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I wish to know if someone could help me with the adjoint multigroup diffusion equation. In particular with the terms that make up the macroscopic removal cross section. Below, both the multigroup diffusion equation and its adjoint are shown, but I'm not sure about the latter. I would be extremely grateful to anyone who could help me out with it.

Thank you very much.

Multigroup Diffusion Equation: $$ -D_i\nabla^2 \phi_i + \left( \Sigma_{c,i} + \Sigma_{f,i} \right) \phi_i + \left( \Sigma_{el,i\rightarrow \left( i + 1 \right)} \right) \phi_i + \sum_{j = i + 1}^{Ng} \left( \Sigma_{el,i\rightarrow j} \right) \phi_i = \left( \Sigma_{el,{\left(i - 1 \right)}\rightarrow i} \right) \phi_{\left(i - 1 \right)} + \sum_{j = 1}^{i - 1} \left( \Sigma_{inel,j\rightarrow i} \right) \phi_j + \chi_i \sum_{j = 1}^{Ng} \left( \nu \Sigma_{f,j} \right) \phi_j $$

Adjoint Diffusion Equation: $$ -D_i\nabla^2 \phi^*_i + \left( \Sigma_{c,i} + \Sigma_{f,i} \right) \phi^*_i + \left( \Sigma_{el,i\rightarrow \left( i + 1 \right)} \right) \phi^*_i + \sum_{j = i + 1}^{Ng} \left( \Sigma_{el,i\rightarrow j} \right) \phi^*_i = \left( \Sigma_{el,i \rightarrow \left( i + 1 \right)} \right) \phi^*_{ \left(i +1 \right)} + \sum_{j = i + 1}^{Ng} \left( \Sigma_{inel,i\rightarrow j} \right) \phi^*_j + \nu \Sigma_{f,i} \sum_{j = 1}^{Ng} \left( \chi_j \phi^*_j \right) $$