1. THE CONTRIBUTION OF WIGNER CRYSTALLIZATION
TO THE EVOLUTION OF WHITE DWARF STARS
Michael L. Montagne

2. ABSTRACT
This paper will review the contribution of Wigner
crystallization to the cooling of white dwarf stars. The basic
properties of white dwarfs are described. The nature of Wigner
crystallization in a one component plasma is characterized. The
effect of this phenomenon on the evolution of white dwarfs is
depicted.

3. INTRODUCTION TO NEUTRON STARS
Stars with less than about 4 M complete their evolutionary
progress by becoming white dwarfs. In the pre-white dwarf stage
mass is reduced to around 1.4 M. At this time the star consists
primarily of carbon or carbon and oxygen with an outer layer of
helium and possibly a thin shell of hydrogen. Due to the low
mass, the star is incapable of significant further fusion
reactions and is supported against gravitational contraction by
the pressure of degenerate electrons.
Since the interior of a dwarf is completely ionized and the
electrons are freed from their atomic bonds the density of a
white dwarf reaches very high levels. The resulting density
constraint will be
p>3AZmp/4πa0
3
=3.2AZg/cm3
.
This will produce densities in the interiors in excess of
104
g/cm3
. Nearly all the mass density is supplied by the atomic
nuclei while nearly all the pressure is supplied by the
degenerate electrons. Thus the hydrostatic and thermal
equilibrium equations are decoupled from one another. In low
mass white dwarfs, the electron pressure is still governed by
non relativistic equations of state. This allows for a
relationship between mass and radius of M~R-3
. The degenerate
electrons provide the primary means of energy transport in the
interior. White dwarfs, like main sequence stars, are
classified into several spectroscopic groups. The
classification system underwent extensive development in the
early 1980's as improved observations provided more detailed

4. information about the temperature, luminosity, and composition
of various white dwarfs. The system rates the stars based
on their primary spectroscopic features in the visible spectrum,
secondary spectroscopic features in any part of the
electromagnetic spectrum, and temperature. The major
classification groups are: DA, stars whose spectra show only
Balmer lines, no He I or metals are present, DB, stars whose
spectra show only He I, no H or metals are present, DC, stars
with a continuous spectrum and no lines deeper than 5% anywhere
in the spectrum, DO stars show strong HeII and may have HeI or H
lines, DZ stars have metal lines only with no H or He, DQ are
stars which show any kind of atomic or molecular carbon anywhere
in their spectrum. Other upper case letters are used to
indicate the secondary characteristics, and numbers from 1-9 are
used to indicate the temperature range.
WHITE DWARF COOLING
Since the star is no longer capable of sustained nuclear
fusion reactions, the evolutionary process of a white dwarf
consists primarily in transporting thermal energy from its
interior to the surface where it is radiated away as
electromagnetic energy. Energy is also transported away by
neutrino emissions. The development of a white dwarf then is a
gradual process of cooling to a final equilibrium state.
White dwarf stars have several different cooling
mechanisms. Each of these makes various contributions to the
release of thermal energy depending on; the phase of the star's
evolution, the composition of the star, and its mass. Although

5. all white dwarfs shed energy as electromagnetic radiation from
their surfaces, there are different means by which that energy
is transported from the interior to the radiative surface. The
degenerate core of a white dwarf is nearly isothermal and
gradually cools as its residual heat escapes through the
nondegenerate surface layers of the star. In general the
relationship between the luminosity L of a white dwarf and the
amount of time that it has been cooling can be expressed by the
simple power law function
tcool L-5/7
(D'antona and Mazzitelli 1990,). The various methods that white
dwarfs exhibit to expel energy include the following mechanisms.
During the late pre-white dwarf stage and the early parts of the
white dwarf's existence, nuclear fusion can still play a
significant role in its energy production. This will take the
form of both p-p fusion and He reactions. Helium burning ceases
very quickly when the star reaches a luminosity of L≤100 LO.
Although theoretical calculations show that p-p reactions can
play a major role in white dwarf luminosity the thinness of the
H shell and the temperature dependence of the reaction make it
unlikely that the appropriate conditions are often met (D'Antona
and Mazzitelli 1990).
The CNO cycle might provide a dramatic result in some cases
however. If the He layer between the H shell and the carbon
oxygen core is sufficiently thin then diffusion could bring the
hydrogen into proximity with the carbon and oxygen. This could
result in an explosive occurrence of the CNO cycle. A self

6. induced nova is produced by the flash of the hydrogen shell.
In the early stages of white dwarf evolution a substantial
amount of thermal energy is generated as a result of
gravitational contraction. Gravitational energy affects
primarily the less degenerate outer portions of the star. As
more energy is released and more of the stellar material becomes
degenerate it eventually reaches a state of equilibrium with its
gravitational field. Gravitational contraction contributes
primarily to raising the Fermi energy of the degenerate
electrons. Only a small fraction of it contributes to the
luminosity of the star (Lamb and Van Horn 1975). It should be
noted however that in the later stages of its evolution the
gravitational contribution to luminosity can experience a
temporary increase. After most of the latent heat has been
radiated away, continued gravitational action on the outer
layers of the atmosphere which are not yet fully degenerate can
contribute as much as 30% of the star's luminosity.
Another significant source of energy loss is neutrino
emission. When CNO burning ceases to support the star the
surface luminosity decreases substantially. The interior
temperature drops only slowly however. In this stage neutrino
emission becomes the dominant means of energy release for the
star. When Log L/L ≈ 1.5 neutrinos can be carrying away as
much as 300% more energy than thermal radiation. (D'Antona and
Mazzitelli 1989).
In the degenerate core which is very close to its final
radius at the beginning of the white dwarf stage, any energy

7. given off by contraction will, according to the virial theorem,
primarily result in an increase in the kinetic energy of the
degenerate electrons. The primary mechanism for reducing the
thermal energy of a white dwarf core is the liberation of
thermal energy during Wigner crystallization.
WIGNER CRYSTALLIZATION OF ONE COMPONENT PLASMAS
Wigner's original work (Wigner 1934) showed that under the
proper conditions of temperature and pressure a one component
plasma can crystalize. A one component plasma consists of ions
or nuclei of charge Ze, mass M, number density rn and temperature
T embedded in a neutralizing background of degenerate electrons.
The Coulomb interactions between the ions can be expressed as
Γ=(Ze)2
/akT
in this equation a is the radius of the system and obeys the
relation (4/3)πa3
=pn.
When Γ<<1 the electrostatic interactions are unable to produce
correlations between the ions and the system behaves as a gas.
As Γ increases the system will first pass through a liquid stage
and then eventually become arranged in to body centered cubic
lattice structures. (Ichimaru 1982)
The total Helmholtz free energy for such a system can be
expressed as a function of Γ. The free energy of the liquid
phase is
F(0)
/NkT = ln{p(2πh2
/MkT)3/2
}-1 = -0.7153+3/2ln(kT)Ry
(Pollack and Hansen 73a). For the solid phase this is
F/NkT = (Fharm
/NkT)-1750/Γ2
=
-0.895929Γ+9/2lnΓ-1.8856+3/2ln(kT)Ry-1750/Γ2

8. (Pollock and Hansen 73b). From these two expressions it is
possible to determine the point of phase transition from liquid
to solid. Using very precise Monte Carlo calculations it has
been determined that the melting line on a graph of pressure p
vs temperature T will be T=Xp1/3
where X=(Ze)2
(4π/3)1/3
/158k.
CRYSTALLIZATION AND WHITE DWARF EVOLUTION
The actual transition within a white dwarf will be a first
order phase change. The release of latent heat during the phase
change results in a decrease in the rate of cooling of the star.
This produces crystallization sequences which are analogous to
the main sequence. The stars of higher mass will have greater
densities and therefore will tend to crystallize at higher
central temperatures.
The phase transition will cause a discontinuous change in
the symmetry properties of the distribution of ions within the
stellar plasma. The abrupt change in the symmetry of the
distribution function of the particles implies a discontinuity
in the average Coulomb interaction energy per particle. In a
white dwarf of .6Mo the crystallization would result in the
release of approximately 5x1045 ergs which would be several
percent of the entire thermal energy of the star. The latent
heat of crystallization can be expressed in terms of the entropy
UCoul of the system as TδS=T(Ssol-Sliq)=δUCoul=-3kT/4. The negative
result is due to the fact that the energy is emitted during
transition from liquid to the solid phase.
The general expression for the pressure of an interacting
Coulomb system (Landau and Lifschitz 1958) gives us this

9. expression for the pressure in the solid phase
P-P0 = 1/3nikT(UCoul/kT)
where P0 = Pi+Pe is the pressure of the noninteracting plasma of
ions plus electrons and UCoul/kT is a function of Γ. It follows
from this that in a phase transition at constant density the
pressure of the solid phase would be less than that of the
liquid phase by 1/4 nik. In point of fact though the white dwarf
is in a state of equilibrium. This means that pressure and
temperature must be continuous across the interface between the
two phases. This results in a discontinuity in the density of a
surface as it crystallizes. This means that there is actually
only a small change in pressure (Van Horn 1967).
Significant convective forces can occur in white dwarfs
such that if they are sufficiently cool deep circulation can
occur allowing convective energy transport through the
nondegenerate He and H layers and mixing of central and outer
components. This phenomenon has an effect on the
crystallization process. Fully explaining the effects of
crystallization requires taking into account the interplay
between the Coulomb interactions in the interior including
crystallization and the convection in the envelope.
Lamb and Van Horn (1975) developed the equations of state
for both the interior and outer layers of a crystallizing pure
12C white dwarf of 1 M. They then used that information to
develop a complete evolutionary model of such a star. They
determined that crystallization would occur when Γ=160±15. They
also found that ion quantum effects will play an important role

10. in defining the location and properties of the phase transition
in the more massive white dwarfs.
In comparing the total heat capacity to temperature at a
typical density of 107.5
gm/cm3
two important aspects were
observed. The ionic heat capacity will be large in the dense
liquid state because of the Coulomb interactions. This implies
that there will be only a small discontinuity in the heat
capacity as the phase transition occurs. There will however be
a rapid decrease in the heat capacity of the crystalline
material when the temperature drops below the Debye temperature.
The crystallization boundary can be treated completely
within the equation of state. When a particular mass shell
crosses the crystallization boundary as determined by the values
for pressure and temperature the thermodynamic quantities
required at that shell can be calculated according to the
crystal rather than the liquid equation of state. The ion
equation of state for the crystal phase is expressed as
Pi=Pp+Pl, Pl=1/3ulp, Pp=1/2upp
where Pi is the total ion pressure, P and u are the pressure and
specific internal energy and the subscripts l and p indicate the
static lattice and phonon contributions. The internal energies
can be approximated as
ul=-ΓkT/AH, up=3kT/AH.
The latent heat energy released during crystallization is
δElatent heat=⌠δulatent heatdm=⌠qdm/AH
From this the total luminosity of a star undergoing
crystallization is determined to be

11. L+Lv=d/dt(Ethermal+Egrav+Elatent heat+Eδp/p=f(L/L)kM/AHdTc/dt
(Lamb and Van Horn 1975). In which the cooling function f(L/L)
gives the effective heat capacity per ion in dimensionless units
(Van Horn 1968). The luminosity L of a white dwarf that has
crystallized will be
Lx≈-kT(d/dt)(Mx/AH)=(kT/AH)(dMr/dp(r))(dpx/dT)(dT/dt)
(Van Horn 68). Models with a low mass tend to have higher
luminosities in the early parts of their evolution. This is
because their history involves less total energy loss.(Shaviv
and Kovetz 1976)
There are five distinct phases in the crystallization
process. In the first the neutrino luminosity falls below the
photon luminosity and the time scales for decrease of the
thermal energy content and the luminosity of the dwarf become
longer. In the second the heat capacity of the ions begins to
become dominant over that of the electrons. As this happens the
dimensionless cooling function approaches the value 3 which is
characteristic of high temperature crystals. The third
constitutes the onset of actual crystallization. The additional
energy supplied by latent heat causes a sharp rise in the value
of the cooling function and a slowing of the luminosity decline.
The fourth phase is marked by an increase in Debye cooling as
the latent heat release declines. This results in a rapid loss
of thermal energy. Finally deep convection reaches the core
producing an even more rapid decline of the central temperature
and luminosity. (Lamb and Van Horn 1975)
Thermodynamic considerations favor the final structure

12. consisting of a mixture of macroscopic grains of pure element.
This is true even for a star consisting of a mixture of carbon
and oxygen. The other possibilities, lattice sights randomly
occupied by different ions or a distribution determined by the
ration of carbon to oxygen are less likely regardless of the
ratio off carbon to oxygen. This greatly simplifies the task of
expressing the thermodynamic quantities in a white dwarf model.
Although one of the elements will solidify sooner than the other
and begin to sediment out, the sedimentation process is slow
compared to the to the total process and the time difference of
sedimentation is minute compared to the time for a given
fraction of stellar mass to crystallize.
CONCLUSION
White dwarf stars lack any internal source of energy
production. Their evolutionary development consists of
releasing the remaining internal energy until a state of thermal
equilibrium with the surrounding space is reached. Energy is
radiated in the form of photons and neutrinos. As the star
cools the major form of energy loss in the core comes in the
form of Wigner crystallization of the one component plasma.
Crystallization alters the thermal properties of the core
material and thus causes the release of large amounts of latent
heat. This causes a temporary increase in the star's luminosity
and a slowing of the overall cooling process. Once the latent
heat has been radiated away, luminosity and temperature begin to
decline very rapidly.

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