Krebs, K.

Archive of European Integration

EU R 444.e
REPRINT
EUROPEAN ATOMIC ENERGY
'Seef'e{ 
Coel
COMMUNITY - EURATOIII
RENORMALISATION EFFECT IN THE
ELECTRONIC SPECIFXQ HEAT OF VANADIUM
by
K. KREBS
1q83
Joint Nuqlear Besearch Centre
Ispra Establishment (Itafy)
Reactor Physlcs Departmen{
Reprlnted from
PHYSICS LETTERS
Vol. 6, No. 1 - 1963
LEGAL NOTICE
This docunent was prepared rurder the sponsorship of the Commission of tbe European
Atomic Energ;y Commurity (EURATOM).
Neither the EURATOM Com.mission, its contractors nor any person acting on their behalf :
lo 
- 
Make any warrarrty or representation, e:qpress or implied, with respect to the accuracy,
completeness, or usefulness of the information contained in this document, or that the use
of arry information, apparatus, method, or process disclosed in this document may not
infringe privately owned rights; or
2o 
- 
Assume any liability with respect to the use of, or for damages resulting from the use of
any information, apparatus, method or process disclosed h this docrrment.
This rofuinl is dntenileil lor rcstricted ilishib*tion only. It refrducx,
by kind. fermission ol thc publisher, an afiicle fuom "PHYSICS
LETTERS", Vol,. 6 - No. I - 1963, 31-33. For lwrther copics fhase
apfly to Nortk-Hol,lanil Publ,ishing Comfany - P.O. Box 103 
-Amsterdam ( N eth.erlanils).
Dieser Sonderdruch ist tiir eine beschrd.nhte Vateilung bestimmt. Die
Widergabc des vorliegenden in ,,PHYSICS LETTERS", Vol.. 6 -
Nr. 1 - 1963, 31-33. crschionenen Aulsalzes efiol,g mit lrwnd,lichcr
Genehmd gun g d,es H eraus geb ers. B estellun gen w eiler u E xempla,re sind
an North-Holl,and Publ,ishdng Company - P.O. Box 103 
- 
Arnsterdant
( N etheilands), zu richten.
Ce tirC-d-part est exclusivement destini d. une d,iltusion restreinte. Il
raprend, aaec l,'uimabla autorisalion d,e l,'dd,iteur, un article fubliC dans
le TPHYSICS LETTERST - Vol.6 - No 1 - 1963, 31-33. Tout autre
eremplaire de cet drticle, d,oil Atue demandC d NortbHolland. Pabl.ishing
Company - P.O. Box 103 
- 
Amsterd,am (Ndkerlan"ds).
Quarto *lratto d ilestinato escl,us'iaamente ad una ilillusione ldmitata.
Esso d stato riprod,o#0, per gmtil,e coruessione ddl'Eil,itore, da sPHY-
SICS LETTERSv, Vol..6 - No 1 - 7963,31-33. Ukcrioti cupie dell,'arti-
colo debbono essere richieste a Norlh-Holland Publishing Company -
P.O. Box 103 
- 
Amsterilam (Netlwlard,s).
Dcze oaeilhuek is slultts ooor beperhte aerspreid,ing bcstemd,. Hd artihel,
is mct wclwillende toesletnming van d,e witgeaer oaergenomen *it ,,PHY-
SICS LETTERS", Vol.6 - w. 1- 1963,31-33. Meer eremplaren
hwnne* bestal,il uorden bi,i North-Holl,and. Publ,ishdng Conpany - P.O.
Box 703 
- 
Amsterd.am (Netherl,anils).
Volume 6, number 1 PHYSICS LETTERS 15 August 1963
RENORMALISATION EFFECT IN
THE ELECTRONIC SPECIFIC HEAT OF VANADIUM
K.I(REBS
Department of Reactor Physics, Euratom C.C.R.,Ispra (Varese'1 ,Italy
Received 24 JuIy 1963
In the Sommerfeld-Bethe theory 1) the electronic)art of the specific heat of a metal is given by the
lensity of states at the Fermi surface. To first or-
ler it T/Tp the electronic specific heat varies lin-
:arly with temperature and may be written in the
brm
c"=!#nzN(Ds)T=t"7, (1)
uhere N(Ep) is the density of states at the Fermi
rurface. This result is valid only if one neglects
he electron-phonon interactirn. The latter leads to
wo renormalisation effects, firstly it modifies the
Iormal mode frequencies of the crystal, and sec-
rndly it modifies the free energy of the electrons.
the first correction is small and amounts to only
r few percent 2,3). The second effect 4,5) may be
'ather large and, depending on the strength of the
rlectron-phonon interaction, may lead to an essen-
ial contribution to the electronic specific heat of
he same order as the Sommerteld value. This in-
eraction part comes from electron transitlons in
he thermally excited region near the Fermi sur-
ace. The perturbation is strong iL kT is small,
.e., this renormalisation effect is temperature
iependent and reaches its maximum at low temper-
.tures.
For completeness we may mention another
ffect, that is, the dependence of the free energ'y
f the crystal on shear effects which iirvolve changes
in the electron distribution 6). The magnitude of
this effect depends on the relation between the
Fermi surface and the Brillouin zone boundary,
being the stronger the closer the Fermi surface
comes to the Brillouin zone. The corresponding
contribution to the specific heat augments the Som-
merfeld value and varies linearly with temperature
in the liquid helium range. This effect is expected
to be appreciable only when the electron energy
differs considerably from the free electron case.
Even neglecting the last effect, we see that the
experimental low-temperature value of 7 is not
always related to the density of states in a simple
way, or in other words the effective y-value be-
comes a function of temperature. Until now a direct
experimental check on this has not been possible,
since it involves the knowledge of the lattice spe-
c,iIic heat, resp., of the lattice vibrational spec-
trum. The calculation of such spectra for metals
is not yet reliable enough, since a lattice vibra-
tional model which takes proper account of the
electrons is still missing 7). Hou,ever, this un-
favourable situation is changing with the possibility
to determine such spectra experimentally by neu-
tron techniques. As a first case we present here
some results on vanadium.
Fig. 1 shows the lattice vibrational spectrum of
vanadium, calculated from the differential inelastic
scattering cross section for slow neutrons 819).
31
Volume 6, number 1 PHYSICS LETTERS 15 August 1963
Fig. 1. Experimental lattice vibrational spectrum
of vanadium at 300oK.
Fig. 2. Lattice specific heat of vanadium, calculated from
the spectrum in fig. 1. The dashed curve takes
accouDt of the difference in the anharmonic contri-
butions between room temperature aud the corre-
sponding temperafures.
Fig. 3. Apparent Debye temperatures. (For the dashed
curve see text,in fig. 2.)
The lattice specific heat derived from this spectrum
is given in fig. 2 *, the corresponding apparent
Debye temperatures are shown in fig. 3..The 0 ver-
sus ? curve has the usual low temperature mini-
mum, which is due to the deviation of the true lat-
tice spectrum from the spectrum of a Debye solid.
Since the neutron data were taken at room tempera-
ture, the speeific heat calculated from the lattice
vibrational spectrum is strictly valid only at this
temperature. For other temperatures the anhar-
monic contribution to the specific heat will be dif-
ferent. We may roughly estimate this modiJication
using Griineisen?s law for the relation between Debye
temperature and volume change. We find
* For the lowest frequencies of the spectrum we have
used the Debye approximation with I = 399o6 10).
32
0(")=e(rn) {r-tu^r}K
(where Ip is the reference temperature, a is the
coefficient of linear expansion, K is the Grtineisen
constant), and calculate the new lattice specific
heat from the corrected apparent Debye tempera-
ture, using the room temperature value as a ref-
erence. The result of this calculation is shown in
figs. 2 and 3 by the dashed curves. Due to the uncer
tainty of this correction we will restrict our follow-
ing discussion to temperahres, say, above 150oK.
150 200 250 300 350 T('K)
Fig. 4. Lattice specific heat of vanadium above 150oK.
-neutrou 
data, ---- caloric values of Cv/lR,
values of (Cy - ys7)/3R using the experi-
mental low-temperature value for y6 (= 2.21 x 10-3
cal mol-1 deg-z;.
In fig. 4 we compare the lattice specific heat
above 150oK derived from the neutron data with
caloric measuremenls 11,14,15). ff we subtract
from the caloric data the electronic contribution
using the low--temperature value 12) , yo = 2.21 x
10-3 cal mol-l d"g-2, we find points w[ich lie for
all temperatures below the lattice specific heat
curve derived from the neutron data. Therefore,
we conclude from fig. 4 that the 7e-value, which
was found at low temperature, is no longer valid
at higher temperature. We propose that this effect
might be attributed to the renormalisation of the
free enefry of the electrons, which is caused by
the electron-phonon interaction.
In order to give at least a qualitative justifica-
tion of this view, we have evaluated the expression
for the free enerry of the electrons, given by
Buckingham and Schafrottr 5). ttreir theory is valid
for free"electrons in a Debye solid and can thus
serve only as an approximative check for a rela-
tively complicated metal like vanadium.
The electronic free energy consists in this ap-
proximation of two terms, namely of the usual
Sommerfeld value g" and of a perturbation contri-
bution 92:
L
pel 
= es + e2 = ps {1 + lt + g(r/e)l (in)t F} ,
t.o
cL
TCK)
- 
cv-ar
o Ret'l)
o Ref.rt)
o Ref.'c)
es = - 612 arlrp) @t)2 ;
z 4 . o b toxtol2
olume 6, number 1 PHYSICS LETTERS 15 August 1963
is the number of free electrons per atom, f is
he interaction gonstant of Frdhlich 13), which for
superconductOr, like vanadium is presumably
arger than (4n)-2; S(r) is a sum of two integrals
'a(t) andf6Q) which carurot be evaluated in closed
)rm:
',@=+i &-{t2x2rcg#
- ,*'t rog lffi} ,
'u(t) 
= *ri *,
o
x {(x1+x2)z tos
(x1+x2)2 
- t-2
(x1 + x2)2
o 100 200 300 T(.K)
Fig. 5. The tunction g(fl, defined iu eqs. (3) and (a) 
,for two 0-values.
Using
ce=- r(azqu/airz) =ts11T, (5)
'e see from fig. 5 that the electronic specific heat
epends on temperature. OnIy for temperatures of
t least the order of the Debye temperature we ob-
rin the usual Sommerfeld expression (1). For low
:mperatures the effective 7-value is higher de-
ending on the interaction constant F.
We may estimate this constant from the differ-
nce between the lattice specific heat which was
alculated from the neutron data and the values
hich were derived from caloric data *, using the
xpgrimental low-temperature value lo = 2.21 x
0-3 cal mol-l deg-2. p"o,,
7oT
assuming n = L, we obtain F = 1.16. This value is
larger than (42)-i - 0.63, as one expects for su-
perconducting elements 13).
Besides the fact that we find at low tempera-
tures an effective 7-value, which is remarkably
higher than the high-temperature value 7s, another
interesting conclusion may be drawn from fig. 5.
If we calculate the Debye temperature of vanadium
from the elastic constants at low temperatures, as
measured by Alers 10) , *" obtain 0eI = 399oK,
which is remarkably different from the caloric
value obtaiaed by Corak et aI. 12), 0cal = 338oK.
The caloric value was derived from the usual C/T
versus T2 plot. This experimental discrepancy
may be caused also by the renormalisation effect
discussed above.
From eqs. (3) and (4) follows for the y-value at
Iow temperatures, assuming that roughly g = cTZ(see fig. 5): r r
7 = 7s {1 + (in)lF + 6(fu)ircr2} . (?)
In eq. (7) appears a T2-term, which generates an
additional 7r-term in the expression for the lattice
specific heat. This term may account for the dif-
ference between elastic and caloric Debye temper-
atures. Estimating c from fig. 5, we find a new
caloric Debye temperature, 0lat 
", 
3?0oK, which
agrees better with the elastic value than the pre-
vious one. In view of the approximations made,
this seems to be a further confirmation for the
existence of an appreciable renormalisation effect
in the electronic specific heat of vanadium.
It is a pleasure to thark Mr. Cocchi and Mr.
De Bruyn from the Cetis Computational Centre for
their help in the evaluation of the two integrals in
eq. (4).
1) A.Sommerfeld and H.A.Bethe, Handb. Physik 24 (1933)
333.
2) G.V.Chester and A,Houghton, Proc. Phys. Soc. (Lon-
don) 73 (1959) 609.
3) G.V.Chester, Adv. Phys. 10 (1961) 357.
4) M.J.Buckingham, Nature 168 (1951) 281.
5) M.J.Buckingham and M.R.Schafroth, Proc. Phys. Soc.
(London) 67 (19s4) 828.
6) H.Jones, Proc. Roy. Soc. (Londou) 240 (1957) 321.
7) K.Krebs, Report EUR 366. e C.C.R. Ispra (1963) .
8) K. C. T\rrbersfield and P. A. Egelstaff, Phys. Rev. 127
(1962) 1017.
9) R. Haas et al., Inelastic scattering of neutrons (I.A.E.A.,
Vienna, 1963) p. 145.
10) G.A.Alers, Phys. Rev. 119 (1960) 1932.
11) K.Clusius, P. Franzosini and U.Piesbergen, Z. Natur-
forsch.15a (1960) 728.
12) W.S.Corak et al ., Phys. Rev. 102 (1956) 656.
13) H. Frtihlich, Phys. Rev. 79 (1950) 845.
14) C.T.Anderson, J. Am. Chem. Soc. 58 (1936) 564.
15) D.R.Stull and G.C.Sinke, Adv. Chem. Ser. 18 (1956)
2L6.
(4)
€24,2 f
,o J
,ro (1+expr1)(l +expx2)
- @r-*il2 r.rt\!t*].
\r7 - xD'
)ur numerical calculation of these two integrals
eads to the curves in fig. 5.
g (r)
* Gdir =+=
C$aloric 17; - Cfeutron(?)
(= 2.21/1.15)
For ? = 300cK.
-/->--
x
t-<---.-
*****
33





Renormalisation effect in the electronic specific heat of vanadium. Euratom Report, 1963. EUR 444.e

EU R 444.eREPRINTEUROPEAN ATOMIC ENERGY'Seef'e{ CoelCOMMUNITY - EURATOIIIRENORMALISATION EFFECT IN THEELECTRONIC SPECIFXQ HEAT OF VANADIUMbyK. KREBS1q83Joint Nuqlear Besearch CentreIspra Establishment (Itafy)Reactor Physlcs Departmen{Reprlnted fromPHYSICS LETTERSVol. 6, No. 1 - 1963LEGAL NOTICEThis docunent was prepared rurder the sponsorship of the Commission of tbe EuropeanAtomic Energ;y Commurity (EURATOM).Neither the EURATOM Com.mission, its contractors nor any person acting on their behalf :lo - Make any warrarrty or representation, e:qpress or implied, with respect to the accuracy,completeness, or usefulness of the information contained in this document, or that the useof arry information, apparatus, method, or process disclosed in this document may notinfringe privately owned rights; or2o - Assume any liability with respect to the use of, or for damages resulting from the use ofany information, apparatus, method or process disclosed h this docrrment.This rofuinl is dntenileil lor rcstricted ilishib*tion only. It refrducx,by kind. fermission ol thc publisher, an afiicle fuom "PHYSICSLETTERS", Vol,. 6 - No. I - 1963, 31-33. For lwrther copics fhaseapfly to Nortk-Hol,lanil Publ,ishing Comfany - P.O. Box 103 -Amsterdam ( N eth.erlanils).Dieser Sonderdruch ist tiir eine beschrd.nhte Vateilung bestimmt. DieWidergabc des vorliegenden in ,,PHYSICS LETTERS", Vol.. 6 -Nr. 1 - 1963, 31-33. crschionenen Aulsalzes efiol,g mit lrwnd,lichcrGenehmd gun g d,es H eraus geb ers. B estellun gen w eiler u E xempla,re sindan North-Holl,and Publ,ishdng Company - P.O. Box 103 - Arnsterdant( N etheilands), zu richten.Ce tirC-d-part est exclusivement destini d. une d,iltusion restreinte. Ilraprend, aaec l,'uimabla autorisalion d,e l,'dd,iteur, un article fubliC dansle TPHYSICS LETTERST - Vol.6 - No 1 - 1963, 31-33. Tout autreeremplaire de cet drticle, d,oil Atue demandC d NortbHolland. Pabl.ishingCompany - P.O. Box 103 - Amsterd,am (Ndkerlan"ds).Quarto *lratto d ilestinato escl,us'iaamente ad una ilillusione ldmitata.Esso d stato riprod,o#0, per gmtil,e coruessione ddl'Eil,itore, da sPHY-SICS LETTERSv, Vol..6 - No 1 - 7963,31-33. Ukcrioti cupie dell,'arti-colo debbono essere richieste a Norlh-Holland Publishing Company -P.O. Box 103 - Amsterilam (Netlwlard,s).Dcze oaeilhuek is slultts ooor beperhte aerspreid,ing bcstemd,. Hd artihel,is mct wclwillende toesletnming van d,e witgeaer oaergenomen *it ,,PHY-SICS LETTERS", Vol.6 - w. 1- 1963,31-33. Meer eremplarenhwnne* bestal,il uorden bi,i North-Holl,and. Publ,ishdng Conpany - P.O.Box 703 - Amsterd.am (Netherl,anils).Volume 6, number 1 PHYSICS LETTERS 15 August 1963RENORMALISATION EFFECT INTHE ELECTRONIC SPECIFIC HEAT OF VANADIUMK.I(REBSDepartment of Reactor Physics, Euratom C.C.R.,Ispra (Varese'1 ,ItalyReceived 24 JuIy 1963In the Sommerfeld-Bethe theory 1) the electronic)art of the specific heat of a metal is given by thelensity of states at the Fermi surface. To first or-ler it T/Tp the electronic specific heat varies lin-:arly with temperature and may be written in thebrmc"=!#nzN(Ds)T=t"7, (1)uhere N(Ep) is the density of states at the Fermirurface. This result is valid only if one neglectshe electron-phonon interactirn. The latter leads towo renormalisation effects, firstly it modifies theIormal mode frequencies of the crystal, and sec-rndly it modifies the free energy of the electrons.the first correction is small and amounts to onlyr few percent 2,3). The second effect 4,5) may be'ather large and, depending on the strength of therlectron-phonon interaction, may lead to an essen-ial contribution to the electronic specific heat ofhe same order as the Sommerteld value. This in-eraction part comes from electron transitlons inhe thermally excited region near the Fermi sur-ace. The perturbation is strong iL kT is small,.e., this renormalisation effect is temperatureiependent and reaches its maximum at low temper-.tures.For completeness we may mention anotherffect, that is, the dependence of the free energ'yf the crystal on shear effects which iirvolve changesin the electron distribution 6). The magnitude ofthis effect depends on the relation between theFermi surface and the Brillouin zone boundary,being the stronger the closer the Fermi surfacecomes to the Brillouin zone. The correspondingcontribution to the specific heat augments the Som-merfeld value and varies linearly with temperaturein the liquid helium range. This effect is expectedto be appreciable only when the electron energydiffers considerably from the free electron case.Even neglecting the last effect, we see that theexperimental low-temperature value of 7 is notalways related to the density of states in a simpleway, or in other words the effective y-value be-comes a function of temperature. Until now a directexperimental check on this has not been possible,since it involves the knowledge of the lattice spe-c,iIic heat, resp., of the lattice vibrational spec-trum. The calculation of such spectra for metalsis not yet reliable enough, since a lattice vibra-tional model which takes proper account of theelectrons is still missing 7). Hou,ever, this un-favourable situation is changing with the possibilityto determine such spectra experimentally by neu-tron techniques. As a first case we present heresome results on vanadium.Fig. 1 shows the lattice vibrational spectrum ofvanadium, calculated from the differential inelasticscattering cross section for slow neutrons 819).31Volume 6, number 1 PHYSICS LETTERS 15 August 1963Fig. 1. Experimental lattice vibrational spectrumof vanadium at 300oK.Fig. 2. Lattice specific heat of vanadium, calculated fromthe spectrum in fig. 1. The dashed curve takesaccouDt of the difference in the anharmonic contri-butions between room temperature aud the corre-sponding temperafures.Fig. 3. Apparent Debye temperatures. (For the dashedcurve see text,in fig. 2.)The lattice specific heat derived from this spectrumis given in fig. 2 *, the corresponding apparentDebye temperatures are shown in fig. 3..The 0 ver-sus ? curve has the usual low temperature mini-mum, which is due to the deviation of the true lat-tice spectrum from the spectrum of a Debye solid.Since the neutron data were taken at room tempera-ture, the speeific heat calculated from the latticevibrational spectrum is strictly valid only at thistemperature. For other temperatures the anhar-monic contribution to the specific heat will be dif-ferent. We may roughly estimate this modiJicationusing Griineisen?s law for the relation between Debyetemperature and volume change. We find* For the lowest frequencies of the spectrum we haveused the Debye approximation with I = 399o6 10).320(")=e(rn) {r-tu^r}K(where Ip is the reference temperature, a is thecoefficient of linear expansion, K is the Grtineisenconstant), and calculate the new lattice specificheat from the corrected apparent Debye tempera-ture, using the room temperature value as a ref-erence. The result of this calculation is shown infigs. 2 and 3 by the dashed curves. Due to the uncertainty of this correction we will restrict our follow-ing discussion to temperahres, say, above 150oK.150 200 250 300 350 T('K)Fig. 4. Lattice specific heat of vanadium above 150oK.-neutrou data, ---- caloric values of Cv/lR,values of (Cy - ys7)/3R using the experi-mental low-temperature value for y6 (= 2.21 x 10-3cal mol-1 deg-z;.In fig. 4 we compare the lattice specific heatabove 150oK derived from the neutron data withcaloric measuremenls 11,14,15). ff we subtractfrom the caloric data the electronic contributionusing the low--temperature value 12) , yo = 2.21 x10-3 cal mol-l d"g-2, we find points w[ich lie forall temperatures below the lattice specific heatcurve derived from the neutron data. Therefore,we conclude from fig. 4 that the 7e-value, whichwas found at low temperature, is no longer validat higher temperature. We propose that this effectmight be attributed to the renormalisation of thefree enefry of the electrons, which is caused bythe electron-phonon interaction.In order to give at least a qualitative justifica-tion of this view, we have evaluated the expressionfor the free enerry of the electrons, given byBuckingham and Schafrottr 5). ttreir theory is validfor free"electrons in a Debye solid and can thusserve only as an approximative check for a rela-tively complicated metal like vanadium.The electronic free energy consists in this ap-proximation of two terms, namely of the usualSommerfeld value g" and of a perturbation contri-bution 92:Lpel = es + e2 = ps {1 + lt + g(r/e)l (in)t F} ,t.ocLTCK)- cv-aro Ret'l)o Ref.rt)o Ref.'c)es = - 612 arlrp) @t)2 ;z 4 . o b toxtol2olume 6, number 1 PHYSICS LETTERS 15 August 1963is the number of free electrons per atom, f ishe interaction gonstant of Frdhlich 13), which forsuperconductOr, like vanadium is presumablyarger than (4n)-2; S(r) is a sum of two integrals'a(t) andf6Q) which carurot be evaluated in closed)rm:',@=+i &-{t2x2rcg#- ,*'t rog lffi} ,'u(t) = *ri *,ox {(x1+x2)z tos(x1+x2)2 - t-2(x1 + x2)2o 100 200 300 T(.K)Fig. 5. The tunction g(fl, defined iu eqs. (3) and (a) ,for two 0-values.Usingce=- r(azqu/airz) =ts11T, (5)'e see from fig. 5 that the electronic specific heatepends on temperature. OnIy for temperatures oft least the order of the Debye temperature we ob-rin the usual Sommerfeld expression (1). For low:mperatures the effective 7-value is higher de-ending on the interaction constant F.We may estimate this constant from the differ-nce between the lattice specific heat which wasalculated from the neutron data and the valueshich were derived from caloric data *, using thexpgrimental low-temperature value lo = 2.21 x0-3 cal mol-l deg-2. p"o,,7oTassuming n = L, we obtain F = 1.16. This value islarger than (42)-i - 0.63, as one expects for su-perconducting elements 13).Besides the fact that we find at low tempera-tures an effective 7-value, which is remarkablyhigher than the high-temperature value 7s, anotherinteresting conclusion may be drawn from fig. 5.If we calculate the Debye temperature of vanadiumfrom the elastic constants at low temperatures, asmeasured by Alers 10) , *" obtain 0eI = 399oK,which is remarkably different from the caloricvalue obtaiaed by Corak et aI. 12), 0cal = 338oK.The caloric value was derived from the usual C/Tversus T2 plot. This experimental discrepancymay be caused also by the renormalisation effectdiscussed above.From eqs. (3) and (4) follows for the y-value atIow temperatures, assuming that roughly g = cTZ(see fig. 5): r r7 = 7s {1 + (in)lF + 6(fu)ircr2} . (?)In eq. (7) appears a T2-term, which generates anadditional 7r-term in the expression for the latticespecific heat. This term may account for the dif-ference between elastic and caloric Debye temper-atures. Estimating c from fig. 5, we find a newcaloric Debye temperature, 0lat ", 3?0oK, whichagrees better with the elastic value than the pre-vious one. In view of the approximations made,this seems to be a further confirmation for theexistence of an appreciable renormalisation effectin the electronic specific heat of vanadium.It is a pleasure to thark Mr. Cocchi and Mr.De Bruyn from the Cetis Computational Centre fortheir help in the evaluation of the two integrals ineq. (4).1) A.Sommerfeld and H.A.Bethe, Handb. Physik 24 (1933)333.2) G.V.Chester and A,Houghton, Proc. Phys. Soc. (Lon-don) 73 (1959) 609.3) G.V.Chester, Adv. Phys. 10 (1961) 357.4) M.J.Buckingham, Nature 168 (1951) 281.5) M.J.Buckingham and M.R.Schafroth, Proc. Phys. Soc.(London) 67 (19s4) 828.6) H.Jones, Proc. Roy. Soc. (Londou) 240 (1957) 321.7) K.Krebs, Report EUR 366. e C.C.R. Ispra (1963) .8) K. C. T\rrbersfield and P. A. Egelstaff, Phys. Rev. 127(1962) 1017.9) R. Haas et al., Inelastic scattering of neutrons (I.A.E.A.,Vienna, 1963) p. 145.10) G.A.Alers, Phys. Rev. 119 (1960) 1932.11) K.Clusius, P. Franzosini and U.Piesbergen, Z. Natur-forsch.15a (1960) 728.12) W.S.Corak et al ., Phys. Rev. 102 (1956) 656.13) H. Frtihlich, Phys. Rev. 79 (1950) 845.14) C.T.Anderson, J. Am. Chem. Soc. 58 (1936) 564.15) D.R.Stull and G.C.Sinke, Adv. Chem. Ser. 18 (1956)2L6.(4)€24,2 f,o J,ro (1+expr1)(l +expx2)- @r-*il2 r.rt\!t*].\r7 - xD')ur numerical calculation of these two integralseads to the curves in fig. 5.g (r)* Gdir =+=C$aloric 17; - Cfeutron(?)(= 2.21/1.15)For ? = 300cK.-/->--xt-<---.-*****33

Renormalisation effect in the electronic specific heat of vanadium. Euratom Report, 1963. EUR 444.e

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