The Adjoint Problem

Solutions to the adjoint transport equation are useful for calculating the change in the response of a system to a small perturbation and for calculating the source that will give a desired response. Adjoint fluxes are being used more and more frequently to compute the sensitivity of a response to changes in the basic cross sections.

The adjoint transport equation can be obtained from the direct or forward equation by transposing the scattering and fission matrices. However, this converts the downscatter calculation into an upscatter calculation. If the order of the groups is then reversed, the cross sections can be used with the regular transport equation; the solution for the direction $\vec{\Omega}$ is just the adjoint flux for direction $-\vec{\Omega}$ numbered backwards.

The existing SN codes perform this operation of transposition and reordering of the groups, but only if the entire cross-section table fits into core storage at once. TRANSX can produce an adjoint set in group ordering, thereby allowing fine-group adjoint solutions to be obtained. When TRANSX adjoint tables are used, the SN code must be run in forward mode, and the adjoint flux is obtained from the SN flux solution $\sigma (g,\vec{\Omega})$ using

\phi^+ (g,\vec{\Omega}) = \phi(N{-}g{+}1,-\vec{\Omega}).

Note: In the absense of a good HTML math mode, the equations have been left in TeX format.

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