Electrostatic Screening of Nuclear Reactions in Stellar Plasmas

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Electrostatic Screening of Nuclear Reactions in
Stellar Plasmas
Erice
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Plan of talk I
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Plan of talk II
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What is the problem? Where it arises? Why bother?
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In core of stars - where the nuclear reactions
take place- the matter is in a form of plasma a
sea of bare protons, bare He nuclei as well as
(almost) fully stipped heavier ions with free
electrons.
An ion in the plasma feels a potential created by
the sum of the interaction of all particles. The
sum of all interactions changes the simple
Coulomb force (and potential).
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The motion of a charge particle in plasma
We search the electrostatic potential of a
proton moving in the plasma
Let
be the single particle distribution function
The electric potential of the proton with speed
vp relative to the laboratory
The undisturbed number density of particles
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The governing equations
Solution first linearize, then Fourrier
transform to get
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For vanishing velocity one finds
Here RD is the Debye radius and it expresses the
rate at which the simple Coulomb potential decays
in plasma
This is the Debye Huckel potential. It is the
effective potential in the plasma. Note the
potential does not depend on velocity (or kinetic
energy) of the particle in this limit.
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The classical picture of the screening Shatzmann
1948 Salpeter 1954.
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The enhancement factor fexp(-Epot/kT) gt 1
But this is all static the potential does not
depend on the relative velocity between the
particles
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Simple questions
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Getting the effect on the nuclear reaction
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If the effective potential does not depend on the
velocities we can take the exponent out of the
integral
This is the Salpeter result.
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Typical numbers
Enhancement factor fexp(G)
Weak screening limit G ltlt 1
fBe7p 1.4-1.5
In WDs (SNIa) the screening correction is about
1018!
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In the solar core ND 3-4
The expansion is not valid and three body and
higher correlations may be important
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Finding the screening potential
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What is the effective potential?
What is the effective interaction between two
particles in the plasma? Does it depend on X and
V? Can the effective potential depend on V when
the basic interaction depends on X only?
Various authors used different approximations to
get the effective potential. The controversy is
about the dependence of the potential on the
velocity.
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We need the effect at the Gamow peak What is the
Gamow peak
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So, let us find the effective potential
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The ensemble average
Take the canonical ensemble
is separable the potential does not depend on
the velocity
Define the operator
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The ensemble average of the potential is
The kinetic energy factors out
The average potential energy of a particle does
not depend on its kinetic energy Ensemble
average the average energy of every particle is
on a long time scale the same. This is so because
every particle must go through all energy states
of the system. Hence, the long time means long
enough for it to go through all states.
But we want only particles in the Gamow peak, we
do not care what happens in other energies.
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Time averages
The time average of the potential energy of a
particle is
In the limit of very long time
The limit does not depend on the particular
particle. All particles have the same limit.
For thermodynamics we need
Time average
Canonical average
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The ergodic assumptionThe two averages are equal
If we assume the ergodic assumption we have
thermodynamics and canonical averaging.
But what we need is
Or alternatively
The sum is only over the relevant periods, when
the kinetic energy is in the Gamow peak.
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The basic question
Mean potential energy of particles in the Gamow
peak
Mean potential energy of a particle
If there is dynamic screening a la Koonin et al.
there should be no equality!
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The Molecular Dynamic calculation
Classical MD 512 - 1024 particles periodic
boundary conditions (Ewald sums)
The returned particle forgets the conditions
around it.
Actual calculations with 2363 93312 to 2403
128,000 particles
You must have a large number of particles to
probe the high energy tail of the Maxwell
Boltzmann distribution
The interaction with up to 1200 particles is
included
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Results for a high density low T case
The main reason for these conditions to enhance
the effects. In the Sun
We have results also for the sun
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A snapshot at the distribution of potential
energies in the system.
n1029/cc, T1.5 x 107K
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The potential energy per particle as a function
of kinetic energy for protons. Also shown, the
standard deviation, the error and the number of
particles. n1029/cc, T1.5 x 107K
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The distribution of the potential energy, ltf2gt1/2
and ltfgt as a function of the kinetic energy of
the proton.
n1029/cc, T1.5 x 107K
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A comparison of the potential energy per particle
distribution for two assumed masses of the proton.
n1029/cc, T1.5 x 107K
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Comparison between the kinetic energy
distribution and the MB distribution.
n1029/cc, T1.5 x 107K
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Conclusions
The ergodic assumption is fully satisfied
The conditional average is equal to the ensemble
one.
There is no dynamic screening
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Is it possible to calculate the screening effect
(the plasma interaction with a scattering pair)
from first principles? ...and in a clear and
obvious way?
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How to calculate directly the screening
Consider the energy of the entire plasma
Consider the total energy of just a pair of
mutually scattering protons (Z1).
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The classical approach
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Define screening energy energy
transfer between the pair and the plasma during a
collision.
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The description of the path of two particles.
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The potential energy of particle 1, The relative
kinetic energy and the distance from the paired
particles as a function of time.
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The long time average potential (The Debye
potential)
This is the mean field.
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The change of the energy of the pair due to
plasma interaction during the collision for
collisions with maximum distance of closest
approach rmaxlt0.25.
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The dependence of the assumed max(rmin) on the
energy exchange. The number in brackets give the
maximum distance of closest approach
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The effect of the proton mass on the screening
energy as a function of the relative kinetic
energy when far away. The red is mp0.001amu and
the blue mp1amu.
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The change in the energy of the center of mass
during The scattering of two protons
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Results for the pp in the core of the Sun
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The Be7p reaction
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The results for the He3He3
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The mean energy exchange between the scattering
particles in the plasma vanishes!
0!
As it should in a plasma in equilibrium, namely
zero!
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Power
Plasma frequency
w
tun
The power spectrum of the fluctuations
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In the classical approach one considers a binary
collision and the interaction of the pair with
many other particles and the interaction is
approximated by a mean field. This mean field is
not dynamic but some long time average of the
interactions.
We find that under the conditions prevailing in
stellar cores the pair interacts with few
particles only. The recoil of the particles
affects the results. The light electrons do not
affect the single scattering.
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conclusions
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So what is the screening?
We saw that the mean screening energy is a
function of relative kinetic energy.
There is a wide distribution of screening around
the average for each energy.
On the other hand, the nuclear cross section is
very non linear
Hence, one has to average over the distribution
of the screening and over the Gamow peak
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Thanks for your attention
End of part I