness.

This gives

AH(bulk, residual)

= 75 f 25

gauss and

Tp

= (y T ) - l

AH of the order of 2

x

10-~s. From the

measured spin surface scattering rate, one can esti- mate the spin flip probability

E

per collision at the surface. In the notation of Dyson /lo/ this is twice the ratio of the spin transit time I1/vF by the surface spin flip time, we find

:

e(Pd surface)

=

1

x

if we take v

= 5 x

lo7 a/s. A similar estimate of

F

E

can be made using the measured resistivity incre- ment due to size effect.

Subject to modification from a computer fit

lineshape analysis (which we have not carried out

in view of the large number of ill defined parame-

JOURNAL DE PHYSIQUE

ters) the g valued has been defined, as mentioned at the peak of the resonance signal. In view of the very large g value distribution /7/ it is tempting to attribute some, if not all, oi5 the asymmetry in the signal shape to an incomplete narrowing of this distribution. However a rapid estimate shows that the narrowing factor must be of the order of 100 at least, ensuring a Lorentzian lineshape. In order to study this dynamical process the frequency depen- dence of the residual linewidth should be investi- gated 1221. The physical origin of this strong nar- rowing that allows our observation of a unique, well defined resonance, is probably due to the strong electron-electron interaction exchange as explained by Freedman and Fredkin 1231.

We have tentatively attributed the negative high field lobe visible on the spin resonance signal of figure 2 to a barely resolved standing spin wave for the following reasons : the classical derivation of spin wave 1121 gives very anisotropic spectra only because the cyclotron motion determines an ani- sotropic diffusion constant in the limit w T > 1, where w = 7 eH is the cyclotron frequency. In our

m c

case however a rough estimate shows that wcr 0.1.

This is more than enough to kill any spin wave pro- pagation in the usual metals where the electron- electron interaction is moderate since the criterion for spin wave resolution is 1121 :

where

-

is the susceptibility enhancement ex- 1 + Bo

pressed in the Fermi liquid notation 181, assuming higher order coefficients to be zero. We hint from the fact that w T > w T and that the enhancement is sizeable, that this criterion is still approximately satisfied in our case. The shift of the spin waves then becomes isotropic and is given by the field perpendicular classical case which we write for the n = 1 spin wave /8/ :

2 ~

k2

2

( l + B o )

2 1

= 1) =

- ' - 2 - -

3 m 2 G w m * 2 ' Bo (3) (

; )

where IQ is the bare electron mass and Mk K

F"""

F

the Fermi momentum. We have checked on samples of 6.1, 7.0 and 8.9 microns that this negative lobe was shifted, in accordance to (3), as the square of

the inverse thickness towards high fields 1241. Ho- wever no n =

2

resonance could be detected, mainly because of severe base line substraction problems

.

In order to use (3) we assume that kF is determined by the volume of the Fermi sea containing 0.36 ellat, thus avoiding Fermi velocity determination. We take kF = 0.897x10~cm-'. We find from (3) /24/ that :

I

and, assuming that !!!-= 3.3 (for sake of estimate) m

we get : B, =

-

0.8

and,

in poor agreement with estimates from the static susceptibility. This last parameter (5) is seen to be quite independent of the choice of m*/m if B %

- ^1.

We check that the condition (2) is fullfil- led for wo T > 0.25. We note that an equivalent man- ner to analyse our data is to evaluate the diffusion coefficient using the conductivity 1251, 1261, 1271.

We see that any improvement in this type of determi- nation rests on our experimental effort to observe better defined spinwaves but also a better understan- ding of the internal dynamics of s and d electrons allowing a knowledge of the diffusion parameters to be used for spinwave propagation.

We have not dealt, in this experimental pre- sentation, with many aspects of the spin resonance of Palladium which merit further investigation, ei- ther for their intrinsic interest for the resonance itself such as the problem of lineshape in reflec- tion /16/ or for the physics of Palladium, such as the connection between the temperature dependent linewidth 1151 and the

AT^

term in the resistivity, characteristic of s-d scattering /5/. As a conclu- sion of this ensemble of observations, the main fact remains the very unexpected "goodwill" of the Pd electron spins to display a rather narrow resonance.

In our interpretation it's, somewhat unusual, cha- racteristics are due to electron-electron interac- tions and hopefully this will provide a way to un- derstand better the physics of transition metals.

ACKNOWLEDGMENTS.- Acknowledgments are gratefully extended to H. Hurdequint for this numerous sugges- tions and collaboration and to G. Alquie thanks to whom this work was started. We thank Mrs. Marguerite Boix for her pertinent help in purifying the samples.

M.T.

Beal-Monod is indebted for illuminating private communications.

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1261 The same problem a r i s e s i n r e f e r e n c e /18/ f o r t h e c a s e of L i and t h e same l i n e o f argument i s u s e d . Note t h a t t h e r e e x i s t s o n l y one t y p e of e l e c t r o n s i n t h a t c a s e

/27/ I t should b e s t r e s s e d t h a t t h e a l m o s t r e s o l v e d s p e c t r a o b s e r v e d i s i n i t s e l f q u i t e remarqua- b l e , i n view of t h e s p r e a d of

mf .

^{I t i s con-}

c e i v a b l e t h a t t h e p h y s i c a l v a r l a b l e would b e Jn t h e combination a p p e a r i n g i n e q u a t i o n (5). We n o t e i n t h a t d e r i v a t i o n

2

d o e s n o t c o n t a i n electron-phonon interaction

m

and t h u s s h o u l d b e compared w i t h s p e c i f i c h e a t r e s u l t s : c . f .

erri in^,

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I V , p . 290.