Acta Astronautica 67 (2010) 1333–1336
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Acta Astronautica
journal homepage: www.elsevier.com/locate/actaastro
Academy Transactions Note
Correlation between transition temperature and crystal structure of
niobium, vanadium, tantalum and mercury superconductors
H.P. Roeser a,n, D.T. Haslam a, J.S. López a, M. Stepper a, M.F. von Schoenermark b,
F.M. Huber b, A.S. Nikoghosyan c
a
b
c
Institute of Space Systems, University of Stuttgart, Pfaffenwaldring 31, 70569 Stuttgart, Germany
German Aerospace Center, GSOC, Muenchnerstr. 20, 82234 Wessling, Germany
Department of Microwave and Telecommunication, Yerevan State University, Alex Manoogian 1, Yerevan 0025, Armenia
a r t i c l e in fo
abstract
Article history:
Received 17 May 2010
Accepted 30 June 2010
Available online 17 July 2010
The bond length (x) of transition metals Nb, V, Ta and Hg represents the shortest atomic
separation in a crystal unit cell. It is suggested that there exists a strong correlation
between (x) and the inverse of the critical superconducting transition temperature Tc in
the form (2x)2 Neff = m2 1/Tc. Here Neff is the number of electrons in the outermost s-shell.
The slope of the fitted straight line has a value of m2 E 3.0 10 18 m2 K.
& 2010 Elsevier Ltd. All rights reserved.
Keywords:
Conventional superconductor
Superconductor crystal structure
1. Introduction
2. Crystal structure and particle in the box concept
Many papers have shown experimental evidence that
the critical transition temperature Tc of any type of
superconductor depends among other criteria on the
crystal structure.
In this paper, we will investigate if there exists a
correlation between the geometry of the crystal structure
and the transition temperature Tc of well known one
component superconductors at normal pressure. It is
assumed that the one component superconductors of
interest are very pure materials, whose critical transition
temperatures have been established by many experiments. To start with, the selected superconductors have a
well known, simple and highly symmetrical crystal
structure.
The energy level estimate for electronic excitation in
an atom is given by
E¼
h2
¼ 6:4 1019 J 4 eV
8me d2A
ð1Þ
where me is the electron mass and dA the atomic diameter,
leading to a typical value of 4 eV for dA =0.3 nm. Eq. (1)
can be considered as the lowest energy level E1 of a
‘‘particle in the box’’ (PiB) with a quantum well width
equal to the diameter of the atom [1].
The next fundamental units/structures following an
atom are a molecule for the gas phase and a crystal unit
cell for the solid phase formed by equal atoms. Each
crystal is characterized by a unit cell, which specifies the
geometrical arrangement of neighbouring atoms. In this
case the energy level estimate for a unit cell should read
in analogy
E
h2
8M a2
ð2Þ
n
Corresponding author. Tel.: + 49 711 685 62375;
fax: + 49 711 685 63596.
E-mail address: [email protected] (H.P. Roeser).
0094-5765/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.actaastro.2010.06.048
where (a) is the lattice constant (typically 0.2 nmoa o1.2
nm). The mass M is not well defined, because one has to
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H.P. Roeser et al. / Acta Astronautica 67 (2010) 1333–1336
consider the number of atoms per unit cell NAUC as well as
the number of participating electrons per atom. Probably
it will also depend on the different properties of conductors, semiconductors and insulators.
If the structure offers straight lines or arrays of
equidistant atoms with separation length (x) throughout
the unit cell we could relate the energy of the unit cell to
the ground state E1 of a one-dimensional PiB, which scales
with the size of the unit cell.
To determine the effectively participating number of
particles Neff, the number of valence electrons Nval or the
electrons in the outermost shell Nos has to be taken into
account. Therefore Eq. (2) should be transformed by
M =Neff Meff into
E1 ¼
h2
Neff 8Meff x2
ð3Þ
On the other hand, a superconductor is characterized
by its transition temperature Tc and the thermal energy
Eth of the unit cell related to it. Therefore Eth is a function
of kTc expressed by Eth = f(kTc). We are interested in a
correlation between Eth and the maximum value of E1
which is given by the shortest atomic separation (x) in a
straight line of equidistant atoms through the whole
crystal. Usually, (x) represents the bond length in the
crystal. The energy E1 is then replaced by f(kTc) and Eq. (3)
can be rewritten as
ð2xÞ2 Neff ¼
h2
2Meff f ðkTc Þ
Fig. 1. The unit cell of niobium, vanadium and tantalum has a bodycentred-cubic (bcc) crystal structure and the shortest distance between
two atoms is given by the bond length between the central and a corner
atom.
ð4Þ
In the following, conventional superconductors with
simple crystal structures will be analyzed by plotting
(2x)2 Neff versus 1/Tc.
Fig. 2. The unit cell of mercury Hg (a) has a rhombohedral crystal
structure with lattice constant a= b = c= 0.3005 nm and a = b = g = 70.5201.
3. Niobium, vanadium and tantalum
Niobium, vanadium and tantalum are type II superconductors and their transition temperatures range from
4.55 K for Ta up to 9.25 K for Nb [2,3]. They are transition
metals characterized by an odd number of protons and an
even number of neutrons in the atomic nucleus. They
have in total an odd number of 5 valence electrons within
the two outer d- and s-electron shells with an incomplete
d-sub-shell. Although the d-shell has higher energy levels,
the experimental observations show that the electrons of
the s-shell are removed first, e.g. when oxidized. Geometry and reactivity of transition metals are described by the
ligand and the crystal field theory [4].
Nb, V and Ta have body-centred cubic (bcc) crystal
structures (space group Im-3m; space group number 229)
with a =b =c and a = b = g = 90.01 as shown in Fig. 1. The
atomic parameters are summarized in Fig. 3. The fundamental bcc crystal cell has 1+ 8 1/8=2 atoms. The
highest value for E1 is given by the shortest atomic
distance
between a corner atom and the centred atom by
1=2
x ¼ a 34
forming a straight line of equidistant atoms
throughout the whole crystal. If coupling between
equidistant atoms is happening along the diagonal lines,
it would include all atoms in a bcc structure by counting
1 þ 2 12 ¼ 2 and all other atoms are participating in
neighbouring unit cells. All atoms at the corner have the
same distance to the body-centred atom so that all
diagonals through the body are equal. The analysis for
the three materials results in (2x)2NeffTc E3 10 18 m2 K
when considering the outermost s-shell electrons only
(Neff =Nos).
4. Rhombohedral mercury Hg (a)
Mercury is a transition metal which has an electron
shell configuration of 5d10 5f0 6s2 and an even number of
80 protons. Compared to V, Nb and Ta the d-sub-shell is
completely filled. The crystal structure of Hg (a) belongs
to the rhombohedral structure (space group R-3m, space
group number 166) with cell parameters a =b= c =0.3005
nm and a = b = g =70.5201 and only one atom per unit cell
(Fig. 2). Hg (a) shows an isotope effect in superconducting
with a temperature variation of 4.18 K 4Tc 44.12 K.
For this investigation Hg (a) with Tc = 4.16 K will be used,
which is related to mixed isotopes with the mass
199.7 mu [1,2,5]. The shortest distance between two
atoms is given by the lattice constant (a) and a
superconducting effect along the lattice edges would
include all atoms and would count as 4 14 ¼ 1 direction.
The three rhombohedral edges represent equal directions.
The analysis results in (2x)2 Neff Tc E3 10 18 m2 K when
H.P. Roeser et al. / Acta Astronautica 67 (2010) 1333–1336
considering
(Neff = Nos).
the
outermost
s-shell
electrons
only
5. Results and summary
Fig. 3 summarizes the atomic and lattice structure data
of the different materials and the result of the structural
analysis. Fig. 4 shows the diagram by plotting (2x)2 Neff
versus the inverse critical transition temperature Tc. The
lattice parameters are very well known with an accuracy
better than 72 10 13 m. Therefore the uncertainty
of the graph is mostly determined by the experimental
values for the transition temperatures. Tc is considered
to be taken at half of the transition interval of the
1335
resistivity-temperature curves in contrast to the onset
transition temperature very often used in literature. The
uncertainty for Tc for all materials is about 70.3%. The
data points can be fitted by a straight line
ð2xÞ2 Neff ¼ m2 1=Tc
ð5Þ
with a slope of m2 =(2.996 70.014) 10 18 [m2 K]. The
intersection with the ordinate axis is remarkably small
( +0.304 10 20 m2) and will be neglected for the
moment. Obviously a correlation is only found when
Neff =Nos. It is surprising that not the number of valence
electrons Nval but the number of the outermost s-shellelectrons Nos is relevant, which is similar to the behaviour
by oxidation as already mentioned in Section 3 [4].
Fig. 3. Atomic and crystal structures of the transition metals V, Nb, Ta and Hg.
Fig. 4. Correlation between the bond length (x) in the unit cell and the critical superconducting transition temperature Tc. Neff is the number of electrons
in the outermost s-shell.
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H.P. Roeser et al. / Acta Astronautica 67 (2010) 1333–1336
For a more general discussion more data points of
other one component superconductors and different
crystal structures will be necessary in order to find out
if they also match the linear behaviour.
So far the results confirm the presumption of a linear
correlation between the geometry of the crystal structure
and the transition temperature which has already been
mentioned by the authors [6]. It also shows a similar
behaviour to high temperature superconductors (HTSC),
where (x) is the separation of doped unit cells, which
correlates with 1/Tc too [7,8]. For HTSCs the slope m1 of
the straight line (2x)2 =m1 1/Tc has a value of
m1 =2.771 10 15[m2 K] and is m1/m2 E925 times larger
compared to m2.
Acknowledgements
We would like to thank H. Hall, J. Vernerey and
E. Messerschmid for discussions and valuable comments.
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