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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 114931, 3865]*) (*NotebookOutlinePosition[ 115644, 3890]*) (* CellTagsIndexPosition[ 115600, 3886]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Effective Medium Approximations", "Title"], Cell["\<\ Created by I Wayan Sudiarta (sudiarta@dal.ca) Dept. of Physics and Atmospheric Science Dalhousie University, Halifax, NS B3H 3J5 Lasr Updated: 4 Feb 2003\ \>", "Subsubtitle"], Cell[CellGroupData[{ Cell["Simple Averaging", "Section"], Cell[BoxData[ \(\(AveDiel[mh_, mi_, fv_] := Sqrt[\((1 - fv)\)*mh^2 + fv*mi^2]; \n AveRef[mh_, mi_, fv_] := \((1 - fv)\)*mh + fv*mi; \n AveOneThird[mh_, mi_, fv_] := \((\((1 - fv)\)*mh^\((2/3)\) + fv*mi^\((2/3)\))\)^\((3/2)\); \n AveInvDiel[mh_, mi_, fv_] := \((\((1 - fv)\)/mh^2 + fv/mi^2)\)^\((\(-1\)/2)\); \n\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ The Bruggeman (BR), the Maxwell-Garnett (MG), and Inverted MG (IMG) Mixing \ Rules\ \>", "Section"], Cell[BoxData[ \(BR[mh_, mi_, fv_] := Module[{ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; meff = FindRoot[ fv*\((mi^2 - m^2)\)/\((mi^2 + 2*m^2)\) + \((1.0 - fv)\)*\((mh^2 - m^2)\)/\((mh^2 + 2*m^2)\), { m, {ma, mb}}, MaxIterations -> 5000, AccuracyGoal \[Rule] 8]; m /. meff]; \n MG[mh_, mi_, fv_] := Module[{ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; meff = FindRoot[ \((m^2 - mh^2)\)/\((m^2 + 2*mh^2)\) - fv*\((mi^2 - mh^2)\)/\((mi^2 + 2*mh^2)\), {m, {ma, mb}}, MaxIterations -> 5000, AccuracyGoal \[Rule] 8]; m /. meff]; \n IMG[mh_, mi_, fv_] := MG[mi, mh, 1.0 - fv]; \)], "Input"], Cell[BoxData[ \(\(BRCube[mh_, mi_, fv_] := Module[{ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; meff = FindRoot[ fv*\((mi^2 - m^2)\)/\((mi^2 + 2.93*m^2)\) + \((1.0 - fv)\)*\((mh^2 - m^2)\)/\((mh^2 + 2.93*m^2)\), { m, {ma, mb}}, MaxIterations -> 5000, AccuracyGoal \[Rule] 8]; m /. meff]; \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Modules for calculating Riccati Bessel functions Note: array [[1]] RB functions, array [[2]] its derivative\ \>", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(rby[z_, nmax_] := Module[{sby, dsby, i, n}, \n\t\tsby\ = \ Table[0.0, {i, 0, nmax}]; \ \ dsby\ = \ Table[0.0, {i, 0, nmax}]; sby[\([1]\)]\ = \ \(-Cos[z]\)/z; \ sby[\([2]\)]\ = \((sby[\([1]\)] - Sin[z])\)/z; \ For[n = 2, n <= nmax, \(n++\), \n\t\t\t\ti = n + 1; \n\t\t\t sby[\([i]\)] = N[\((2*n - 1)\)*sby[\([i - 1]\)]/z - sby[\([i - 2]\)], 20]; \n \ \ \ \ \ \ ]; \n\t\tdsby[\([1]\)]\ = \((Sin[z] + Cos[z]/z)\)/z; For[n = 1, n <= nmax, \(n++\), \n\t\t\t\ti = n + 1; \t dsby[\([i]\)] = sby[\([i - 1]\)] - \((n + 1)\)*sby[\([i]\)]/z; \n \t\t]; {z*Delete[sby, 1], Delete[sby + z*dsby, 1]}]; \n\n rbj[z_, nmax_] := Module[{sbj, dsbj, i, n, niter, f, f0, f1, sa, sb, cs}, \n\t\t niter\ = \ IntegerPart[nmax + Sqrt[100 + Abs[z]]]; sbj\ = \ Table[0.0, {i, 0, nmax}]; \ \ dsbj\ = \ Table[0.0, {i, 0, nmax}]; sbj[\([1]\)]\ = \ Sin[z]/z; \ \ sbj[\([2]\)]\ = \((sbj[\([1]\)] - Cos[z])\)/z; \ \ \n\t\t sa\ = \ sbj[\([1]\)]; \n\t\tsb\ = \ sbj[\([2]\)]; \n\t\tf0\ = 0.0; \n\t\tf1\ = 1.0*10^\((\(-10\))\); \n\t\t For[n = niter, n >= 0, \(n--\), \n\t\t\t\ti = n + 1; \n\t\t\t\t f\ = N[\((2.0*n + 3.0)\)*f1/z - f0, 20]; \t If[n <= nmax, \(sbj[\([i]\)] = f; \), ]; \n\t\t\t\tf0 = f1; \t f1 = f; \n\t\t]; \n\ \ \ \ \ If[Abs[sa] >= Abs[sb], cs = sa/f, cs = sb/f0]; \n\ \ \ \ For[n = 0, n <= nmax, \(n++\), \n\t\ \ \ \ \ \ i = n + 1; \n \t\ \ \ \ \ \ sbj[\([i]\)] = cs*sbj[\([i]\)]; \n\ \ \ ]; \n\ \ dsbj[\([1]\)]\ = \((Cos[z] - Sin[z]/z)\)/z; \n\ \ For[n = 1, n <= nmax, \(n++\), \n\t\ \ \ i = n + 1; \n\t\ \ \ \ dsbj[\([i]\)] = sbj[\([i - 1]\)] - \((n + 1.0)\)*sbj[\([i]\)]/z; \n]; {Delete[z*sbj, 1], Delete[sbj + z*dsbj, 1]}]; \n cj[z_] := Conjugate[z]; \)], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(dsby\)\" is similar to \ existing symbol \"\!\(sby\)\"."\)], "Message"], Cell[BoxData[ \(General::"spell" \( : \ \) "Possible spelling error: new symbol name \"\!\(dsbj\)\" is similar to \ existing symbols \!\({dsby, sbj}\)."\)], "Message"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Extended EMAs", "Section"], Cell[CellGroupData[{ Cell["Aggregate Structure - Spheres Model", "Subsection", FontFamily->"Times New Roman"], Cell[BoxData[ \(CorEBR[meff_, mh_, mi_, fv_, r1_, r2_] := Module[\n\t\t{i, mr, mmed, msph, k, k2, y, nstop, lambda, meff2, an, bn, psi, psid, psi2, psi2d, \n\ \ \ \ \ \ \ \ xi, xid, A, B}, \n\t\t lambda\ = \ 2*Pi; \n\t\t (*\ sphere\ 1\ *) \n\t\tmmed\ = meff; \n \t\tmsph\ = \ mh; \n\t\tk = mmed*2*Pi/lambda; \n\t\t k2 = msph*2*Pi/lambda; \n\t\tx\ = k*r1; \n\t\ty\ = k2*r1; \n\t\t nstop = IntegerPart[Abs[x]\ + \ 4.0*Abs[x]^0.3333\ + \ 2.0]; \n\t\t mr\ = \ msph/mmed; \n\t\t{psi, psid} = rbj[x, nstop]; \n \t\t{psi2, psi2d} = rbj[y, nstop]; \n \t\t{xi, xid}\ = rbj[x, nstop] + I*rby[x, nstop]; \n\t\t an = Table[ \((mr*psi2[\([i]\)]*psid[\([i]\)] - psi[\([i]\)]*psi2d[\([i]\)]) \)/\((\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mr*psi2[\([i]\)]*xid[\([i]\)] - xi[\([i]\)]*psi2d[\([i]\)]) \), {i, 1, nstop}]; \n\t\t bn = Table[ \((psi2[\([i]\)]*psid[\([i]\)] - mr*psi[\([i]\)]*psi2d[\([i]\)]) \)/\((\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ psi2[\([i]\)]*xid[\([i]\)] - mr*xi[\([i]\)]*psi2d[\([i]\)]) \), {i, 1, nstop}]; \n\t\t A\ = Sum[ \((2*i + 1)\)* \((\n\t\t\t\t\t\t\t\t bn[\([i]\)]* \((cj[k]*cj[psid[\([i]\)]]*xi[\([i]\)] - k*cj[psi[\([i]\)]]*xid[\([i]\)])\) + \n \t\t\t\t\t\t\t\t an[\([i]\)]* \((k*cj[psid[\([i]\)]]*xi[\([i]\)] - cj[k]*xid[\([i]\)]*cj[psi[\([i]\)]])\))\), {i, 1, nstop}]; \n\t\t (*\ sphere\ 2\ *) \n\t\tmsph\ = \ mi; \n\t\t k = mmed*2*Pi/lambda; \n\t\tk2 = msph*2*Pi/lambda; \n\t\tx\ = k*r2; \n\t\ty\ = k2*r2; \n\t\t nstop = IntegerPart[Abs[x]\ + \ 4.0*Abs[x]^0.3333\ + \ 2.0]; \n\t\t mr\ = \ msph/mmed; \n\t\t{psi, psid} = rbj[x, nstop]; \n \t\t{psi2, psi2d} = rbj[y, nstop]; \n \t\t{xi, xid}\ = rbj[x, nstop] + I*rby[x, nstop]; \n\t\t an = Table[ \((mr*psi2[\([i]\)]*psid[\([i]\)] - psi[\([i]\)]*psi2d[\([i]\)]) \)/\((\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mr*psi2[\([i]\)]*xid[\([i]\)] - xi[\([i]\)]*psi2d[\([i]\)]) \), {i, 1, nstop}]; \n\t\t bn = Table[ \((psi2[\([i]\)]*psid[\([i]\)] - mr*psi[\([i]\)]*psi2d[\([i]\)]) \)/ \((\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ psi2[\([i]\)]*xid[\([i]\)] - mr*xi[\([i]\)]*psi2d[\([i]\)]) \), {i, 1, nstop}]; \n\t\t B\ = Sum[ \((2*i + 1)\)* \((\n\t\t\t\t\t\t\t\t bn[\([i]\)]* \((cj[k]*cj[psid[\([i]\)]]*xi[\([i]\)] - k*cj[psi[\([i]\)]]*xid[\([i]\)])\) + \n \t\t\t\t\t\t\t\t an[\([i]\)]* \((k*cj[psid[\([i]\)]]*xi[\([i]\)] - cj[k]*xid[\([i]\)]*cj[psi[\([i]\)]])\))\), {i, 1, nstop}]; \n\t\t eq\ = \((1 - fv)\)*A/\((4*Pi*r1^3/3)\) + fv*B/\((4*Pi*r2^3/3)\); \n \t\teq]; \n EBR[mh_, mi_, fv_, r1_, r2_] := Module[{m, ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; \n\t\tmf = FindRoot[CorEBR[m, mh, mi, fv, r1, r2], {m, {ma, mb}}, AccuracyGoal -> 8, MaxIterations -> 5000]; m /. mf]; \n TRYEBR[mh_, mi_, fv_, r1_, r2_] := Module[{m, ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; \n\t\tmf = FindMinimum[ Abs[CorEBR[u + v*I, mh, mi, fv, r1, r2]], { u, {Re[mb], Re[mb] + 0.01}}, {v, {Im[mb], Im[mb] + 0.01}}, AccuracyGoal -> 10, MaxIterations -> 10000]; \((u + v*I)\) /. mf[\([2]\)]]; \)], "Input"], Cell[BoxData[""], "Input"], Cell[BoxData[ \(UnCorEBR[meff_, mh_, mi_, fv_, r1_, r2_] := Module[\n\t\t{i, mr, mmed, msph, k, k2, y, nstop, lambda, meff2, an, bn, psi, psid, psi2, psi2d, \n\ \ \ \ \ \ \ \ xi, xid, A, B}, \n\t\t lambda\ = \ 2*Pi; \n\t\t (*\ sphere\ 1\ *) \n\t\tmmed\ = meff; \n \t\tmsph\ = \ mh; \n\t\tk = mmed*2*Pi/lambda; \n\t\t k2 = msph*2*Pi/lambda; \n\t\tx\ = k*r1; \n\t\ty\ = k2*r1; \n\t\t nstop = IntegerPart[Abs[x]\ + \ 4.0*Abs[x]^0.3333\ + \ 2.0]; \n\t\t mr\ = \ msph/mmed; \n\t\t{psi, psid} = rbj[x, nstop]; \n \t\t{psi2, psi2d} = rbj[y, nstop]; \n \t\t{xi, xid}\ = rbj[x, nstop] + I*rby[x, nstop]; \n\t\t an = Table[ \((mr*psi2[\([i]\)]*psid[\([i]\)] - psi[\([i]\)]*psi2d[\([i]\)]) \)/\((\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mr*psi2[\([i]\)]*xid[\([i]\)] - xi[\([i]\)]*psi2d[\([i]\)]) \), {i, 1, nstop}]; \n\t\t bn = Table[ \((psi2[\([i]\)]*psid[\([i]\)] - mr*psi[\([i]\)]*psi2d[\([i]\)]) \)/\((\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ psi2[\([i]\)]*xid[\([i]\)] - mr*xi[\([i]\)]*psi2d[\([i]\)]) \), {i, 1, nstop}]; \n\t\t A\ = Sum[ \((2*i + 1)\)*\((an[\([i]\)] + bn[\([i]\)])\), {i, 1, nstop}]; \n \t\t (*\ sphere\ 2\ *) \n\t\tmsph\ = \ mi; \n\t\t k = mmed*2*Pi/lambda; \n\t\tk2 = msph*2*Pi/lambda; \n\t\tx\ = k*r2; \n\t\ty\ = k2*r2; \n\t\t nstop = IntegerPart[Abs[x]\ + \ 4.0*Abs[x]^0.3333\ + \ 2.0]; \n\t\t mr\ = \ msph/mmed; \n\t\t{psi, psid} = rbj[x, nstop]; \n \t\t{psi2, psi2d} = rbj[y, nstop]; \n \t\t{xi, xid}\ = rbj[x, nstop] + I*rby[x, nstop]; \n\t\t an = Table[ \((mr*psi2[\([i]\)]*psid[\([i]\)] - psi[\([i]\)]*psi2d[\([i]\)]) \)/\((\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mr*psi2[\([i]\)]*xid[\([i]\)] - xi[\([i]\)]*psi2d[\([i]\)]) \), {i, 1, nstop}]; \n\t\t bn = Table[ \((psi2[\([i]\)]*psid[\([i]\)] - mr*psi[\([i]\)]*psi2d[\([i]\)]) \)/\((\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ psi2[\([i]\)]*xid[\([i]\)] - mr*xi[\([i]\)]*psi2d[\([i]\)]) \), {i, 1, nstop}]; \n\t\t B\ = Sum[ \((2*i + 1)\)*\((an[\([i]\)] + bn[\([i]\)])\), {i, 1, nstop}]; \n \t\teq\ = \((1 - fv)\)*A/\((4*Pi*r1^3/3)\) + fv*B/\((4*Pi*r2^3/3)\); \n\t\teq]; \n UEBR[mh_, mi_, fv_, r1_, r2_] := Module[{m, ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; \n\t\tmf = FindRoot[UnCorEBR[m, mh, mi, fv, r1, r2], {m, {ma, mb}}, AccuracyGoal \[Rule] 16, MaxIterations -> 5000]; \n\t\tm /. mf]; \nTRYUEBR[mh_, mi_, fv_, r1_, r2_] := Module[{m, ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; \n\t\tmf = FindMinimum[ Abs[UnCorEBR[u + v*I, mh, mi, fv, r1, r2]], { u, {Re[mb], Re[mb] + 0.01}}, {v, {Im[mb], Im[mb] + 0.01}}, AccuracyGoal -> 10, MaxIterations -> 10000]; \((u + v*I)\) /. mf[\([2]\)]]; \)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Separated Grain Structure - Coated Spheres Model", "Subsection", FontFamily->"Times New Roman"], Cell[CellGroupData[{ Cell[BoxData[ \(\(CorEMG[meff_, mh_, mi_, fv_, ri_] := Module[\n\t\t{i, mr, mmed, msph, k, k2, r1, r2, nstop, lambda, an, bn, psi, psid, psi2, psi2d, \n\ \ \ \ \ \ \ \ xi, xid, An, Bn, x, x1, x2}, \n\t\tr1\ = \ ri; \ \n\ \ \ \ \ \ r2\ = \ r1/\((fv^\((1/3)\))\); \n\t\tlambda\ = \ 2*Pi; \n\t\t mmed\ = meff; \n\t\tm1\ = \ mi; \n\t\tm2\ = \ mh; \n\t\t k = mmed*2*Pi/lambda; \n\t\tk1 = m1*2*Pi/lambda; \n\t\t k2\ = \ m2*2*Pi/lambda; \n\t\tx1\ = k*r1; \n\t\tx2\ = \ k*r2; \n \t\tnstop = IntegerPart[Abs[x2]\ + \ 4.0*Abs[x2]^0.3333\ + \ 2.0]; \n\t\tmr1\ = \ m1/mmed; \n\t\tmr2\ = \ m2/mmed; \n \t\t{psi11, psi11d} = rbj[mr1*x1, nstop]; \n \t\t{psi21, psi21d} = rbj[mr2*x1, nstop]; \n \t\t{psi12, psi12d} = rbj[mr1*x2, nstop]; \n \t\t{psi22, psi22d} = rbj[mr2*x2, nstop]; \n \t\t{psi2, psi2d} = rbj[x2, nstop]; \n \t\t{xi2, xi2d}\ = rbj[x2, nstop] + I*rby[x2, nstop]; \n \t\t{xi22, xi22d}\ = rbj[mr2*x2, nstop] + I*rby[mr2*x2, nstop]; \n \t\t{xi21, xi21d}\ = rbj[mr2*x1, nstop] + I*rby[mr2*x1, nstop]; \n \t\tAn\ = \ Table[\n\t\t\t\t \((mr2*psi11d[\([i]\)]*psi21[\([i]\)]\n\t\t\t\t\t\t\t\t - mr1*psi11[\([i]\)]*psi21d[\([i]\)])\)/ \((mr2*psi11d[\([i]\)]*xi21[\([i]\)]\n\t\t\t\t\t\t\t\t - mr1*psi11[\([i]\)]*xi21d[\([i]\)])\), {i, 1, nstop}]; \n\t\t Bn\ = \ Table[\n\t\t\t\t \((mr2*psi21d[\([i]\)]*psi11[\([i]\)]\n\t\t\t\t\t\t\t\t - mr1*psi21[\([i]\)]*psi11d[\([i]\)])\)/ \((mr2*xi21d[\([i]\)]*psi11[\([i]\)]\n\t\t\t\t\t\t\t\t - mr1*xi21[\([i]\)]*psi11d[\([i]\)])\), {i, 1, nstop}]; \n\t\t an = Table[\n\t\t\t\t \((psi2[\([i]\)]* \((psi22d[\([i]\)] - An[\([i]\)]*xi22d[\([i]\)])\)\n \t\t\t\t\t - mr2*psi2d[\([i]\)]* \((psi22[\([i]\)] - An[\([i]\)]*xi22[\([i]\)])\))\)/\t \((xi2[\([i]\)]* \((psi22d[\([i]\)] - An[\([i]\)]*xi22d[\([i]\)])\)\n \t\t\t\t\t - mr2*xi2d[\([i]\)]* \((psi22[\([i]\)] - An[\([i]\)]*xi22[\([i]\)])\))\), {i, 1, nstop}]; \n\t\t bn = Table[\n\t\t\t\t \((mr2*psi2[\([i]\)]* \((psi22d[\([i]\)] - Bn[\([i]\)]*xi22d[\([i]\)])\)\n \t\t\t\t\t - psi2d[\([i]\)]* \((psi22[\([i]\)] - Bn[\([i]\)]*xi22[\([i]\)])\))\)/\t \((mr2*xi2[\([i]\)]* \((psi22d[\([i]\)] - Bn[\([i]\)]*xi22d[\([i]\)])\)\n \t\t\t\t\t - xi2d[\([i]\)]* \((psi22[\([i]\)] - Bn[\([i]\)]*xi22[\([i]\)])\))\), {i, 1, nstop}]; \n\t eq\ = Sum[ \((2*i + 1)\)* \((\n\t\t\t\t\t\t\t\t bn[\([i]\)]* \((cj[k]*cj[psi2d[\([i]\)]]*xi2[\([i]\)] - k*cj[psi2[\([i]\)]]*xi2d[\([i]\)])\) + \n \t\t\t\t\t\t\t\t an[\([i]\)]* \((k*cj[psi2d[\([i]\)]]*xi2[\([i]\)] - cj[k]*xi2d[\([i]\)]*cj[psi2[\([i]\)]])\))\), {i, 1, nstop}]; \n\t\teq]; \)\)], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(psi11d\)\" is similar \ to existing symbol \"\!\(psi11\)\"."\)], "Message"], Cell[BoxData[ \(General::"spell" \( : \ \) "Possible spelling error: new symbol name \"\!\(psi21d\)\" is similar \ to existing symbols \!\({psi21, psi2d}\)."\)], "Message"], Cell[BoxData[ \(General::"spell" \( : \ \) "Possible spelling error: new symbol name \"\!\(psi12\)\" is similar to \ existing symbols \!\({psi2, psi21}\)."\)], "Message"], Cell[BoxData[ \(General::"spell" \( : \ \) "Possible spelling error: new symbol name \"\!\(psi12d\)\" is similar \ to existing symbols \!\({psi12, psi21d, psi2d}\)."\)], "Message"], Cell[BoxData[ \(General::"stop" \( : \ \) "Further output of \!\(General :: \"spell\"\) will be suppressed during \ this calculation."\)], "Message"] }, Open ]], Cell[BoxData[ \(\(EMG[mh_, mi_, fv_, r1_] := Module[{m, ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; \n\t\tmf = FindRoot[CorEMG[m, mh, mi, fv, r1], {m, {ma, mb}}, AccuracyGoal -> 8, MaxIterations -> 5000]; m /. mf]; \)\)], "Input"], Cell[BoxData[ \(\(UnCorEMG[meff_, mh_, mi_, fv_, ri_] := Module[\n\t\t{i, mr, mmed, msph, k, k2, r1, r2, nstop, lambda, an, bn, psi, psid, psi2, psi2d, \n\ \ \ \ \ \ \ \ xi, xid, An, Bn, x, x1, x2}, \n\t\tr1\ = \ ri; \ \n\ \ \ \ \ \ r2\ = \ r1/\((fv^\((1/3)\))\); \n\t\tlambda\ = \ 2*Pi; \n\t\t mmed\ = meff; \n\t\tm1\ = \ mi; \n\t\tm2\ = \ mh; \n\t\t k = mmed*2*Pi/lambda; \n\t\tk1 = m1*2*Pi/lambda; \n\t\t k2\ = \ m2*2*Pi/lambda; \n\t\tx1\ = k*r1; \n\t\tx2\ = \ k*r2; \n \t\tnstop = IntegerPart[Abs[x2]\ + \ 4.0*Abs[x2]^0.3333\ + \ 2.0]; \n\t\tmr1\ = \ m1/mmed; \n\t\tmr2\ = \ m2/mmed; \n \t\t{psi11, psi11d} = rbj[mr1*x1, nstop]; \n \t\t{psi21, psi21d} = rbj[mr2*x1, nstop]; \n \t\t{psi12, psi12d} = rbj[mr1*x2, nstop]; \n \t\t{psi22, psi22d} = rbj[mr2*x2, nstop]; \n \t\t{psi2, psi2d} = rbj[x2, nstop]; \n \t\t{xi2, xi2d}\ = rbj[x2, nstop] + I*rby[x2, nstop]; \n \t\t{xi22, xi22d}\ = rbj[mr2*x2, nstop] + I*rby[mr2*x2, nstop]; \n \t\t{xi21, xi21d}\ = rbj[mr2*x1, nstop] + I*rby[mr2*x1, nstop]; \n \t\tAn\ = \ Table[\n\t\t\t\t \((mr2*psi11d[\([i]\)]*psi21[\([i]\)]\n\t\t\t\t\t\t\t\t - mr1*psi11[\([i]\)]*psi21d[\([i]\)])\)/ \((mr2*psi11d[\([i]\)]*xi21[\([i]\)]\n\t\t\t\t\t\t\t\t - mr1*psi11[\([i]\)]*xi21d[\([i]\)])\), {i, 1, nstop}]; \n\t\t Bn\ = \ Table[\n\t\t\t\t \((mr2*psi21d[\([i]\)]*psi11[\([i]\)]\n\t\t\t\t\t\t\t\t - mr1*psi21[\([i]\)]*psi11d[\([i]\)])\)/ \((mr2*xi21d[\([i]\)]*psi11[\([i]\)]\n\t\t\t\t\t\t\t\t - mr1*xi21[\([i]\)]*psi11d[\([i]\)])\), {i, 1, nstop}]; \n\t\t an = Table[\n\t\t\t\t \((psi2[\([i]\)]* \((psi22d[\([i]\)] - An[\([i]\)]*xi22d[\([i]\)])\)\n \t\t\t\t\t - mr2*psi2d[\([i]\)]* \((psi22[\([i]\)] - An[\([i]\)]*xi22[\([i]\)])\))\)/\t \((xi2[\([i]\)]* \((psi22d[\([i]\)] - An[\([i]\)]*xi22d[\([i]\)])\)\n \t\t\t\t\t - mr2*xi2d[\([i]\)]* \((psi22[\([i]\)] - An[\([i]\)]*xi22[\([i]\)])\))\), {i, 1, nstop}]; \n\t\t bn = Table[\n\t\t\t\t \((mr2*psi2[\([i]\)]* \((psi22d[\([i]\)] - Bn[\([i]\)]*xi22d[\([i]\)])\)\n \t\t\t\t\t - psi2d[\([i]\)]* \((psi22[\([i]\)] - Bn[\([i]\)]*xi22[\([i]\)])\))\)/\t \((mr2*xi2[\([i]\)]* \((psi22d[\([i]\)] - Bn[\([i]\)]*xi22d[\([i]\)])\)\n \t\t\t\t\t - xi2d[\([i]\)]* \((psi22[\([i]\)] - Bn[\([i]\)]*xi22[\([i]\)])\))\), {i, 1, nstop}]; \n\t eq\ = Sum[ \((2*i + 1)\)*\((an[\([i]\)] + bn[\([i]\)])\), {i, 1, nstop}]; \n \t\teq]; \)\)], "Input"], Cell[BoxData[ \(\(UEMG[mh_, mi_, fv_, r1_] := Module[{m, ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; \n\t\tmf = FindRoot[UnCorEMG[m, mh, mi, fv, r1], {m, {ma, mb}}, AccuracyGoal -> 8, MaxIterations -> 5000]; m /. mf]; \)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Energy Based Extended EMA - Aggregate Structures", "Subsection", FontFamily->"Times New Roman"], Cell[BoxData[ \(CorEnergyBR[meff_, mh_, mi_, fv_, r1_, r2_] := Module[\n\t\t{i, mr, mmed, msph, k, k2, y, nstop, lambda, meff2, an, bn, psi, psid, psi2, psi2d, \n\ \ \ \ \ \ \ \ xi, xid, A, B}, \n\t\t lambda\ = \ 2*Pi; \n\t\t (*\ sphere\ 1\ *) \n\t\tmmed\ = meff; \n \t\tmsph\ = \ mh; \n\t\tk = mmed*2*Pi/lambda; \n\t\t k2 = msph*2*Pi/lambda; \n\t\tx\ = k*r1; \n\t\ty\ = k2*r1; \n\t\t nstop = IntegerPart[Abs[x]\ + \ 4.0*Abs[x]^0.3333\ + \ 2.0]; \n\t\t mr\ = \ msph/mmed; \n\t\t{psi, psid} = rbj[x, nstop]; \n \t\t{psi2, psi2d} = rbj[y, nstop]; \n \t\t{xi, xid}\ = rbj[x, nstop] + I*rby[x, nstop]; \n\t\t an = Table[ \((mr*psi2[\([i]\)]*psid[\([i]\)] - psi[\([i]\)]*psi2d[\([i]\)]) \)/\((\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mr*psi2[\([i]\)]*xid[\([i]\)] - xi[\([i]\)]*psi2d[\([i]\)]) \), {i, 1, nstop}]; \n\t\t bn = Table[ \((psi2[\([i]\)]*psid[\([i]\)] - mr*psi[\([i]\)]*psi2d[\([i]\)]) \)/\((\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ psi2[\([i]\)]*xid[\([i]\)] - mr*xi[\([i]\)]*psi2d[\([i]\)]) \), {i, 1, nstop}]; \n\t\t A\ = Im[ Conjugate[k]* Sum[\((2*i + 1)\)* \((bn[\([i]\)]*cj[psid[\([i]\)]]*xi[\([i]\)] + cj[bn[\([i]\)]]*psi[\([i]\)]*cj[xid[\([i]\)]] - an[\([i]\)]*cj[psi[\([i]\)]]*xid[\([i]\)] - cj[an[\([i]\)]]*cj[xi[\([i]\)]]*psid[\([i]\)] + \n \t\t\t\t\t\t\t Abs[an[\([i]\)]]^2*xid[\([i]\)]*cj[xi[\([i]\)]] - Abs[bn[\([i]\)]]^2*xi[\([i]\)]*cj[xid[\([i]\)]])\), {i, 1, nstop}]]; \n\t\t (*\ sphere\ 2\ *) \n\t\tmsph\ = \ mi; \n\t\tk = mmed*2*Pi/lambda; \n\t\tk2 = msph*2*Pi/lambda; \n\t\t x\ = k*r2; \n\t\ty\ = k2*r2; \n\t\t nstop = IntegerPart[Abs[x]\ + \ 4.0*Abs[x]^0.3333\ + \ 2.0]; \n\t\t mr\ = \ msph/mmed; \n\t\t{psi, psid} = rbj[x, nstop]; \n \t\t{psi2, psi2d} = rbj[y, nstop]; \n \t\t{xi, xid}\ = rbj[x, nstop] + I*rby[x, nstop]; \n\t\t an = Table[ \((mr*psi2[\([i]\)]*psid[\([i]\)] - psi[\([i]\)]*psi2d[\([i]\)]) \)/\((\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ mr*psi2[\([i]\)]*xid[\([i]\)] - xi[\([i]\)]*psi2d[\([i]\)]) \), {i, 1, nstop}]; \n\t\t bn = Table[ \((psi2[\([i]\)]*psid[\([i]\)] - mr*psi[\([i]\)]*psi2d[\([i]\)]) \)/\((\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ psi2[\([i]\)]*xid[\([i]\)] - mr*xi[\([i]\)]*psi2d[\([i]\)]) \), {i, 1, nstop}]; \n\t\t B\ = Im[ Conjugate[k]* Sum[\((2*i + 1)\)* \((bn[\([i]\)]*cj[psid[\([i]\)]]*xi[\([i]\)] + cj[bn[\([i]\)]]*psi[\([i]\)]*cj[xid[\([i]\)]] - an[\([i]\)]*cj[psi[\([i]\)]]*xid[\([i]\)] - cj[an[\([i]\)]]*cj[xi[\([i]\)]]*psid[\([i]\)] + \n \t\t\t\t\t\t\t Abs[an[\([i]\)]]^2*xid[\([i]\)]*cj[xi[\([i]\)]] - Abs[bn[\([i]\)]]^2*xi[\([i]\)]*cj[xid[\([i]\)]])\), {i, 1, nstop}]]; \n\t\t eq\ = \((1 - fv)\)*A/\((4*Pi*r1^3/3)\) + fv*B/\((4*Pi*r2^3/3)\); \n \t\teq]; \n EnergyBR[mh_, mi_, fv_, r1_, r2_] := Module[{m, ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; \n\t\tmeff = FindRoot[CorEnergyBR[m, mh, mi, fv, r1, r2], {m, {ma, mb}}, AccuracyGoal -> 5, MaxIterations -> 5000]; m /. meff]; \)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Energy Based Extended EMA - Separated Grain Structures", "Subsection", FontFamily->"Times New Roman"], Cell[BoxData[ \(\(CorEnergyMG[meff_, mh_, mi_, fv_, ri_] := Module[\n\t\t{i, mr, mmed, msph, k, k2, r1, r2, nstop, lambda, an, bn, psi, psid, psi2, psi2d, \n\ \ \ \ \ \ \ \ xi, xid, An, Bn, x, x1, x2}, \n\t\tr1\ = \ ri; \ \n\ \ \ \ \ \ r2\ = \ r1/\((fv^\((1/3)\))\); \n\t\tlambda\ = \ 2*Pi; \n\t\t mmed\ = meff; \n\t\tm1\ = \ mi; \n\t\tm2\ = \ mh; \n\t\t k = mmed*2*Pi/lambda; \n\t\tk1 = m1*2*Pi/lambda; \n\t\t k2\ = \ m2*2*Pi/lambda; \n\t\tx1\ = k*r1; \n\t\tx2\ = \ k*r2; \n \t\tnstop = IntegerPart[Abs[x2]\ + \ 4.0*Abs[x2]^0.3333\ + \ 2.0]; \n\t\tmr1\ = \ m1/mmed; \n\t\tmr2\ = \ m2/mmed; \n \t\t{psi11, psi11d} = rbj[mr1*x1, nstop]; \n \t\t{psi21, psi21d} = rbj[mr2*x1, nstop]; \n \t\t{psi12, psi12d} = rbj[mr1*x2, nstop]; \n \t\t{psi22, psi22d} = rbj[mr2*x2, nstop]; \n \t\t{psi2, psi2d} = rbj[x2, nstop]; \n \t\t{xi2, xi2d}\ = rbj[x2, nstop] + I*rby[x2, nstop]; \n \t\t{xi22, xi22d}\ = rbj[mr2*x2, nstop] + I*rby[mr2*x2, nstop]; \n \t\t{xi21, xi21d}\ = rbj[mr2*x1, nstop] + I*rby[mr2*x1, nstop]; \n \t\tAn\ = \ Table[\n\t\t\t\t \((mr2*psi11d[\([i]\)]*psi21[\([i]\)]\n\t\t\t\t\t\t\t\t - mr1*psi11[\([i]\)]*psi21d[\([i]\)])\)/ \((mr2*psi11d[\([i]\)]*xi21[\([i]\)]\n\t\t\t\t\t\t\t\t - mr1*psi11[\([i]\)]*xi21d[\([i]\)])\), {i, 1, nstop}]; \n\t\t Bn\ = \ Table[\n\t\t\t\t \((mr2*psi21d[\([i]\)]*psi11[\([i]\)]\n\t\t\t\t\t\t\t\t - mr1*psi21[\([i]\)]*psi11d[\([i]\)])\)/ \((mr2*xi21d[\([i]\)]*psi11[\([i]\)]\n\t\t\t\t\t\t\t\t - mr1*xi21[\([i]\)]*psi11d[\([i]\)])\), {i, 1, nstop}]; \n\t\t an = Table[\n\t\t\t\t \((psi2[\([i]\)]* \((psi22d[\([i]\)] - An[\([i]\)]*xi22d[\([i]\)])\)\n \t\t\t\t\t - mr2*psi2d[\([i]\)]* \((psi22[\([i]\)] - An[\([i]\)]*xi22[\([i]\)])\))\)/\t \((xi2[\([i]\)]* \((psi22d[\([i]\)] - An[\([i]\)]*xi22d[\([i]\)])\)\n \t\t\t\t\t - mr2*xi2d[\([i]\)]* \((psi22[\([i]\)] - An[\([i]\)]*xi22[\([i]\)])\))\), {i, 1, nstop}]; \n\t\t bn = Table[\n\t\t\t\t \((mr2*psi2[\([i]\)]* \((psi22d[\([i]\)] - Bn[\([i]\)]*xi22d[\([i]\)])\)\n \t\t\t\t\t - psi2d[\([i]\)]* \((psi22[\([i]\)] - Bn[\([i]\)]*xi22[\([i]\)])\))\)/\t \((mr2*xi2[\([i]\)]* \((psi22d[\([i]\)] - Bn[\([i]\)]*xi22d[\([i]\)])\)\n \t\t\t\t\t - xi2d[\([i]\)]* \((psi22[\([i]\)] - Bn[\([i]\)]*xi22[\([i]\)])\))\), {i, 1, nstop}]; \n\t eq\ = Im[ Conjugate[k]* Sum[\((2*i + 1)\)* \((bn[\([i]\)]*cj[psi2d[\([i]\)]]*xi2[\([i]\)] + cj[bn[\([i]\)]]*psi2[\([i]\)]*cj[xi2d[\([i]\)]] - an[\([i]\)]*cj[psi2[\([i]\)]]*xi2d[\([i]\)] - cj[an[\([i]\)]]*cj[xi2[\([i]\)]]*psi2d[\([i]\)] + \n \t\t\t\t\t\t\t Abs[an[\([i]\)]]^2*xi2d[\([i]\)]*cj[xi2[\([i]\)]] - Abs[bn[\([i]\)]]^2*xi2[\([i]\)]*cj[xi2d[\([i]\)]])\), { i, 1, nstop}]]; \n\t\teq]; \)\)], "Input"], Cell[BoxData[""], "Input"], Cell[BoxData[ \(\(EnergyMG[mh_, mi_, fv_, r1_] := Module[{m, ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; \n\t\tmf = FindRoot[CorEnergyMG[m, mh, mi, fv, r1], {m, {ma, mb}}, AccuracyGoal -> 8, MaxIterations -> 5000]; m /. mf]; \)\)], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Multiple Scattering Methods (EFA, QCA, QCA-CP)", "Section"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\nEFA[mh_, mi_, fv_, xi_] := Module[{eh, ei, eeff, ed}, \n\t\teh\ = \ mh^2; \ ei = mi^2; \n\t\t ed\ = \((ei - eh)\)/eh; \n\t\t eeff\ = eh*\((1 + \((3*ed*fv/\((ed + 3)\))\) + I*2*xi^2 ed^2*fv/\((\((ed + 3)\)^2)\))\); \n\t\tSqrt[eeff]]; \n\nQCAPY[mh_, mi_, fv_, xi_] := Module[{eh, ei, eeff, ed}, \n\t\teh\ = \ mh^2; \ ei = mi^2; \n\t\t ed\ = \((ei - eh)\)/eh; \n\t\t eeff\ = eh*\((1 + \((3*ed*fv/\((ed*\((1 - fv)\) + 3)\))\) + I*2*xi^2 ed^2*fv* \((1 - fv)\)^4/ \((\((ed*\((1 - fv)\) + 3)\)^2*\((1 + 2*fv)\)^2)\))\); \n \t\tSqrt[eeff]]; \n QCACPPY[mh_, mi_, fv_, xi_] := Module[{ma, mb, ea, eb, eh, ei, ee}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; \n\t\teh = mh^2; ei = mi^2; ea = ma^2; eb = mb^2; \n\t\t ed = \((ei - eh)\)/eh; \n\t\t eeff = ee /. FindRoot[\n\t\t\t\t ee - eh*\(( 1 + \((3*ed*fv* \((ee/eh)\)/\((ed*\((1 - fv)\) + 3*\((ee/eh)\))\)) \) + I*2*xi^2 ed^2*fv*\((ee/eh)\)^\((5/2)\)* \((1 - fv)\)^4/ \((\((ed*\((1 - fv)\) + 3*\((ee/eh)\))\)^2* \((1 + 2*fv)\)^2)\))\)\n\t\t\t\t, {ee, eb}, MaxIterations -> 5000, AccuracyGoal \[Rule] 8]; Sqrt[eeff]]; \)\)], "Input"], Cell[BoxData[ \(General::"spell1" \( : \ \) "Possible spelling error: new symbol name \"\!\(eeff\)\" is similar to \ existing symbol \"\!\(meff\)\"."\)], "Message"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Combination of MG and BR", "Section"], Cell[BoxData[ \(\(MGBR[mh_, mi_, fv_] := Module[{ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; meff = FindRoot[ fv*\((mi^2 - m^2)\)/\((mi^2 + 2*m^2)\) + \((1.0 - fv)\)*\((mh^2 - m^2)\)/\((mh^2 + 2*m^2)\)\n\t\t\t\t\t\t + \n\t\t\t\t\t \((\((mh^2 - m^2)\)*\((mi^2 + 2*mh^2)\) + fv*\((2*mh^2 + m^2)\)*\((mi^2 - mh^2)\))\)/\n\t\t\t\t\t\t \((\((mh^2 + 2*m^2)\)*\((mi^2 + 2*mh^2)\) + 2*fv*\((mh^2 - m^2)\)*\((mi^2 - mh^2)\))\)\n \t\t\t\t\t\ , {m, {ma, mb}}, MaxIterations -> 5000, AccuracyGoal \[Rule] 8]; m /. meff]; \)\)], "Input"], Cell[CellGroupData[{ Cell["Combination of EMG and EBR", "Subsection"], Cell[BoxData[ \(\(EMGEBR[mh_, mi_, fv_, r1_, r2_] := Module[{m, ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; \n\t\tmf = FindMinimum[ Abs[CorEBR[u + v*I, mh, mi, fv, r1, r2] + \((fv/\((4*Pi*r2^3/3)\))\)*CorEMG[u + v*I, mh, mi, fv, r2]], { u, {Re[mb], Re[mb] + 0.01}}, {v, {Im[mb], Im[mb] + 0.01}}, AccuracyGoal -> 10, MaxIterations -> 10000]; \((u + v*I)\) /. mf[\([2]\)]]; \)\)], "Input"], Cell[BoxData[ \(\(TRYEMGEBR[mh_, mi_, fv_, r1_, r2_] := Module[{m, ma, mb}, \n\t\t If[Abs[mh] <= Abs[mi], ma\ = mh; \(mb = mi; \), ma = mi; mb = mh; ]; \n\t\tmf = FindRoot[ CorEBR[m, mh, mi, fv, r1, r2] + \((fv/\((4*Pi*r2^3/3)\))\)*CorEMG[m, mh, mi, fv, r2], { m, {ma, mb}}, AccuracyGoal \[Rule] 8, MaxIterations -> 5000]; \n \t\tm /. mf]; \)\)], 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