### Abstract

We revisit the problem of de Haas-van Alphen (dHvA) diamagnetic
susceptibility
oscillations in a
thin, free-electron film trapped in a synthetic harmonic potential well. A treatment of
this phenomenon at zero temperature was announced many years ago by Childers and Pincus
(designated hereafter as CP), and we traverse initially much the same ground,
but from a slightly different analytic perspective. That difference hinges around our use,
in calculating the Helmholtz free
energy
F, of an inverse Laplace transform, Bromwich-type contour integral representation
for the sharp distribution cutoff at Fermi level μ. The contour integral permits closed-form
summation all at once over the discrete orbital Landau energy levels transverse to the
magnetic field,
and the energy
associated with the in-plane canonical momenta ℏ k
x
and ℏ k
z
. Following such summation/integration, pole/residue pairs
appear in the plane of complex transform variable s, a fourth-order pole at origin
s = 0, and an infinite ladder, both up and down, of simple poles along
the imaginary axis. The residue sum from the infinite pole ladder automatically engenders
a Fourier series
with period one in dimensionless variable μ/ ℏ ω (with effective angular frequency ω
suitably defined), series which admits closed-form summation as a cubic polynomial within any given
periodicity slot. Such periodicity corresponds to Landau levels slipping sequentially
beneath Fermi
level μ as the ambient magnetic field
H declines in strength, and is manifested by the dHvA
pulsations in diamagnetic
susceptibility.
The coëxisting steady contribution from the pole at origin has a similar cubic structure
but is opposite in sign, inducing a competition whose outcome is a net magnetization that
is merely quadratic in any given periodicity slot, modulated by a slow amplitude growth.
Apart from some minor notes of passing discord, these simple algebraic structures confirm
most of the CP formulae, and their graphic display reveals a numerically
faithful portrait of the oscillatory
dHvA
diamagnetic
susceptibility
phenomenon. The calculations on view have a merely proof-of-principle aim, with no
pretense at all of being exhaustive. The zero-temperature results hold moreover the key to
the entire panorama of finite-temperature thermodynamics with T > 0.
Indeed, thanks to the elegant work of Sondheimer and Wilson, one can promote the
classical, Maxwell-Boltzmann partition function Z ( s ),
via an inverse Laplace transform of its ratio to s
^{ 2}, directly
into the required, Fermi-Dirac Helmholtz free energy
F at finite temperature T > 0. While the underlying
cubic polynomial
commonality continues to bestow decisive algebraic advantages, the evolving formulae are
naturally more turgid than their zero-temperature counterparts. Nevertheless we do retain
control over them by exhibiting their retrenchment into precisely these antecedents. So
fortified, we undertake what is at once both a drastic and yet a simple-minded step of
successive approximation, a step which clears the path toward numerical evaluation of the
finite-temperature diamagnetic
susceptibility. We
are rewarded finally with a persistent dHvA periodicity imprint, but with its
peaks increasingly flattened and its valleys filled in response to temperature rise, all
as one would expect on physical grounds.

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