Peretti, J.

Archive of European Integration


«Γι mm 
LEGAL 
JBüfeÉM . 
This document was prepared under the sponsorship 
tomic Energy Community (EURATOM). 
Neither the Eura tom Commission, its contractors nor any person acting on their behalf : 
Make any warranty or representation, express or implied, with respect to the accuracy, 
completeness, or usefulness of the information contained in this document, or tha t the use 
of any information, apparatus, method, or process disclosed in this document may not 
infringe privately owned rights; or raüp^r«' 
Assume any liability with respect to the use of, or for damages resulting from the use oi 
any information, apparatus, method or process disclosed in this document. Ti 
m) 3ïi 
omrnission of the E u r 
This reprint is intended for restricted distribution only. It reproduces, 
by kind permission of the publisher, an article from "PHYSICS 
LETTERS", Vol. 2, No. 6 ­ 1962, 275­276. For further copies 
olease apply to North­Holland Publishing Co., P.O. Box 103 — ' ¡rs π» ¡ι IBI) ytiiTiiif ι ηπΒ 
Amsterdam (Nederland). 
Dieser Sonderdruck ist für eine beschränkte Verteilung bestimmt. Die 
Wiedergabe des vorliegenden in­ ,,PHYSICS LETTERS" Vol. 2; 
Nr. 6 ­ 1962, pages 275­276 erschienenen Aufsatzes erfolgt mit 
freundlicher Genehmigung des Herausgebers. Bestellungen weiterer 
Exemplare sind an North­Holland Publishing Co., P.O. Box 103 ­
Amsterdam (Nederland) zu richten 
Ce tiré­à­part est exclusivement destiné à une diffusion restreinte. 
Il reprend, avec l'aimable autorisation de l'éditeur, un article publié 
dans «PHYSICS LETTERS», Vol. 2, N° 6 ­ 1962, 275­276. 
Tout autre exemplaire de cet article doit être demandé à North­
Holland Publishing Co., P.O. Box 103 — Amsterdam (Nederland).. 
i^wñwm^. ■BSwWfc t^rF^WWWftlÄmi; AVi¿ >¥w'v 5DSÍa;¿ «ilirMMHlannSj?'!! t· illa 
Questo estratto 'è destinato esclusivamente ad una diffusione limitata. 
Esso è stato riprodotto, per gentile concessione dell'Editore, da 
«PHYSICS LETTERS», Vol. 2, N° 6 ­ 1962, 275­276. Ulte­
riori copie dell'articolo debbono essere richieste a North­Holland 
Publishing Co., P.O. Box 103 — Amsterdam (Nederland). 
Deze overdruk is slechts voor beperkte verspreiding bestemd. Het artikel 
is met welwillende toestemming van de uitgever overgenomen uit 
„PHYSICS LETTERS", Vol. 2, no. 6 ­ 1962, 275­276. Meer 
exemplaren kunnen besteld worden bij North­Holland Publishing 
Co., P.O. Box X03 — Amsterdam (Nederland). 
fm 
¡t!'.·» I 
E U R 2 2 6 . e 
R E P R I N T 
SUPERCONDUCTIVITY OF TRANSITION ELEMENTS by J. PERETTI 
European Atomic Energy Community - EURATOM 
Joint Nuclear Research Center 
Ispra Establishment - Italy 
Reactors Physics Department 
Reprinted from "Physics Letters" Vol. 2, N. 6 - 1962 p. 275-276 
Summary not available 
E U R 2 2 6 . e 
R E P R I N T 
SUPERCONDUCTIVITY OF TRANSITION ELEMENTS by J. P E R E T T I 
European Atomic Energy Community - EURATOM 
Joint Nuclear Research Center 
Ispra Establishment - Italy 
Reactors Physics Department 
Reprinted from "Physics Letters" Vol. 2, N. 6 - 1962 p. 275-276 
Summary not available 
E U R 2 2 6 . e 
REPRINT 
SUPERCONDUCTIVITY OF TRANSITION ELEMENTS by J. PERETTI 
European Atomic Energy Community - EURATOM 
Joint Nuclear Research Center 
Ispra Establishment - Italy 
Reactors Physics Department 
Reprinted from "Physics Letters" Vol. 2, N. 6 - 1962 p. 27Õ-276 
Summary not available 

Volume 2, number 6 PHYSICS L E T T E R S 15 October 1962 
S U P E R C O N D U C T I V I T Y O F T R A N S I T I O N E L E M E N T S 
J . PERETTI 
Euratom, Department of Reactor Physics, Ispra, Varese, Italy 
Received 29 September 1962 
Since the discovery that no isotope effect exis ts 
for transit ion meta l s like Ru and Os 1), there has 
been some speculation as to whether , in some in­
s tances , the electron-phonon interaction is or i s 
not completely responsible for superconductivity. 
Empir ical evidence has led Matthias 2) to propose 
that there should be two types of superconductivity, 
one valid for non-transi t ion meta l s and due to the 
electron-phonon interact ion, the other suitable for 
transit ion meta l s and based on another mechanism, 
producing no isotope effect (Ru and Os) , or a par t ia l 
one (Mo 3)). His views seem to be supported also by 
the recent observations of Bucher et al . 4) and of 
Blaugher et a l . 5) that the pa rame te r V' =N(0)V/y 
i s increasing very rapidly with the electronic con­
centration for Nb-Ru and Nb-Mo alloys. These 
findings a re at var iance with the ideas expressed 
by Pines 6) based on the BCS theory Ό of supercon­
ductivity. Garland 8) has thus suggested that the 
Coulomb interaction between the s e lectrons is 
screened in transi t ion meta l s by the s-d electron 
cloud. He introduces a d ie lect r ic constant c(q, ω) 
and using a very specific model , he i s able to prove 
that in a certain domain e(q, ω) is negative, giving 
thus an at t ract ive interaction between the s e l ec ­
t rons . Unfortunately, his model leads to an energy 
gap much too large (0.03 eV) and to a f i rs t o rder 
transition at Tc. 
The purpose of this note i s to show that gener­
ally s-d interaction leads to superconductivity, and 
thus to support the idea of Garland. We use the 
Hamiltonian of Suhl 9 ) : 
H =H0+Hs+Hd + Hsd , (D 
where 
Ho - Ζ f/es cka cka + çkd dL· dka 
k ,Cf 
H* = - ¿ , v*{k'k,) <k+ <k- c-k'- c+k'+ > 
H« = ­ ~ki vd(k,k') d*+k+ dlk_ d_k,_ d+k,+ , 
Hsd = ­ Σ [vsd(k>k') ck+ c­k­ d­k<­ d+k<+ 
+ Vds(k,k<) d%k+ d*_k_ c_k,_ c+k,+] , 
where V(k,k') a r e the mat r ix elements (real) for the 
different in terac t ions , c,c*, d,d* the F e r m i opera ­
to r s for the s and d bands. V < 0 means a repuls ion, 
V > 0 an at t ract ion. The e's a r e the Bloch energ ies . 
We perform the Bogoliubov canonical t r ans fo rma­
tion as in ref. 9): 
Ck+ = uk eko + vk e^i , 
c_¿_ = ­vk e%0 + uk ekl , 
dk+ = nk fko + ck fkl > 
d­k­ = ~ek fko + ak fkl . 
and wri te that at the absolute zero the coefficients 
of £&0e&i and fkofkl a r e zer°> thus obtaining 
*ks 2ukvk - (sk + Nk) ("I - vb = ° . 
*kdûtfk - (Dk+Mk) (a¡ - of) = 0 , 
where 
Sk = A Vs(k,k') uk, vk, , 
(3) 
Dk =1 Vd(k,k<)fìk,0k, , 
k' 
Mk = Z Vds{k,k')uk, vk, , k' 
Nk = r vsd(k'*)ak<0k' ■ 
ft' 
The compatibility equations obtained from eqs . (2) 
and (3) a r e : 
V Sk<+Nk< 
Â ? ' ¿"A­'s 
(4a) 
275 
Volume 2, number 6 P H Y S I C S L E T T E R S 15 October 1962 
Dk 7¿k,k') 
Dk ' +Mk' 
2 Q k'd 
¡r· S, ι + N. t 
t vds(*'k') \rh, k 
k' ¿wk's 
(4b) 
Σ vs¿k,k·) Dp + Mk, 2Γ2 k'd 
ι 
^ks = [4s+(Sk+ Nk)2f>0 
Ω ¡fed = [4d^Dk + Mk)f 
The ground state energy is 
ff<°) = ­ ï 1 Ω71 sa 
(5) 
(6) 
and the elementary excitations are 
fe^o^o + ^ l ^ l ) 
+ Vkd{fk0fko+fklfkl)> (7) 
//<*) = Σ nte0 
In the limit of weak interactions S^, D^, M^, Nk 
are independent of & and given by: 
S =(S+N)VS Nshi[€1\S+N\~1] , 
D = (D +M)Vd Nd)ii[<L2 \D+M \ ~l] , 
M = (S +N) VsdNs In [eg \D+N | ­ 1 ] , 
Ν = (D + M) Vsd Nd In [ e 3 \D+M | _ 1] , 
(8) 
where Vs, Vd, Vsd are averages of the interactions 
over the Fermi surface, Ns, Ndthe density of s and 
d levels at the Fermi surface, e j , *2, f3, the dis­
tance from Fermi surface where interactions be­
come negligible. By a careful inspection it is pos­
sible to show that, provided thatiVgFg andNdVd 
are of the order of 10"1, or less , eqs. (8) admit 
always a non­trivial solution, whatever the sign of 
the interactions *. Because of eq. (6), anon­tr ivial 
solution gives always a ground state of lower ener­
gy than the normal state. This shows that on the 
basis of the Hamiltonian (1) one can predict that 
* We assume that Vsci £ 0; V^ φ VSV¿\ NSVS In e 1/83 f 1 and N¿V¿ In €2/83 φ 1. 
all the transition elements are superconductors. 
The argument fails naturally for those elements 
which exhibit a magnetic ordering since in this case 
it is not possible to perform the Bogoliubov t rans­
formation (see, e.g., Vonsovskii and Svirskii 10)). 
An order of magnitude for the energy gaps can 
be calculated for the simple case where Vs = V^ = 0 
It is found that the solution of eqs. (8) is 
5 =D = 0 
\M\ = (Nd/Nsy e 3 exp (­ l/\Vsd\JNs Nd) , 
\N\ (Ns/Ndf €3 exp (­ 1/1 Vsd\JNs Nd) 
(9) 
Assuming that Vsd is a Coulomb interaction, it is 
roughly equal to ­ ^e^KY~^ud/vs, where vd is the 
"effective" volume occupied by d electrons in each 
shell, and νs the volume of each cell. For vd/vs = 
IO"2 we find that | Vsd \JNsNd is of the order of 0.1, 
which gives a reasonable value for the energy gaps. 
Finally we notice that in eq. (9), e 3 is not pro­
portional to any vibration frequency, i.e., no iso­
tope effect is predicted; that | Vsd \*JNsNd is not 
proportional to γ since the latter is proportional to 
Ns + Nd. By the usual procedure of Bogoliubov et 
al. 11) the transition at Tc can be proved to be a 
second order transition. 
References 
1) T . H . Gebal le . Β. T. Matthias and E. Corenzwit , P h y s . 
Rev . L e t t e r s 6 (1961) 275. 
T . H . Geballe and B . T. Mat thias , IBM J . R e s e a r c h 
Develop. 6 (1962) 256. 
2) B . T . M a t t h i a s , J . Appi. P h y s . Suppl. 31 (1960) 235. 
B . T . M a t t h i a s , IBM J . R e s e a r c h Develop. 6 (1962) 
250. 
3) T . H . G e b a l l e and B . T . M a t t h i a s , P r o c . Vffl Int. Conf. 
on Low T e m p e r a t u r e Phys i c s , London 1962. 
4) E. Bucher , J . Muheim and J . Mül ler , P r o c . Vlllth Int. 
Conf. on Low T e m p e r a t u r e P h y s i c s , London 1962. 
5) R . D . B l a u g h e r et a l . , P r o c . VHIth Int. Conf. on Low 
T e m p . P h y s i c s , London 1962. 
6) D . P i n e s , P h y s . Rev . 109 (1958) 280. 
7) J . B a r d e e n , L . N . Cooper and J . R . S c h r i e f f e r , P h y s . 
Rev . 108 (1957) 1175. 
8) J . W . G a r l a n d , P r o c . Vmth Int. Conf. on Low T e m p e r ­
a tu re Phys i c s , London 1962. 
9) H.Suhl , B . T . M a t t h i a s and L . R . W a l k e r , P h y s . Rev. 
L e t t e r s 3 (1959) 552. 
10) S. V. Vonsovskii and M . S . S v i r s k i i , Doklady Akad. Nauk 
SSSR 122 (1958) 204. 
11) N . N . Bogoliubov, V. V .Tolmachev and D.V.Sh i rkov , 
A new method in the theory of superconduct ivi ty (New 
York, 1959) p . 100. 
* * * * * 
276 

:;ï 
§m¡ 
Sh 


SUPERCONDUCTIVITY OF TRANSITION ELEMENTS. EUR 226.e

«Γι mm LEGAL JBüfeÉM . This document was prepared under the sponsorship tomic Energy Community (EURATOM). Neither the Eura tom Commission, its contractors nor any person acting on their behalf : Make any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this document, or tha t the use of any information, apparatus, method, or process disclosed in this document may not infringe privately owned rights; or raüp^r«' Assume any liability with respect to the use of, or for damages resulting from the use oi any information, apparatus, method or process disclosed in this document. Ti m) 3ïi omrnission of the E u r This reprint is intended for restricted distribution only. It reproduces, by kind permission of the publisher, an article from "PHYSICS LETTERS", Vol. 2, No. 6 ­ 1962, 275­276. For further copies olease apply to North­Holland Publishing Co., P.O. Box 103 — ' ¡rs π» ¡ι IBI) ytiiTiiif ι ηπΒ Amsterdam (Nederland). Dieser Sonderdruck ist für eine beschränkte Verteilung bestimmt. Die Wiedergabe des vorliegenden in­ ,,PHYSICS LETTERS" Vol. 2; Nr. 6 ­ 1962, pages 275­276 erschienenen Aufsatzes erfolgt mit freundlicher Genehmigung des Herausgebers. Bestellungen weiterer Exemplare sind an North­Holland Publishing Co., P.O. Box 103 ­Amsterdam (Nederland) zu richten Ce tiré­à­part est exclusivement destiné à une diffusion restreinte. Il reprend, avec l'aimable autorisation de l'éditeur, un article publié dans «PHYSICS LETTERS», Vol. 2, N° 6 ­ 1962, 275­276. Tout autre exemplaire de cet article doit être demandé à North­Holland Publishing Co., P.O. Box 103 — Amsterdam (Nederland).. i^wñwm^. ■BSwWfc t^rF^WWWftlÄmi; AVi¿ >¥w'v 5DSÍa;¿ «ilirMMHlannSj?'!! t· illa Questo estratto 'è destinato esclusivamente ad una diffusione limitata. Esso è stato riprodotto, per gentile concessione dell'Editore, da «PHYSICS LETTERS», Vol. 2, N° 6 ­ 1962, 275­276. Ulte­riori copie dell'articolo debbono essere richieste a North­Holland Publishing Co., P.O. Box 103 — Amsterdam (Nederland). Deze overdruk is slechts voor beperkte verspreiding bestemd. Het artikel is met welwillende toestemming van de uitgever overgenomen uit „PHYSICS LETTERS", Vol. 2, no. 6 ­ 1962, 275­276. Meer exemplaren kunnen besteld worden bij North­Holland Publishing Co., P.O. Box X03 — Amsterdam (Nederland). fm ¡t!'.·» I E U R 2 2 6 . e R E P R I N T SUPERCONDUCTIVITY OF TRANSITION ELEMENTS by J. PERETTI European Atomic Energy Community - EURATOM Joint Nuclear Research Center Ispra Establishment - Italy Reactors Physics Department Reprinted from "Physics Letters" Vol. 2, N. 6 - 1962 p. 275-276 Summary not available E U R 2 2 6 . e R E P R I N T SUPERCONDUCTIVITY OF TRANSITION ELEMENTS by J. P E R E T T I European Atomic Energy Community - EURATOM Joint Nuclear Research Center Ispra Establishment - Italy Reactors Physics Department Reprinted from "Physics Letters" Vol. 2, N. 6 - 1962 p. 275-276 Summary not available E U R 2 2 6 . e REPRINT SUPERCONDUCTIVITY OF TRANSITION ELEMENTS by J. PERETTI European Atomic Energy Community - EURATOM Joint Nuclear Research Center Ispra Establishment - Italy Reactors Physics Department Reprinted from "Physics Letters" Vol. 2, N. 6 - 1962 p. 27Õ-276 Summary not available Volume 2, number 6 PHYSICS L E T T E R S 15 October 1962 S U P E R C O N D U C T I V I T Y O F T R A N S I T I O N E L E M E N T S J . PERETTI Euratom, Department of Reactor Physics, Ispra, Varese, Italy Received 29 September 1962 Since the discovery that no isotope effect exis ts for transit ion meta l s like Ru and Os 1), there has been some speculation as to whether , in some in­s tances , the electron-phonon interaction is or i s not completely responsible for superconductivity. Empir ical evidence has led Matthias 2) to propose that there should be two types of superconductivity, one valid for non-transi t ion meta l s and due to the electron-phonon interact ion, the other suitable for transit ion meta l s and based on another mechanism, producing no isotope effect (Ru and Os) , or a par t ia l one (Mo 3)). His views seem to be supported also by the recent observations of Bucher et al . 4) and of Blaugher et a l . 5) that the pa rame te r V' =N(0)V/y i s increasing very rapidly with the electronic con­centration for Nb-Ru and Nb-Mo alloys. These findings a re at var iance with the ideas expressed by Pines 6) based on the BCS theory Ό of supercon­ductivity. Garland 8) has thus suggested that the Coulomb interaction between the s e lectrons is screened in transi t ion meta l s by the s-d electron cloud. He introduces a d ie lect r ic constant c(q, ω) and using a very specific model , he i s able to prove that in a certain domain e(q, ω) is negative, giving thus an at t ract ive interaction between the s e l ec ­t rons . Unfortunately, his model leads to an energy gap much too large (0.03 eV) and to a f i rs t o rder transition at Tc. The purpose of this note i s to show that gener­ally s-d interaction leads to superconductivity, and thus to support the idea of Garland. We use the Hamiltonian of Suhl 9 ) : H =H0+Hs+Hd + Hsd , (D where Ho - Ζ f/es cka cka + çkd dL· dka k ,Cf H* = - ¿ , v*{k'k,) <k+ <k- c-k'- c+k'+ > H« = ­ ~ki vd(k,k') d*+k+ dlk_ d_k,_ d+k,+ , Hsd = ­ Σ [vsd(k>k') ck+ c­k­ d­k<­ d+k<+ + Vds(k,k<) d%k+ d*_k_ c_k,_ c+k,+] , where V(k,k') a r e the mat r ix elements (real) for the different in terac t ions , c,c*, d,d* the F e r m i opera ­to r s for the s and d bands. V < 0 means a repuls ion, V > 0 an at t ract ion. The e's a r e the Bloch energ ies . We perform the Bogoliubov canonical t r ans fo rma­tion as in ref. 9): Ck+ = uk eko + vk e^i , c_¿_ = ­vk e%0 + uk ekl , dk+ = nk fko + ck fkl > d­k­ = ~ek fko + ak fkl . and wri te that at the absolute zero the coefficients of £&0e&i and fkofkl a r e zer°> thus obtaining *ks 2ukvk - (sk + Nk) ("I - vb = ° . *kdûtfk - (Dk+Mk) (a¡ - of) = 0 , where Sk = A Vs(k,k') uk, vk, , (3) Dk =1 Vd(k,k<)fìk,0k, , k' Mk = Z Vds{k,k')uk, vk, , k' Nk = r vsd(k'*)ak<0k' ■ ft' The compatibility equations obtained from eqs . (2) and (3) a r e : V Sk<+Nk< Â ? ' ¿"A­'s (4a) 275 Volume 2, number 6 P H Y S I C S L E T T E R S 15 October 1962 Dk 7¿k,k') Dk ' +Mk' 2 Q k'd ¡r· S, ι + N. t t vds(*'k') \rh, k k' ¿wk's (4b) Σ vs¿k,k·) Dp + Mk, 2Γ2 k'd ι ^ks = [4s+(Sk+ Nk)2f>0 Ω ¡fed = [4d^Dk + Mk)f The ground state energy is ff<°) = ­ ï 1 Ω71 sa (5) (6) and the elementary excitations are fe^o^o + ^ l ^ l ) + Vkd{fk0fko+fklfkl)> (7) //<*) = Σ nte0 In the limit of weak interactions S^, D^, M^, Nk are independent of & and given by: S =(S+N)VS Nshi[€1\S+N\~1] , D = (D +M)Vd Nd)ii[<L2 \D+M \ ~l] , M = (S +N) VsdNs In [eg \D+N | ­ 1 ] , Ν = (D + M) Vsd Nd In [ e 3 \D+M | _ 1] , (8) where Vs, Vd, Vsd are averages of the interactions over the Fermi surface, Ns, Ndthe density of s and d levels at the Fermi surface, e j , *2, f3, the dis­tance from Fermi surface where interactions be­come negligible. By a careful inspection it is pos­sible to show that, provided thatiVgFg andNdVd are of the order of 10"1, or less , eqs. (8) admit always a non­trivial solution, whatever the sign of the interactions *. Because of eq. (6), anon­tr ivial solution gives always a ground state of lower ener­gy than the normal state. This shows that on the basis of the Hamiltonian (1) one can predict that * We assume that Vsci £ 0; V^ φ VSV¿\ NSVS In e 1/83 f 1 and N¿V¿ In €2/83 φ 1. all the transition elements are superconductors. The argument fails naturally for those elements which exhibit a magnetic ordering since in this case it is not possible to perform the Bogoliubov t rans­formation (see, e.g., Vonsovskii and Svirskii 10)). An order of magnitude for the energy gaps can be calculated for the simple case where Vs = V^ = 0 It is found that the solution of eqs. (8) is 5 =D = 0 \M\ = (Nd/Nsy e 3 exp (­ l/\Vsd\JNs Nd) , \N\ (Ns/Ndf €3 exp (­ 1/1 Vsd\JNs Nd) (9) Assuming that Vsd is a Coulomb interaction, it is roughly equal to ­ ^e^KY~^ud/vs, where vd is the "effective" volume occupied by d electrons in each shell, and νs the volume of each cell. For vd/vs = IO"2 we find that | Vsd \JNsNd is of the order of 0.1, which gives a reasonable value for the energy gaps. Finally we notice that in eq. (9), e 3 is not pro­portional to any vibration frequency, i.e., no iso­tope effect is predicted; that | Vsd \*JNsNd is not proportional to γ since the latter is proportional to Ns + Nd. By the usual procedure of Bogoliubov et al. 11) the transition at Tc can be proved to be a second order transition. References 1) T . H . Gebal le . Β. T. Matthias and E. Corenzwit , P h y s . Rev . L e t t e r s 6 (1961) 275. T . H . Geballe and B . T. Mat thias , IBM J . R e s e a r c h Develop. 6 (1962) 256. 2) B . T . M a t t h i a s , J . Appi. P h y s . Suppl. 31 (1960) 235. B . T . M a t t h i a s , IBM J . R e s e a r c h Develop. 6 (1962) 250. 3) T . H . G e b a l l e and B . T . M a t t h i a s , P r o c . Vffl Int. Conf. on Low T e m p e r a t u r e Phys i c s , London 1962. 4) E. Bucher , J . Muheim and J . Mül ler , P r o c . Vlllth Int. Conf. on Low T e m p e r a t u r e P h y s i c s , London 1962. 5) R . D . B l a u g h e r et a l . , P r o c . VHIth Int. Conf. on Low T e m p . P h y s i c s , London 1962. 6) D . P i n e s , P h y s . Rev . 109 (1958) 280. 7) J . B a r d e e n , L . N . Cooper and J . R . S c h r i e f f e r , P h y s . Rev . 108 (1957) 1175. 8) J . W . G a r l a n d , P r o c . Vmth Int. Conf. on Low T e m p e r ­a tu re Phys i c s , London 1962. 9) H.Suhl , B . T . M a t t h i a s and L . R . W a l k e r , P h y s . Rev. L e t t e r s 3 (1959) 552. 10) S. V. Vonsovskii and M . S . S v i r s k i i , Doklady Akad. Nauk SSSR 122 (1958) 204. 11) N . N . Bogoliubov, V. V .Tolmachev and D.V.Sh i rkov , A new method in the theory of superconduct ivi ty (New York, 1959) p . 100. * * * * * 276 :;ï §m¡ Sh 

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