Eur. Phys. J. B 39, 427–431 (2004)
DOI: 10.1140/epjb/e2004-00213-y
T
HE
E
UROPEAN
P
HYSICAL
J
OURNAL
B
Superconducting transition temperatures of the elements related
to elastic constants
G.G.N. Angilella
1,a
,N.H.March
2,3
, and R. Pucci
1
1
Dipartimento di Fisica e Astronomia, Universit`a di Catania, and Istituto Nazionale per la Fisica della Materia,
UdR di Catania, Via S. Sofia 64, 95123 Catania, Italy
2
Department of Physics, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium
3
Oxford University, Oxford, UK
Received 11 February 2004 / Received in final form 13 May 2004
Published online 23 July 2004 –
c
EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2004
Abstract. For a given crystal structure, say body-centred-cubic, the many-body Hamiltonian H in which
nuclear and electron motions are to be treated from the outset on the same footing, has parameters, for
the elements, which can be classified as (i) atomic mass M, (ii) atomic number Z, characterizing the
external potential in which electrons move, and (iii) bcc lattice spacing, or equivalently one can utilize
atomic volume, Ω. Since the thermodynamic quantities can be determined from H, we conclude that T
c
,
the superconducting transition temperature, when it is non-zero, may be formally expressed as T
c
=
T
(M)
c
(Z, Ω). One piece of evidence in support is that, in an atomic number vs. atomic volume graph, the
superconducting elements lie in a well defined region. Two other relevant points are that (a) T
c
is related
by BCS theory, though not simply, to the Debye temperature, which in turn is calculable from the elastic
constants C
11
, C
12
,andC
44
, the atomic weight and the atomic volume, and (b) T
c
for five bcc transition
metals is linear in the Cauchy deviation C
∗
=(C
12
− C
44
)/(C
12
+ C
44
). Finally, via elastic constants, mass
density and atomic volume, a correlation between C
∗
and the Debye temperature is established for the
five bcc transition elements.
PACS. 74.62.-c Transition temperature variations – 74.70.Ad Metals; alloys and binary compounds
1 Background and outline
We have recently been concerned with both empirical and
theoretical relations between the superconducting tran-
sition temperature T
c
of high-T
c
cuprates and of heavy
Fermion materials [1–3]. The generally complex crystallo-
graphic structure of such compounds makes it difficult to
identify useful correlations between their superconducting
properties (such as T
c
) and the elastic properties of the
lattice. This is not the case of several superconducting
elements with a definite and relatively simple crystallo-
graphic structure, e.g. characterized by only a few non-
zero components of the elastic tensor. Although any such
correlation applying to the ‘simple’ superconducting ele-
ments may not be immediately generalized to other un-
conventional superconductors, they are anyway expected
to focus on the relevant variables which would be worth-
while studying, both experimentally and theoretically, also
in the new classes of superconductors.
Following the Bardeen-Cooper-Schrieffer (BCS) the-
ory [4] of the metallic elements, firmly rooted in electron-
phonon interaction as the basic mechanism resulting in
a
e-mail: giuseppe.angilella@ct.infn.it
the formation of Cooper pairs, questions have come up
regarding the role of strong electron-electron interactions
in both the high-T
c
cuprates and heavy Fermion systems.
Here, our basic philosophy will be to insist that if we
were able to solve the many-body Schr¨odinger equation
for the (considered infinite) superconducting materials,
then by treating the motion of nuclei and electrons on the
same footing, plus full inclusion of electron-electron inter-
actions, such uncertainties involved in separating electron-
lattice and Coulomb repulsions between electrons would
be bypassed.
Having said that, let us take as the simplest start-
ing point the metallic elements. Then, the input infor-
mation into any computer programme to treat these ele-
ments would be as follows. First, of course, we should need
to specify the structure. To be definite, below we shall
single out the body-centred cubic (bcc) lattice, but every-
thing that follows would be equally applicable to the more
closely packed face-centred cubic (fcc) structure. Once the
structure is specified, one would need to insert the atomic
volume Ω (or, of course, essentially equivalently, the lat-
tice parameter a). Then, the external potential created by
the nuclei must be specified, which requires the atomic