Difference between revisions of "Solution details"
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<math> n= R_k (r') Pk(\mu) = \exp{(\lambda_L z)}\sum_{k=0}^{\infty} A_k r'^{-1/2} K_{k+1/2} (\lambda_L r') P_k(\mu) </math> | <math> n= R_k (r') Pk(\mu) = \exp{(\lambda_L z)}\sum_{k=0}^{\infty} A_k r'^{-1/2} K_{k+1/2} (\lambda_L r') P_k(\mu) </math> | ||
− | In case of GEM preamplifer r >>a, since each hole has a radius of 50 um, and the electron streams appear in a mm scale distance. So k's value increases, the coefficient <math> \lambda_L </math> will decreases to | + | In case of GEM preamplifer r >>a, since each hole has a radius of 50 um, and the electron streams appear in a mm scale distance. So k's value increases, the coefficient <math> \lambda_L a </math> will decreases to a value that makes all the higher order terms negligible compared to the first (monopole) and the second (dipole) term. So the solution can be written for k= 0 and k = 1 as : |
<math> n= R_k (r') Pk(\mu) = \exp{(\lambda_L z)}\sum_{k=0}^{\infty} A_k r'^{-1/2} K_{k+1/2} (\lambda_L r') P_k(\mu) </math> | <math> n= R_k (r') Pk(\mu) = \exp{(\lambda_L z)}\sum_{k=0}^{\infty} A_k r'^{-1/2} K_{k+1/2} (\lambda_L r') P_k(\mu) </math> |
Revision as of 22:59, 26 October 2013
Asymptotic solution details for Boltzmann equation for a hole has a uniform electric field
n + - n = 0
Steps to solve Boltzmann equation <ref name="Huxley"> Huxley, L. G. H. Leonard George Holden, The diffusion and drift of electrons in gases, John Wiley and sons, 1974 , call number QC793.5.E628 H89 </ref>
for the previous equation let consider the asymptotic solution has the form:
so
where
and
In spherical coordinates:
which is symmetric in direction.
Assuming
the solution of the zenith angle direction is the Legendre polynomial, and can be written as:
and
so,
The modified Bessel functions, first and second kind, are the solutions for the previous equation but the boundary conditions determines which one to use, in this case ,
, and
as
.
so only the modified Bessel of second kind K_k are the non-zaro terms. so the the general solution for the equation can be written as :
Solution Analysis
The general form of the previous equation and its solution are defined as the following:
and
and
where
and are the modified Bessel of the first and the second kind.In case of of solving for the density number outside the hole, then the solution contains only the modified Bessel of the second kind
, Also applying the boundary conditions below , when , and as where a is the hole radius.implies that
if k is odd, so the solution can be written as :
In case of GEM preamplifer r >>a, since each hole has a radius of 50 um, and the electron streams appear in a mm scale distance. So k's value increases, the coefficient
will decreases to a value that makes all the higher order terms negligible compared to the first (monopole) and the second (dipole) term. So the solution can be written for k= 0 and k = 1 as :
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