Question

15.4.1

The one-dimensional Fermi age equation for the diffusion of neutrons slowing down in some medium (such as graphite) is 15.4.1 a2q(x, T) aq(x, T) ax2 Here q is the number of neutrons that slow down, falling below some given energy per second per unit volume. The Fermi age, T, is a measure of the energy loss If q (x, 0) S8(x), corresponding to a plane source of neutrons at x0, emitting S neutrons per unit area per second, derive the solution x2/4T q S /4πτ Hint. Replace q(x, T) with 1 p(k, τ)- 27T q (x, τ)eίk dx . -00 infinite medium This is analogous to the diffusion of heat in an

It's in Mathematical Methods for Physicists 6e, Arfken ch15

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Answer 1

defination of Fousies apedhes Let us se the t9onsflm and its FOusies tsansldm of xt) luith Vatiable thon hunt) ts nverse Fousies onpm ikx Fat)e dx 2 TT V 2 TT ikx (i Pont) 2 TT Honde the Feulies onsm of (ix PC,t)

gven aflying Fousier tharalerbh Sidey GRYPCK, p(k,r) this is bne dimensi pnal dientiel equatin -kdT J#Jr unti geating dr beth Side On p:e e Cluhere us tegeation rtad C given 9(0)= S Sx) ok,o)= S (by definition f Feubiet tsanydn) than m S= ee hence PCK, T)= SET

nd9x,) by taking inae Foue tharafem a)= We ikx 2 IT e dk Se C 2 TT = (ax)b dk 2 1T -X (2rju 2 1T dk k 4TV 2 7T x/4r dk r (k - ix 21 9(x,r) = S 2 TT 21T xl4T S e ( Jk 2 TT

o thgawion inteial ddmula -k/2a dk= J2TT Honce 2T X4T TT S e gx,T)= 2 T X74 4 TIT