solutionhw7 - Solutions to Homework Set 7 1 Critical mass...

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Solutions to Homework Set 7 1) Critical mass : We expand n ( x, t ) = X m =1 a m ( t ) sin mπx L , and also μ = X m, odd 4 μ sin mπx L , 0 < x < L. Substituting in the given equation, and using the linear independence of the sine functions, then gives ˙ a m ( t ) = λ - Dm 2 π 2 L 2 ! a m ( t ) + 4 μ , where the last term is present only when m is odd. Let us define α m = λ - Dm 2 π 2 L 2 ! . The solution to the evolution equation is either a m ( t ) = a m (0) + 4 μ mπα m e α m t - 4 μ mπα m , or a m ( t ) = a m (0) e α m t , depending on whether m is odd or even. The m = 1 mode is the first to go unstable, and this happens as soon as α 1 > 0, i.e. when L > L crit where L crit = π s D λ . To find the equilibrium distribution we solve D d 2 n dx 2 + λn + μ = 0 , with the boundary conditions n (0) = n ( L ) = 0. This is an inhomogenenous ODE with con- stant coefficients. It is therefore most easily solved by combining a complementary function n CF ( x ) = A cos q λ/D ( x - L/ 2) , which has been chosen to be symmetric about the midpoint of the slab, with the particular integral n PI ( x ) = - μ λ . 1
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To satisfy the boundary conditions we take n equilibrium ( x ) = μ λ cos q λ/D ( x - L/ 2) cos q λ/D ( L/ 2) - 1 .
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