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We want to transform this polarization sum to a \ general {x,y,z} coordinate system and integrate over photon angles. To this \ end, we introduce the matrix \"a\" that transforms from coordinates {x,y,x} \ to coordinates {x',y',z'}" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(a\ = \ {{Cos[\[Phi]], Sin[\[Phi]], 0}, {\(-Cos[\[Theta]]\)\ Sin[\[Phi]], Cos[\[Theta]]\ Cos[\[Phi]], Sin[\[Theta]]}, {Sin[\[Theta]]\ Sin[\[Phi]], \(-Sin[\[Theta]]\)\ \ Cos[\[Phi]], Cos[\[Theta]]}}\)], "Input", CellLabel->"In[1]:="], Cell[BoxData[ \({{Cos[\[Phi]], Sin[\[Phi]], 0}, {\(-Cos[\[Theta]]\)\ Sin[\[Phi]], Cos[\[Theta]]\ Cos[\[Phi]], Sin[\[Theta]]}, {Sin[\[Theta]]\ Sin[\[Phi]], \(-Cos[\[Phi]]\)\ Sin[\ \[Theta]], Cos[\[Theta]]}}\)], "Output", CellLabel->"Out[1]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(a // MatrixForm\)], "Input", CellLabel->"In[2]:="], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\(Cos[\[Phi]]\), \(Sin[\[Phi]]\), "0"}, {\(\(-Cos[\[Theta]]\)\ Sin[\[Phi]]\), \(Cos[\[Theta]]\ \ Cos[\[Phi]]\), \(Sin[\[Theta]]\)}, {\(Sin[\[Theta]]\ Sin[\[Phi]]\), \(\(-Cos[\[Phi]]\)\ \ Sin[\[Theta]]\), \(Cos[\[Theta]]\)} }, RowSpacings->1, ColumnSpacings->1, ColumnAlignments->{Left}], "\[NoBreak]", ")"}], Function[ BoxForm`e$, MatrixForm[ BoxForm`e$]]]], "Output", CellLabel->"Out[2]//MatrixForm="] }, Open ]], Cell["Let Q be the quadrupole matrix in {xyz} system", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Q\ = \ {{Qxx, Qxy, Qxz}, {Qxy, Qyy, Qyz}, {Qxz, Qyz, Qzz}}\)], "Input",\ CellLabel->"In[3]:="], Cell[BoxData[ \({{Qxx, Qxy, Qxz}, {Qxy, Qyy, Qyz}, {Qxz, Qyz, Qzz}}\)], "Output", CellLabel->"Out[3]="] }, Open ]], Cell[TextData[{ "Let R be the quadrupole matrix in the rotated system (", Cell[BoxData[ \(TraditionalForm\`k\&^\)]], " along z'}" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(R\ = \ a . Q . Transpose[a]\)], "Input", CellLabel->"In[4]:="], Cell[BoxData[ \({{Cos[\[Phi]]\ \((Qxx\ Cos[\[Phi]] + Qxy\ Sin[\[Phi]])\) + Sin[\[Phi]]\ \((Qxy\ Cos[\[Phi]] + Qyy\ Sin[\[Phi]])\), \(-Cos[\[Theta]]\)\ Sin[\[Phi]]\ \((Qxx\ \ Cos[\[Phi]] + Qxy\ Sin[\[Phi]])\) + Cos[\[Theta]]\ Cos[\[Phi]]\ \((Qxy\ Cos[\[Phi]] + Qyy\ Sin[\[Phi]])\) + Sin[\[Theta]]\ \((Qxz\ Cos[\[Phi]] + Qyz\ Sin[\[Phi]])\), Sin[\[Theta]]\ Sin[\[Phi]]\ \((Qxx\ Cos[\[Phi]] + Qxy\ Sin[\[Phi]])\) - Cos[\[Phi]]\ Sin[\[Theta]]\ \((Qxy\ Cos[\[Phi]] + Qyy\ Sin[\[Phi]])\) + Cos[\[Theta]]\ \((Qxz\ Cos[\[Phi]] + Qyz\ Sin[\[Phi]])\)}, {Cos[\[Phi]]\ \((Qxy\ Cos[\[Theta]]\ \ Cos[\[Phi]] + Qxz\ Sin[\[Theta]] - Qxx\ Cos[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Phi]]\ \((Qyy\ Cos[\[Theta]]\ Cos[\[Phi]] + Qyz\ Sin[\[Theta]] - Qxy\ Cos[\[Theta]]\ Sin[\[Phi]])\), \(-Cos[\[Theta]]\)\ Sin[\ \[Phi]]\ \((Qxy\ Cos[\[Theta]]\ Cos[\[Phi]] + Qxz\ Sin[\[Theta]] - Qxx\ Cos[\[Theta]]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ Cos[\[Phi]]\ \((Qyy\ Cos[\[Theta]]\ Cos[\[Phi]] + Qyz\ Sin[\[Theta]] - Qxy\ Cos[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Theta]]\ \((Qyz\ Cos[\[Theta]]\ Cos[\[Phi]] + Qzz\ Sin[\[Theta]] - Qxz\ Cos[\[Theta]]\ Sin[\[Phi]])\), Sin[\[Theta]]\ Sin[\[Phi]]\ \((Qxy\ Cos[\[Theta]]\ Cos[\[Phi]] + Qxz\ Sin[\[Theta]] - Qxx\ Cos[\[Theta]]\ Sin[\[Phi]])\) - Cos[\[Phi]]\ Sin[\[Theta]]\ \((Qyy\ Cos[\[Theta]]\ Cos[\[Phi]] + Qyz\ Sin[\[Theta]] - Qxy\ Cos[\[Theta]]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ \((Qyz\ Cos[\[Theta]]\ Cos[\[Phi]] + Qzz\ Sin[\[Theta]] - Qxz\ Cos[\[Theta]]\ Sin[\[Phi]])\)}, {Cos[\[Phi]]\ \((Qxz\ \ Cos[\[Theta]] - Qxy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxx\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Phi]]\ \((Qyz\ Cos[\[Theta]] - Qyy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxy\ Sin[\[Theta]]\ Sin[\[Phi]])\), \(-Cos[\[Theta]]\)\ Sin[\ \[Phi]]\ \((Qxz\ Cos[\[Theta]] - Qxy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxx\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ Cos[\[Phi]]\ \((Qyz\ Cos[\[Theta]] - Qyy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxy\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Theta]]\ \((Qzz\ Cos[\[Theta]] - Qyz\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxz\ Sin[\[Theta]]\ Sin[\[Phi]])\), Sin[\[Theta]]\ Sin[\[Phi]]\ \((Qxz\ Cos[\[Theta]] - Qxy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxx\ Sin[\[Theta]]\ Sin[\[Phi]])\) - Cos[\[Phi]]\ Sin[\[Theta]]\ \((Qyz\ Cos[\[Theta]] - Qyy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxy\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ \((Qzz\ Cos[\[Theta]] - Qyz\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxz\ Sin[\[Theta]]\ Sin[\[Phi]])\)}}\)], "Output", CellLabel->"Out[4]="] }, Open ]], Cell["Introduce the complex conjugates of Q and R", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Qcc\ = \ Conjugate[Q]\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ \({{Conjugate[Qxx], Conjugate[Qxy], Conjugate[Qxz]}, {Conjugate[Qxy], Conjugate[Qyy], Conjugate[Qyz]}, {Conjugate[Qxz], Conjugate[Qyz], Conjugate[Qzz]}}\)], "Output", CellLabel->"Out[5]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Rcc\ = \ a\ . \ Qcc\ . \ Transpose[a]\)], "Input", CellLabel->"In[6]:="], Cell[BoxData[ \({{Cos[\[Phi]]\ \((Conjugate[Qxx]\ Cos[\[Phi]] + Conjugate[Qxy]\ Sin[\[Phi]])\) + Sin[\[Phi]]\ \((Conjugate[Qxy]\ Cos[\[Phi]] + Conjugate[ Qyy]\ Sin[\[Phi]])\), \(-Cos[\[Theta]]\)\ Sin[\[Phi]]\ \ \((Conjugate[Qxx]\ Cos[\[Phi]] + Conjugate[Qxy]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ Cos[\[Phi]]\ \((Conjugate[Qxy]\ Cos[\[Phi]] + Conjugate[Qyy]\ Sin[\[Phi]])\) + Sin[\[Theta]]\ \((Conjugate[Qxz]\ Cos[\[Phi]] + Conjugate[Qyz]\ Sin[\[Phi]])\), Sin[\[Theta]]\ Sin[\[Phi]]\ \((Conjugate[Qxx]\ Cos[\[Phi]] + Conjugate[Qxy]\ Sin[\[Phi]])\) - Cos[\[Phi]]\ Sin[\[Theta]]\ \((Conjugate[Qxy]\ Cos[\[Phi]] + Conjugate[Qyy]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ \((Conjugate[Qxz]\ Cos[\[Phi]] + Conjugate[Qyz]\ Sin[\[Phi]])\)}, {Cos[\[Phi]]\ \((Conjugate[ Qxy]\ Cos[\[Theta]]\ Cos[\[Phi]] + Conjugate[Qxz]\ Sin[\[Theta]] - Conjugate[Qxx]\ Cos[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Phi]]\ \((Conjugate[Qyy]\ Cos[\[Theta]]\ Cos[\[Phi]] + Conjugate[Qyz]\ Sin[\[Theta]] - Conjugate[ Qxy]\ Cos[\[Theta]]\ Sin[\[Phi]])\), \(-Cos[\[Theta]]\)\ \ Sin[\[Phi]]\ \((Conjugate[Qxy]\ Cos[\[Theta]]\ Cos[\[Phi]] + Conjugate[Qxz]\ Sin[\[Theta]] - Conjugate[Qxx]\ Cos[\[Theta]]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ Cos[\[Phi]]\ \((Conjugate[ Qyy]\ Cos[\[Theta]]\ Cos[\[Phi]] + Conjugate[Qyz]\ Sin[\[Theta]] - Conjugate[Qxy]\ Cos[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Theta]]\ \((Conjugate[Qyz]\ Cos[\[Theta]]\ Cos[\[Phi]] + Conjugate[Qzz]\ Sin[\[Theta]] - Conjugate[Qxz]\ Cos[\[Theta]]\ Sin[\[Phi]])\), Sin[\[Theta]]\ Sin[\[Phi]]\ \((Conjugate[ Qxy]\ Cos[\[Theta]]\ Cos[\[Phi]] + Conjugate[Qxz]\ Sin[\[Theta]] - Conjugate[Qxx]\ Cos[\[Theta]]\ Sin[\[Phi]])\) - Cos[\[Phi]]\ Sin[\[Theta]]\ \((Conjugate[ Qyy]\ Cos[\[Theta]]\ Cos[\[Phi]] + Conjugate[Qyz]\ Sin[\[Theta]] - Conjugate[Qxy]\ Cos[\[Theta]]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ \((Conjugate[Qyz]\ Cos[\[Theta]]\ Cos[\[Phi]] + Conjugate[Qzz]\ Sin[\[Theta]] - Conjugate[ Qxz]\ Cos[\[Theta]]\ Sin[\[Phi]])\)}, {Cos[\[Phi]]\ \ \((Conjugate[Qxz]\ Cos[\[Theta]] - Conjugate[Qxy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxx]\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Phi]]\ \((Conjugate[Qyz]\ Cos[\[Theta]] - Conjugate[Qyy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[ Qxy]\ Sin[\[Theta]]\ Sin[\[Phi]])\), \(-Cos[\[Theta]]\)\ \ Sin[\[Phi]]\ \((Conjugate[Qxz]\ Cos[\[Theta]] - Conjugate[Qxy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxx]\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ Cos[\[Phi]]\ \((Conjugate[Qyz]\ Cos[\[Theta]] - Conjugate[Qyy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxy]\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Theta]]\ \((Conjugate[Qzz]\ Cos[\[Theta]] - Conjugate[Qyz]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxz]\ Sin[\[Theta]]\ Sin[\[Phi]])\), Sin[\[Theta]]\ Sin[\[Phi]]\ \((Conjugate[Qxz]\ Cos[\[Theta]] - Conjugate[Qxy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxx]\ Sin[\[Theta]]\ Sin[\[Phi]])\) - Cos[\[Phi]]\ Sin[\[Theta]]\ \((Conjugate[Qyz]\ Cos[\[Theta]] - Conjugate[Qyy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxy]\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ \((Conjugate[Qzz]\ Cos[\[Theta]] - Conjugate[Qyz]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxz]\ Sin[\[Theta]]\ Sin[\[Phi]])\)}}\)], "Output", CellLabel->"Out[6]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Rzx\ = \ R[\([3, 1]\)]\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \(Cos[\[Phi]]\ \((Qxz\ Cos[\[Theta]] - Qxy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxx\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Phi]]\ \((Qyz\ Cos[\[Theta]] - Qyy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxy\ Sin[\[Theta]]\ Sin[\[Phi]])\)\)], "Output", CellLabel->"Out[7]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Rzy\ = \ R[\([3, 2]\)]\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ \(\(-Cos[\[Theta]]\)\ Sin[\[Phi]]\ \((Qxz\ Cos[\[Theta]] - Qxy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxx\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ Cos[\[Phi]]\ \((Qyz\ Cos[\[Theta]] - Qyy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxy\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Theta]]\ \((Qzz\ Cos[\[Theta]] - Qyz\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxz\ Sin[\[Theta]]\ Sin[\[Phi]])\)\)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Rzxcc\ = \ Rcc[\([3, 1]\)]\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ \(Cos[\[Phi]]\ \((Conjugate[Qxz]\ Cos[\[Theta]] - Conjugate[Qxy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxx]\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Phi]]\ \((Conjugate[Qyz]\ Cos[\[Theta]] - Conjugate[Qyy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxy]\ Sin[\[Theta]]\ Sin[\[Phi]])\)\)], "Output", CellLabel->"Out[9]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Rzycc\ = \ Rcc[\([3, 2]\)]\)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \(\(-Cos[\[Theta]]\)\ Sin[\[Phi]]\ \((Conjugate[Qxz]\ Cos[\[Theta]] - Conjugate[Qxy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxx]\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ Cos[\[Phi]]\ \((Conjugate[Qyz]\ Cos[\[Theta]] - Conjugate[Qyy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxy]\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Theta]]\ \((Conjugate[Qzz]\ Cos[\[Theta]] - Conjugate[Qyz]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxz]\ Sin[\[Theta]]\ Sin[\[Phi]])\)\)], "Output", CellLabel->"Out[10]="] }, Open ]], Cell[TextData[{ "The polarization sum is ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ FormBox[\(\(|\)\(Rzx\)\( | \^2\)\), "TraditionalForm"], "+"}], "|", "Rzy", \( | \^2\)}], ";"}], TraditionalForm]]], " this is now to be integrated over angles. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(polsum\ = \ Rzx\ Rzxcc\ + \ Rzy\ Rzycc\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(\((Cos[\[Phi]]\ \((Qxz\ Cos[\[Theta]] - Qxy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxx\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Phi]]\ \((Qyz\ Cos[\[Theta]] - Qyy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxy\ Sin[\[Theta]]\ Sin[\[Phi]])\))\)\ \((Cos[\[Phi]]\ \ \((Conjugate[Qxz]\ Cos[\[Theta]] - Conjugate[Qxy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxx]\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Phi]]\ \((Conjugate[Qyz]\ Cos[\[Theta]] - Conjugate[Qyy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[ Qxy]\ Sin[\[Theta]]\ Sin[\[Phi]])\))\) + \((\(-Cos[\ \[Theta]]\)\ Sin[\[Phi]]\ \((Qxz\ Cos[\[Theta]] - Qxy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxx\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ Cos[\[Phi]]\ \((Qyz\ Cos[\[Theta]] - Qyy\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxy\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Theta]]\ \((Qzz\ Cos[\[Theta]] - Qyz\ Cos[\[Phi]]\ Sin[\[Theta]] + Qxz\ Sin[\[Theta]]\ Sin[\[Phi]])\))\)\ \ \((\(-Cos[\[Theta]]\)\ Sin[\[Phi]]\ \((Conjugate[Qxz]\ Cos[\[Theta]] - Conjugate[Qxy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxx]\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Cos[\[Theta]]\ Cos[\[Phi]]\ \((Conjugate[Qyz]\ Cos[\[Theta]] - Conjugate[Qyy]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[Qxy]\ Sin[\[Theta]]\ Sin[\[Phi]])\) + Sin[\[Theta]]\ \((Conjugate[Qzz]\ Cos[\[Theta]] - Conjugate[Qyz]\ Cos[\[Phi]]\ Sin[\[Theta]] + Conjugate[ Qxz]\ Sin[\[Theta]]\ Sin[\[Phi]])\))\)\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(phiint = \ Integrate[polsum, {\[Phi], 0, 2\ Pi}]\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ \(1\/4\ \[Pi]\ \((4\ Qxz\ Conjugate[Qxz]\ Cos[\[Theta]]\^2 + 4\ Qyz\ Conjugate[ Qyz]\ Cos[\[Theta]]\^2 + \((\((Qxx - Qyy)\)\ Conjugate[Qxx] + 4\ Qxy\ Conjugate[Qxy] + \((\(-Qxx\) + Qyy)\)\ Conjugate[ Qyy])\)\ Sin[\[Theta]]\^2)\) + 1\/16\ \[Pi]\ \((16\ Qxz\ Conjugate[Qxz]\ Cos[2\ \[Theta]]\^2 + 16\ Qyz\ Conjugate[ Qyz]\ Cos[2\ \[Theta]]\^2 + \((\((3\ Qxx + Qyy - 4\ Qzz)\)\ Conjugate[Qxx] + 4\ Qxy\ Conjugate[Qxy] + Qxx\ Conjugate[Qyy] + 3\ Qyy\ Conjugate[Qyy] - 4\ Qzz\ Conjugate[Qyy] - 4\ Qxx\ Conjugate[Qzz] - 4\ Qyy\ Conjugate[Qzz] + 8\ Qzz\ Conjugate[ Qzz])\)\ Sin[2\ \[Theta]]\^2)\)\)], "Output", CellLabel->"Out[12]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(angint\ = \ Integrate[Sin[\[Theta]]\ phiint, {\[Theta], 0, Pi}]\)], "Input", CellLabel->"In[13]:="], Cell[BoxData[ \(4\/15\ \[Pi]\ \((\((2\ Qxx - Qyy - Qzz)\)\ Conjugate[Qxx] + 6\ Qxy\ Conjugate[Qxy] + 6\ Qxz\ Conjugate[Qxz] - \((Qxx - 2\ Qyy + Qzz)\)\ Conjugate[Qyy] + 6\ Qyz\ Conjugate[Qyz] - \((Qxx + Qyy - 2\ Qzz)\)\ Conjugate[ Qzz])\)\)], "Output", CellLabel->"Out[13]="] }, Open ]], Cell["\<\ This result can be simplified by eliminating Qzz. The resulting contribution \ is\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(angint = FullSimplify[angint /. \ Qzz \[Rule] \ \(-Qxx\) - Qyy]\)], "Input", CellLabel->"In[14]:="], Cell[BoxData[ \(4\/5\ \[Pi]\ \((\((2\ Qxx + Qyy)\)\ Conjugate[Qxx] + 2\ Qxy\ Conjugate[Qxy] + 2\ Qxz\ Conjugate[Qxz] + \((Qxx + 2\ Qyy)\)\ Conjugate[Qyy] + 2\ Qyz\ Conjugate[Qyz])\)\)], "Output", CellLabel->"Out[14]="] }, Open ]], Cell["\<\ The spherical components of the quadrupole tensor are given by the \ equations\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(eqns\ = \ {Q0 \[Equal] \(-\((Qxx + Qyy)\)\)/2, Q1 == \(-\((1/Sqrt[6])\)\) \((Qxz + I\ Qyz)\), Q1m == \((1/Sqrt[6])\) \((Qxz - I\ Qyz)\)\ , Q2 \[Equal] \((1/Sqrt[24])\) \((Qxx - Qyy + \ 2\ I\ Qxy)\), Qm2 \[Equal] \((1/Sqrt[24])\) \((Qxx - Qyy - \ 2\ I\ Qxy)\)}\)], "Input", CellLabel->"In[15]:="], Cell[BoxData[ \({Q0 \[Equal] 1\/2\ \((\(-Qxx\) - Qyy)\), Q1 \[Equal] \(-\(\(Qxz + \[ImaginaryI]\ Qyz\)\/\@6\)\), Q1m \[Equal] \(Qxz - \[ImaginaryI]\ Qyz\)\/\@6, Q2 \[Equal] \(Qxx + 2\ \[ImaginaryI]\ Qxy - Qyy\)\/\(2\ \@6\), Qm2 \[Equal] \(Qxx - 2\ \[ImaginaryI]\ Qxy - Qyy\)\/\(2\ \@6\)}\)], \ "Output", CellLabel->"Out[15]="] }, Open ]], Cell[TextData[{ "We can obtain the quadrupole tensor components Qij in terms spherical \ basis components ", Cell[BoxData[ \(TraditionalForm\`Q\_\[CapitalLambda]\)]], " by solving the above" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(sol\ \ = \ Solve[eqns, {Qxz, Qyz, Qxy, Qxx, Qyy}]\)], "Input", CellLabel->"In[16]:="], Cell[BoxData[ \({{Qxy \[Rule] \(-\(1\/2\)\)\ \[ImaginaryI]\ \((\@6\ Q2 - \@6\ Qm2)\), Qxz \[Rule] 1\/2\ \((\(-\@6\)\ Q1 + \@6\ Q1m)\), Qyz \[Rule] \[ImaginaryI]\ \@\(3\/2\)\ \((Q1 + Q1m)\), Qxx \[Rule] 1\/2\ \((\(-2\)\ Q0 + \@6\ Q2 + \@6\ Qm2)\), Qyy \[Rule] 1\/2\ \((\(-2\)\ Q0 - \@6\ Q2 - \@6\ Qm2)\)}}\)], "Output",\ CellLabel->"Out[16]="] }, Open ]], Cell["\<\ Finally, we can express the angular integrated polarization sum in a \ spherical basis as\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(angint0\ = \ FullSimplify[angint /. \ sol] // First\)], "Input", CellLabel->"In[17]:="], Cell[BoxData[ \(24\/5\ \[Pi]\ \((Abs[Q0]\^2 + Abs[Q1]\^2 + Abs[Q1m]\^2 + Abs[Q2]\^2 + Abs[Qm2]\^2)\)\)], "Output", CellLabel->"Out[17]="] }, Open ]], Cell[BoxData[ \(Clear[Q0, Q1, Q1m, Q2, Q2m, Qxx, Qyy, Qzz, Qxy, Qxz, Qyz]\)], "Input", CellLabel->"In[18]:="] }, Open ]] }, FrontEndVersion->"5.2 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 685}}, WindowSize->{704, 549}, WindowMargins->{{65, Automatic}, {49, Automatic}}, StyleDefinitions -> "ArticleModern.nb" ] (******************************************************************* Cached data follows. 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