ORNL-1771 - Oak Ridge National Laboratory

ANP QUARTERLY PROGRESS REPORT
of collisions in dV, for which the particle emerges in the solid angle da with direction 2 is equal to
where
fmtl (P,&) dV, rtw = do $$$ X(d0,&9 e-ar r2 dr dw, f,,,(QId,) 6oo ,
r = [ ( x - xo)2 + (y - yo) 2 + (z - zo) 2 ,i
Here ,Amo is the solid angle subtended at 0 by dV, or by dV,/r2; by putting r2 & dfijo = ~ V Q then ,
Let ySt(i;) bethe normalized spherical surface harmonics. For convenience, these functions can be ordered
in a single series Yc:
Yo = Yo,o; Y, = Y . Y
1,-1 ' 2 = Y1,o; Y, = Y1,,; Y, = Y2,-2; Y5 = Y2,-1; ...
Let
From Eg. 1,
fm(Q, z) = f,,,(Q) yc(d)
Introducing a Fourier transformation of both sides and using
k,x + k2y + k,z = kr cos (k,t) ,
With
gives
where
-
dV, = d(u - xo) d(y - yo) d(z - zo) .
The direction 2, is the direction from Q to P, so that, as indicated, the transform of the right side of Eq. 1
breaks up into a sum of products of Fourier transforms. The last integrals on the right can evidently be
evaluated independently of the position 0. By expanding S in the form
SG,, e, = Aqp Yp(Go) Y,(3 ,