But in this BCS reference case, the conservation of momentum in itself
does not imply the conservation of energy. The BCS reference creates a
lattice(not geometrically defined) lowered in energy by a gap in which
Cooper electron pairs move without resistance (They are elastic) But I
do not see from the BCS reference the basic application of conservation
of energy and momentum to achieve the required elastic characteristic.
My approach is to actually define the lattice amenable to conservation
of energy and momentum. It is a hexagonal lattice with each cell having
base B and height A and volume(cavity) = 2*sqrt(3)*A*B^2 and wave
KB = pi/B
KC = 4pi/(3sqrt(3)A)
KDs = (8pi^3/cavity)^(1/3)
KDn = (6pi^2/cavity)^(1/3
The hexagonal lattice has a necessary space filling property wherein its
real and reciprocal are equal compatible with the following:
p = mv = hK and E = (1/2)mv^2 = h^2*K^2/(2m)
The elastic(conservation of energy and momentum) conditions are as
gs(KB + KC) = KDs + KDn (conservation of momentum)
KB^2 + KC^2 = gs(KDs^2 + KDn'^2) (conservation of energy)
The elastic condition (conservation of energy and momentum) is met with
the following numbers.
B/A = 2.379146658937169267…
gs = 1.0098049781877999262…
by other means a mass(mT) is derived
m= 110.107178208 x electron mass
The hexagonal model can be scaled keeping B/A a constant.
And finally a critical superconductor temperature (Tc) is defined:
E = Boltzmann Constant * Tc = h^2*KB^2/(2m)
time = 2*B/vdx
This superconductor critical Tc model is in general agreement with over
100 experimentally observed superconductors, nuclear quark parameters,
and celestial observations including the rotation time(8.22 hr) of the
interstellar interloper Oumuamua indicating this superconductivity
concept permeates the entire universe.
As a final note, this superconducting model conforms to the Einstein
stress energy tensor that is an elastic criterion.
Richard D Saam