This is the oldest amongst my publications which I was able to recover in digital form so far. PC’s appeared at our Department in 1987, but some texts written in 1987-90 are now lost with deteriorated floppys, old Winchesters & such. This text is complete, except Title and Abstract Pages which were then manufactured with some special IBM typewriter. So now I retyped Title Page, and also the English Abstract; not the Hungarian & Russian ones.

For documentary purposes I did not improve even trivial mistypes; but I corrected formulae. Namely, old MS-DOS 3.0 did not produce half of Greek alphabet and most of logical signs. In the original these were produced by pen. Also, the old and new ASCII tables are not identical.

From the paper you can learn
two things. First, that even Superconductivity, generally described by full
Field Theory (of Cooper pairs) can partially be understood using only parts of
Quantum Mechanics. Second, that how tricky are *we* (I mean, Martinás and
I), to recognise and do this. Earlier we wrote a bigger and even more
interesting paper about this programme; but that was in 1984, too early for a
file. That is Ref. 3.

The text was published in Acta Physica Hungarica in due course. If you check it, you will see an inverse author sequence there. But the preprint (this text) has priority and it was ready even before sending the material to Acta.

___________________________________________________

KFKI-1987-21/E

PREPRINT

**ELEMENTARY QUANTUM
PHYSICAL DESCRIPTION**

**OF TRIPLET
SUPERCONDUCTORS**

B. LUKÁCS and K. MARTINÁS^{*}

Central Research Institute for Physics

H-1525 Budapest 114, P.O.B.49, Hungary

^{*}Department
for Low Temperature Physics,

Roland Eötvös University,

H-1088 Puskin u. 5-7., Budapest, Hungary

**ABSTRACT**

The basic
bulk properties of Type I and II superconductors (as the ef-fects of temperature
and external magnetic field) has been obtained by using only three elementary
facts of quantum physics, Bohr’s quantum principle, uncertainty relation, and Pauli’s
exclusion principle, instead of the full quantum theory. The same method for k > 1 triplet superconductors gives
that, if spin-1 Cooper pairs are bound, the perfect conductor state with B = H
is thermodynamically preferred to the diamagnetic one at any H ą 0 magnetic field; this perfect conductor
state ceases to exist at H_{c2}.

**1. INTRODUCTION**

There is an interesting discrepancy between the generality of superconductivity on one hand (more than 50% of metals is proven to possess a superconducting phase and the theoretical complexity of microscopic explanation on the other. Weisskopf [1] already has demonstrated that a partial but quite detailed understanding can be achieved by using full quantum mechanics but not quantum field theory. Here we use an even more simplified description of superconductivity (for details see Refs. 2 and 3) to investigate triplet superconductors; only elementary constants and data of the lattice ions will be used, and of course the results are expected to be correct only up to number constants of order of unity.

**2. ELEMENTARY QUANTUM
PHYSICS OF SUPERCONDUCTORS**

Consider an ideal
metallic lattice with positive ions and a free electron gas. In first
approximation the opposite charges compensate each other. In second, the moving
electron disturbs the ion lattice causing an effective positive charge near to
its path, which, on a second electron moving collinearly, acts by a potential
U~-(m/M)^{1/2}.(e^{2}/r),where e is the elementary charge, M is
the ion mass, and m is the electron mass. If this were a classical potential,
there would be a bound state of the electrons with a characteristic energy 600
K, too high in metallic superconductor context. Nevertheless, the uncertainty
principle gives a simple, correct estimation. There are momentum and position
uncertainties, so the ground state energy of a pair can be written as

E = (3/2)(¶p^{2})/m -
(m/M)^{1/2}(e^{2}/¶x) (1)

¶p¶xł~~h~~/2

Hence, looking for energy minimum, one obtains

E = -(2/3M)(me^{2}/~~h~~)^{2} (2)

While the M dependence of this energy does not show the correct isotope effect, its numerical value is in the correct order of magnitude, 2 K for a metal of atomic mass 50. So one can conclude that, via lattice oscillations, two electron states may appear with a binding energy.

Since the creation
of such pairs is energetically favoured, one expects the sea of pairs in the
T=0 ground state. Elementary symmetry and quantum considerations yield that the
Cooper pair consists of two electrons being as collinear as possible in order
to maximize the attraction; but, on the other hand, it is a resonance with
finite size x=2v_{F}~~h~~/E_{b} and a
minimal momentum uncertainty p_{o}=E_{b}/v_{F}, which
forbids exactly zero total momentum. The optimal compromise is a state with
total momentum p_{o}; it cannot be smaller, and the binding is weaker
if it is greater.The individual electrons of the pairs are continuously
replaced by those of other pairs, and this procedure leads to a preference of
spin-0 pairs; for further details see Ref. 3.

Consider now an
external effect not disrupting but modifying the superconducting state. It can
only change the total momentum of the pairs, as there are no further parameters
to be modified. The change of the total pair momentum appears in an excess
uncertainty: the disturbed quasiparticle possesses a greater size x_{d},

p_{o}^{2}
= (¶p_{d})^{2}
+ (~~h~~/x_{d})^{2} (3)

Then the new binding energy is

E_{b} = E_{bo}(1
- (¶p_{d}/p_{o})^{2})^{1/2} (4)

In case of thermal excitations
(but not disruption) of Cooper pairs T is comparable or smaller than E_{b},
so the thermal excitation energy is within the energy uncertainty of the pairs,
so it seems that the Fermi distribution of the individual electrons does not
influence the possible excitations, i.e. a Boltzmann approximation can be
used, so

¶p_{T} = k_{B}T/v_{F} (5)

For a usual Cooper pair, being a particle of 0 spin and 0 momentum, the magnetic field can interact only with the individual electrons. (Spin-1 pairs will be discussed at the end of this Chapter.) The interaction via momentum leads to

¶p_{HM} = p_{o}(H/H_{c2}) (6)

where H_{c2}=F_{o}/2px^{2} is the upper critical
magnetic field.

In case of interaction via spin the field H can produce a change in the spin orientation, leading to a change in the potential energy

¶V = 2(e~~h~~/mc)H (7)

while the kinetic energy of the Cooper pair changes by

¶E = (¶p)^{2}/m (8)

In dynamic equilibrium they have to be in the same order of magnitude, so

(¶p)^{2} » p_{o}^{2}(H/H_{c2}) (9)

Combining the thermal and magnetic effects, the binding energy is as follows [3]:

E_{b}(T,H)
=E_{b}(0,0){1-(T/T_{c})^{2}-(H/H_{c2})^{2}-(H/H_{c2})}^{1/2} (10)

For spin-1 Cooper
pairs the magnetic field can interact also with the pair as a whole, and the
released interaction energy can be transferred into e.g. lattice vibrations,
which is an external heat reservoir for the electron gas, therefore this
interaction will not change the binding energy. Then, repeating, *mutatis
mutandis*, the above steps, one gets eq. (10) without its last term:

E_{b}(T,H)
= E_{b}(0,0){1-(T/T_{c})^{2}-(H/H_{c2})^{2}}^{1/2} (11)

**3. THERMODYNAMICS OF
THE SUPERCONDUCTING STATE**

Since superconducting samples are handled at constant temperature and magnetic field, the actual state is selected by the minimum of the Gibbs potential G

G = E - TS - BH/4p (12)

The energy of a superconducting state can be approximated as

E_{s }=
E_{n}- (1/4)VN(E_{F})E_{b}^{2} (13)

[3] where N is the state density; for a cold Fermi gas NµÖE, so, using eq. (11) and the definitions

H_{c} = F_{o}/2plx (14)

l^{2} = mc^{2}/4pne^{2}

one gets

E_{s}=
E_{n} -(H_{c}^{2}/8p){1-(T/T_{c})^{2}-(H/H_{c2})^{2}-(H/H_{c2})}^{1/2}(15)

Again, this is for spin-0 pairs; for spin-1 ones the last term is absent.

Now let us indeed
select the actual state by the minimum of G. For this one has to compare
states of different structure. The list contains at least the following ones:
normal (E=E_{n}, B=H); Meissner (E=E_{s}, B=0); mixed (E=E_{s}(B),
B<H) and a perfect conductor" (E=E_{s}(B), B=H).

For comparison the Gibbs potentials of the usual states (i.e. the first three ones) can be found in Ref. 3. So we have to deal here only with the "perfect conductor". Its Gibbs potential can be directly obtained by using eqs. (12), (15) (with or without the last term, according to the pair spin) and the definition of that state.

After trivial calculation, for spin-0 Cooper pairs:

G_{p} =
G_{M} - (H_{c1}H + H^{2}/k^{2} - H^{2}) (16)

where k is the GL
dimensionless parameter. The Meissner state is below the normal one until H_{c1},
and in this whole range the bracketed term is positive. Above H_{c1}
one could easily show that G_{p}>G_{mixed}. This is just the
standard result that the "perfect conductor" state is not realised.

However, consider the case of spin-1 pairs. Here, according to eq. (11), the term linear in H is absent, that is

G_{p} =
G_{M} - (H^{2}/8p){1 - k^{-2}} (17)

Now, obviously, this
means that for the cases k>1 the "perfect conductor" state is always
preferred to the Meissner state. The decrease in G preferring the mixed state
to the Meissner one is due to the penetration [3] already hap-pened here too.
Thus, if this system has a superconducting state, then that state is a perfect
conductor until H_{c2}.

**4. CONCLUSIONS**

Here we have demonstrated that fundamental quantum principles and thermodynamics do not rule out the possibility of a perfect conductor state, i.e. superconduction without diamagnetism. In fact, such states are rather predicted, but only when the Cooper pairs exist in spin-1 state. This is just the case of triplet superconductors [4], [5].

The authors would like to thank Dr. Gy.Wolf for technical help.

**REFERENCES**

[1] V. F. Weisskopf, Contemp.
Phys. __22__, 375 (1981)

[2] K. Martinás and I. Kirschner, Acta Phys.
Hung. __40__, 297 (1976)

[3] K. Martinás and B. Lukács, KFKI-1984-77

[4] K. Svozil, Phys.
Rev. __B33__, 602 (1986)

[5] P. B. Allen and B.
Mitrovic, Solid State Phys. __37__, 1 (1983)

**My HomePage, with some other studies**, if you are curious.