The Plasma Effect on the Rate of Nuclear Reactions The connection with relaxation processes, Diffusion, scattering etc
Quick derivation of the Fokker Planck equation Let a particle with energy E- E changes its energy by E during time t be the probability that Then in steady state the distribution must satisfy t is the collision time (but it can be any time interval. It is easier and simpler this way) The distribution must yield itself for any t. Note: the integration is over all possible changes of energy during the collision E= E(E)
Expanding the integrand yields: Now substitute the expansion into the first integral
Define now These will be the coefficients in the Fokker-Planck equation Clearly The left hand are two functions of the energy and to know them we have to solve for the details of the collision between the relevant particles.
Substituting the results back we get: This is the Fokker-Planck equation for the plasma. The coefficients, which depend on the type of collision or interaction between the particles, must be evaluated in a separate calculation. The equation describes the plasma relaxation. If we consider steady state then the eq. becomes which can be easily integrated to
In equilibrium we must have: As is a positive function, it means that must change sign. For some of the energies there is energy gain while for the other there must be energy loss. In the Salpeter classical theory A particle does not gain, nor does it lose energy in a collision. The potential is rigid as if it had an infinite mass. In reality, irrespective if the plasma is in steady state or not. Particles collide with one another and exchange Energy.
Define as the probability that a particle with energy E in will gain energy E in as it comes from infinity, scatters from a given particle and reaches the distance of closest approach. Similarly, define as the probability that a particle with energy E out (at the point of closest approach) will gain energy E out as it moves out and separates.
The probability for the particle to come close and then separate and gain energy E= E in + E out is: where The classical approximation is where U is the mean electrostatic energy of the particle Or
The traditional FP eq. is expressed in terms of changes in velocity So that The FP eq. is then where is the collision time. In the case of pure and uniform (no correlations) Hydrogen plasma: Non vanishing contribution to the integral comes only from hence particles with different velocities experience different velocity change.
Define a generalized potenial H so that with In the case of uniform Hydrogen plasma and ignoring the electrons one can show that: where n the number density, m the mass of the particles and the reduced mass. H is called the first Rosenbluth potential. The second Rosenbluth potential yields the second coefficient of the FP eq.
The dynamic friction wants all particles to have the same speed- The average speed. The diffusion in velocity space wants the particles to have a uniform distribution in velocity space. A steady state is a balance between these two trends. The screening is intimately associated with the dynamic friction of the system.
Define P ij ( ) the probability that a particle in energy bin i will change its energy and move to bin j in collision time . Similarly, define P down (i, ) the probability that a particles in energy bin i will lose energy (irrespective of the final energy bin) Similarly, define P up (i, ) the probability that a particles in energy bin i will gain energy (irrespective of the final energy bin) The collision time is a parameter. We calculate the probabilities using our MD method. The probabilities are effectively products of the screening effect!
Brownian motion of a particle Assume a particle with mass m immersed in a medium which acts on the particle and scatters it. Further, assume that there is a conservative force, a friction force and a stochastic force acting on the particle. The equation of motion is: L(t) is a fluctuating or stochastic force, now known as Langevin force. In the classical formulation, the friction is written as: - v but it can have other forms. By stochastic we mean that has a finite memory.
Langevin The Langevin equation Fd - dynamic friction Q(t) - stochastic force A clear connection between the Fokker-Planck and Langevin eq. Assume that two protons scatter (head on) in plasma. The proper equation should be: The original Langevin eq.
The magnitude of the effect depends on what one assumes for ln
The solution of the Langevin equation is: (for example a particle Scattered by random magnetic fields) and Here:
In the particular case that the system tends to equilibrium we must have that D is the diffusion coefficient (in case it is isotropic) This is a manifestation of the Fluctuation- Dissipation Theorem. It is possible to show that the Langevin equation and the Fokker-Planck are connected as follows: where m(v,t) is the integral of f(x,v,t) over x.
where as before But now, f(v) is NOT the Maxwell-Boltzmann distribution but the un-relaxed distribution. The boundary condition is the t=0 distribution. The solution will give the distribution as a function of time, the spread etc
Conclusions: The screening is intimately connected to relaxation processes in the plasma. One cannot derive the screening from thermodynamics but one has to resort to kinetic equations. The Langevin equation and the Fokker-Planck equation confirm the numerical results obtained by means of Molecular Dynamics. The value of the screening depends on the composition of the plasma as well as on density and temperature.
Few interesting dates: Langevin - Brownian motion 1906 Chandrasekar - dynamic friction in stellar clusters 1943 Schatzman- screening in White Dwarf 1943 Salpeter - the screening theory 1954(weak limit)-1960(strong limit) Van Horn & Salpeter-unified thermodynamic screening theory1960-1965 DeWitt et al -detailed thermodynamic calculations of screening 1970-1990 (Ichimaru et al 1980-1990,Book Shaviv NJ & Shaviv G- the present theory 1997-2002 Rosenbluth,Jadd & McDonalds; The Rosenbluth potentials 1956
Astrophysical conclusion: With the new screening (which is still approximate and not for all compositions) the B 8 flux falls inside the parameter range of the astrophysical uncertainties. What did we neglect? Light elements -> additional macroscopic mixing processes Heavy elements accretion during the early solar formation …………………. Deviations from the Maxwell Boltzmann distributions in the tail (slowing down, magnetic fields…) ………………….
RMP nuclear data: Salpeter screening vs. limit B
Present status: If the approximate screening factors are used - no problem with the amplitude. Must obtain the screening factor for Helium rich compositions. This is not yet the final answer The important point: The screening correction does not have the same sign for all reactions
The solar neutrino problem: It is impossible to vary the unknown astrophysical input Parameters and to bound the neatrino flux The new screening allow a variation in various unknown Astrophysical parameters in an effort to obtain the right B 8 flux. However, the ratio of Be 7 /B 8 neutrino fluxes is not explained. yet