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There are two velocity groups and two regions (core \ and reflector) giving the following four diffusion equations for the neutron \ flux:\n\t", Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["D", "C1"], RowBox[{ SuperscriptBox["\[Del]", "2"], SubscriptBox["\[CapitalPhi]", "C1"]}]}], " ", "\[Equal]", " ", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ SubscriptBox["\[CapitalSigma]", "aC1"], "+", SubscriptBox["\[CapitalSigma]", RowBox[{"12", "C"}]]}], ")"}], SubscriptBox["\[CapitalPhi]", "C1"]}], " ", "-", RowBox[{ FractionBox["1", "k"], RowBox[{"(", RowBox[{ RowBox[{ SubscriptBox["\[Nu]", "1"], " ", SubscriptBox["\[CapitalSigma]", "f1"], SubscriptBox["\[CapitalPhi]", "C1"]}], "+", " ", RowBox[{ SubscriptBox["\[Nu]", "2"], " ", SubscriptBox["\[CapitalSigma]", "f2"], SubscriptBox["\[CapitalPhi]", "C2"]}]}], ")"}]}]}]}]], CellChangeTimes->{{3.394072752487823*^9, 3.394072759115205*^9}, { 3.3940727946620293`*^9, 3.394072816655294*^9}, {3.3941094509049177`*^9, 3.394109569378408*^9}}, EmphasizeSyntaxErrors->True], " [1a]\n\t", Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["D", "R1"], RowBox[{ SuperscriptBox["\[Del]", "2"], SubscriptBox["\[CapitalPhi]", "R1"]}]}], " ", "==", " ", RowBox[{ RowBox[{"(", RowBox[{ SubscriptBox["\[CapitalSigma]", "aR1"], "+", " ", SubscriptBox["\[CapitalSigma]", RowBox[{"12", "R"}]]}], ")"}], SubscriptBox["\[CapitalPhi]", "R1"]}]}]], CellChangeTimes->{{3.394072752487823*^9, 3.394072759115205*^9}, { 3.3940727946620293`*^9, 3.394072816655294*^9}, {3.3941094509049177`*^9, 3.394109569378408*^9}}, EmphasizeSyntaxErrors->True], " [1b]\n\t", Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["D", "C2"], RowBox[{ SuperscriptBox["\[Del]", "2"], SubscriptBox["\[CapitalPhi]", "C2"]}]}], " ", "==", " ", RowBox[{ SubscriptBox["\[CapitalSigma]", "aC2"], SubscriptBox["\[CapitalPhi]", "C2"]}]}]], CellChangeTimes->{{3.394072752487823*^9, 3.394072759115205*^9}, { 3.3940727946620293`*^9, 3.394072816655294*^9}, {3.3941094509049177`*^9, 3.394109569378408*^9}}, EmphasizeSyntaxErrors->True], " - ", Cell[BoxData[ RowBox[{ SubscriptBox["\[CapitalSigma]", "sC1"], SubscriptBox["\[CapitalPhi]", "C1"]}]], CellChangeTimes->{{3.394072752487823*^9, 3.394072759115205*^9}, { 3.3940727946620293`*^9, 3.394072816655294*^9}, {3.3941094509049177`*^9, 3.394109569378408*^9}}, EmphasizeSyntaxErrors->True], " [1c]\n\t", Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["D", "R2"], RowBox[{ SuperscriptBox["\[Del]", "2"], SubscriptBox["\[CapitalPhi]", "R2"]}]}], " ", "==", " ", RowBox[{ RowBox[{ SubscriptBox["\[CapitalSigma]", "aR2"], SubscriptBox["\[CapitalPhi]", "R2"]}], "-", RowBox[{ SubscriptBox["\[CapitalSigma]", "sR1"], SubscriptBox["\[CapitalPhi]", "R1"]}]}]}]], CellChangeTimes->{{3.394072752487823*^9, 3.394072759115205*^9}, { 3.3940727946620293`*^9, 3.394072816655294*^9}, {3.3941094509049177`*^9, 3.394109569378408*^9}}, EmphasizeSyntaxErrors->True], " [1d]\nIn the \ following we will solve for the conditions that give k= 1. For slab symmetry \ ", Cell[BoxData[ RowBox[{ SuperscriptBox["\[Del]", "2"], "\[CapitalPhi]"}]], CellChangeTimes->{{3.394072752487823*^9, 3.394072759115205*^9}, { 3.3940727946620293`*^9, 3.394072816655294*^9}, {3.3941094509049177`*^9, 3.394109569378408*^9}}, EmphasizeSyntaxErrors->True], " -> ", Cell[BoxData[ RowBox[{ FractionBox[ SuperscriptBox["d", "2"], SuperscriptBox["dx", "2"]], " ", "\[CapitalPhi]"}]], CellChangeTimes->{{3.394072752487823*^9, 3.394072759115205*^9}, { 3.3940727946620293`*^9, 3.394072816655294*^9}, {3.3941094509049177`*^9, 3.394109569378408*^9}}, EmphasizeSyntaxErrors->True], ". The (8) boundary conditions are:\n\t", Cell[BoxData[ SubscriptBox["\[CapitalPhi]", "C1"]], CellChangeTimes->{{3.394072752487823*^9, 3.394072759115205*^9}, { 3.3940727946620293`*^9, 3.394072816655294*^9}, {3.3941094509049177`*^9, 3.394109569378408*^9}}, EmphasizeSyntaxErrors->True], "'[0] = ", Cell[BoxData[ RowBox[{ SubscriptBox["\[CapitalPhi]", "C2"], "'"}]], CellChangeTimes->{{3.394072752487823*^9, 3.394072759115205*^9}, { 3.3940727946620293`*^9, 3.394072816655294*^9}, {3.3941094509049177`*^9, 3.394109569378408*^9}}, EmphasizeSyntaxErrors->True], "[0] = 0 [2a,b]\n\t", Cell[BoxData[ RowBox[{ SubscriptBox["\[CapitalPhi]", "C1"], "[", FractionBox["a", "2"], "]"}]], CellChangeTimes->{{3.394072752487823*^9, 3.394072759115205*^9}, { 3.3940727946620293`*^9, 3.394072816655294*^9}, {3.3941094509049177`*^9, 3.394109569378408*^9}}, EmphasizeSyntaxErrors->True], " = ", Cell[BoxData[ RowBox[{" ", RowBox[{ SubscriptBox["\[CapitalPhi]", "R1"], "[", FractionBox["a", "2"], "]"}]}]], 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Cell[BoxData[ RowBox[{ SubscriptBox["\[CapitalPhi]", "R1"], "[", "\[Infinity]", "]"}]], CellChangeTimes->{{3.394072752487823*^9, 3.394072759115205*^9}, { 3.3940727946620293`*^9, 3.394072816655294*^9}, {3.3941094509049177`*^9, 3.394109569378408*^9}}, EmphasizeSyntaxErrors->True], " = ", Cell[BoxData[ RowBox[{ SubscriptBox["\[CapitalPhi]", "R2"], "[", "\[Infinity]", "]"}]], CellChangeTimes->{{3.394072752487823*^9, 3.394072759115205*^9}, { 3.3940727946620293`*^9, 3.394072816655294*^9}, {3.3941094509049177`*^9, 3.394109569378408*^9}}, EmphasizeSyntaxErrors->True], " = ", Cell[BoxData["0"], CellChangeTimes->{{3.394072752487823*^9, 3.394072759115205*^9}, { 3.3940727946620293`*^9, 3.394072816655294*^9}, {3.3941094509049177`*^9, 3.394109569378408*^9}}, EmphasizeSyntaxErrors->True], " [2g,h]\n\t \nThe general \ solutions, satisfying [2a,b] and [2g,h] and normalized by the first \ sine-term, are:" }], "Text", CellChangeTimes->{{3.394109632723322*^9, 3.394109660752946*^9}, { 3.394111522198735*^9, 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