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 — documentation:tutorials:nio_ligand_field:fy_l23m45 [2016/10/10 09:41] (current) 2016/10/09 15:36 Maurits W. Haverkort created 2016/10/09 15:36 Maurits W. Haverkort created Line 1: Line 1: + {{indexmenu_n>​6}} + ====== FY $L_{2,​3}M_{4,​5}$ ====== + ### + The absorption cross section is in principle measured using transmission. Transmission experiments in the soft-x-ray regime can be difficult as the absorption is quite high. Alternatively one can measure the reflectivity,​ which allows one to retrive the complete conductivity tensor using ellipsometry. As the x-ray wave-length is not large compared to the sample thickness this does not return the average sample absorption, but gives spatial information as well. Known as resonant scattering or reflectometry a beautiful technique, but might be overkill in some situations. A simple but effective way to measure the absorption cross section is to use the total electron yield, which can be measured by grounding the sample via an Ampare meter. + ### + + ### + An alternative is to measure the fluorescence yield. Although not proportional to the absorption cross section \cite{Kurian:​2012de,​vanVeenendaal:​1996tb,​deGroot:​1994tz} an extremely useful technique that contains similar information as absorption. Actually it is often more sensitive to differences in the ground-state and shows more detail in the spectral features. The calculation of fluorescence yield is similar to the calculation of absorption. + ### + + ### + In the following example we calculate the excitation of a $2p$ electron into the $3d$ shell of Ni in NiO. ($L_{2,3}$ edge). We integrate over the decay of a $3d$ electron into the $2p$ orbital (removing an electron from the $3d$-shell i.e. $M_{4,5}$) We thus look at the $L_{2,​3}$-$M_{4,​5}$ FY spectra. (note that one should always list both the excitation as well as the decay channel as the spectra change between different channels). ​ + ### + + ### + The input file is: + ​ + -- This example calculates the fluorescence yield spectra for NiO (L23M45, i.e. 2p to + -- 3d excitation and decay from 3d back to 2p) within the ligand field theory approximation + + -- We use the definitions of all operators and basis orbitals as defined in the file + -- 44_50_include and can afterwards directly continue by creating the Hamiltonian + -- and calculating the spectra + + dofile("​Include.Quanty"​) + + -- The parameters and scheme needed is the same as the one used for XAS + + -- We follow the energy definitions as introduced in the group of G.A. Sawatzky (Groningen) + -- J. Zaanen, G.A. Sawatzky, and J.W. Allen PRL 55, 418 (1985) + -- for parameters of specific materials see + -- A.E. Bockquet et al. PRB 55, 1161 (1996) + -- After some initial discussion the energies U and Delta refer to the center of a configuration + -- The L^10 d^n   ​configuration has an energy 0 + -- The L^9  d^n+1 configuration has an energy Delta + -- The L^8  d^n+2 configuration has an energy 2*Delta+Udd + -- + -- If we relate this to the onsite energy of the L and d orbitals we find + -- 10 eL +  n    ed + n(n-1) ​    U/2 == 0 + --  9 eL + (n+1) ed + (n+1)n ​    U/2 == Delta + --  8 eL + (n+2) ed + (n+1)(n+2) U/2 == 2*Delta+U + -- 3 equations with 2 unknowns, but with interdependence yield: + -- ed = (10*Delta-nd*(19+nd)*U/​2)/​(10+nd) + -- eL = nd*((1+nd)*Udd/​2-Delta)/​(10+nd) + -- + -- For the final state we/they defined + -- The 2p^5 L^10 d^n+1 configuration has an energy 0 + -- The 2p^5 L^9  d^n+2 configuration has an energy Delta + Udd - Upd + -- The 2p^5 L^8  d^n+3 configuration has an energy 2*Delta + 3*Udd - 2*Upd + -- + -- If we relate this to the onsite energy of the p and d orbitals we find + -- 6 ep + 10 eL +  n    ed + n(n-1) ​    Udd/2 + 6 n     Upd == 0 + -- 6 ep +  9 eL + (n+1) ed + (n+1)n ​    Udd/2 + 6 (n+1) Upd == Delta + -- 6 ep +  8 eL + (n+2) ed + (n+1)(n+2) Udd/2 + 6 (n+2) Upd == 2*Delta+Udd + -- 5 ep + 10 eL + (n+1) ed + (n+1)(n) ​  Udd/2 + 5 (n+1) Upd == 0 + -- 5 ep +  9 eL + (n+2) ed + (n+2)(n+1) Udd/2 + 5 (n+2) Upd == Delta+Udd-Upd + -- 5 ep +  8 eL + (n+3) ed + (n+3)(n+2) Udd/2 + 5 (n+3) Upd == 2*Delta+3*Udd-2*Upd + -- 6 equations with 3 unknowns, but with interdependence yield: + -- epfinal = (10*Delta + (1+nd)*(nd*Udd/​2-(10+nd)*Upd) / (16+nd) + -- edfinal = (10*Delta - nd*(31+nd)*Udd/​2-90*Upd) / (16+nd) + -- eLfinal = ((1+nd)*(nd*Udd/​2+6*Upd)-(6+nd)*Delta) / (16+nd) + -- + -- + -- + -- note that ed-ep = Delta - nd * U and not Delta + -- note furthermore that ep and ed here are defined for the onsite energy if the system had + -- locally nd electrons in the d-shell. In DFT or Hartree Fock the d occupation is in the end not + -- nd and thus the onsite energy of the Kohn-Sham orbitals is not equal to ep and ed in model + -- calculations. + -- + -- note furthermore that ep and eL actually should be different for most systems. We happily ignore this fact + -- + -- We normally take U and Delta as experimentally determined parameters + + -- number of electrons (formal valence) + nd = 8 + -- parameters from experiment (core level PES) + Udd     ​= ​ 7.3 + Upd     ​= ​ 8.5 + Delta   ​= ​ 4.7 + -- parameters obtained from DFT (PRB 85, 165113 (2012)) + F2dd    = 11.14 + F4dd    =  6.87 + F2pd    =  6.67 + G1pd    =  4.92 + G3pd    =  2.80 + tenDq   ​= ​ 0.56 + tenDqL ​ =  1.44 + Veg     ​= ​ 2.06 + Vt2g    =  1.21 + zeta_3d =  0.081 + zeta_2p = 11.51 + Bz      =  0.000001 + H112    =  0 + + ed      = (10*Delta-nd*(19+nd)*Udd/​2)/​(10+nd) + eL      = nd*((1+nd)*Udd/​2-Delta)/​(10+nd) + + epfinal = (10*Delta + (1+nd)*(nd*Udd/​2-(10+nd)*Upd)) / (16+nd) + edfinal = (10*Delta - nd*(31+nd)*Udd/​2-90*Upd) / (16+nd) + eLfinal = ((1+nd)*(nd*Udd/​2+6*Upd) - (6+nd)*Delta) / (16+nd) + + F0dd    = Udd + (F2dd+F4dd) * 2/63 + F0pd    = Upd + (1/15)*G1pd + (3/70)*G3pd + + Hamiltonian =  F0dd*OppF0_3d + F2dd*OppF2_3d + F4dd*OppF4_3d + zeta_3d*Oppldots_3d + Bz*(2*OppSz_3d + OppLz_3d) + H112 * (OppSx_3d+OppSy_3d+2*OppSz_3d)/​sqrt(6) + tenDq*OpptenDq_3d + tenDqL*OpptenDq_Ld + Veg * OppVeg + Vt2g * OppVt2g + ed * OppN_3d + eL * OppN_Ld + ​ + XASHamiltonian =  F0dd*OppF0_3d + F2dd*OppF2_3d + F4dd*OppF4_3d + zeta_3d*Oppldots_3d + Bz*(2*OppSz_3d + OppLz_3d)+ H112 * (OppSx_3d+OppSy_3d+2*OppSz_3d)/​sqrt(6) + tenDq*OpptenDq_3d + tenDqL*OpptenDq_Ld + Veg * OppVeg + Vt2g * OppVt2g + edfinal * OppN_3d + eLfinal * OppN_Ld + epfinal * OppN_2p + zeta_2p * Oppcldots + F0pd * OppUpdF0 + F2pd * OppUpdF2 + G1pd * OppUpdG1 + G3pd * OppUpdG3  ​ + + -- we now can create the lowest Npsi eigenstates:​ + Npsi=3 + -- in order to make sure we have a filling of 8 electrons we need to define some restrictions + StartRestrictions = {NF, NB, {"​000000 00 1111111111 0000000000",​8,​8},​ {"​111111 11 0000000000 1111111111",​18,​18}} + + psiList = Eigensystem(Hamiltonian,​ StartRestrictions,​ Npsi) + oppList={Hamiltonian,​ OppSsqr, OppLsqr, OppJsqr, OppSx_3d, OppLx_3d, OppSy_3d, OppLy_3d, OppSz_3d, OppLz_3d, Oppldots_3d,​ OppF2_3d, OppF4_3d, OppNeg_3d, OppNt2g_3d, OppNeg_Ld, OppNt2g_Ld, OppN_3d} + + -- print of some expectation values + print(" ​ #    <​E> ​     <​S^2> ​   <​L^2> ​   <​J^2> ​   <​S_x^3d>​ <​L_x^3d>​ <​S_y^3d>​ <​L_y^3d>​ <​S_z^3d>​ <​L_z^3d>​ <​l.s> ​   <​F[2]> ​  <​F[4]> ​  <​Neg^3d>​ <​Nt2g^3d><​Neg^Ld>​ <​Nt2g^Ld><​N^3d>"​);​ + for i = 1,#psiList do + io.write(string.format("​%3i ",i)) + for j = 1,#oppList do + expectationvalue = Chop(psiList[i]*oppList[j]*psiList[i]) + io.write(string.format("​%8.3f ",​expectationvalue)) + end + io.write("​\n"​) + end + + -- We calculate the x-ray absorption spectra for z, right circular and left circular polarized light for the ground-state. (3 spectra in total) + XASSpectra = CreateSpectra(XASHamiltonian,​ {T2p3dz, T2p3dr, T2p3dl}, psiList, {{"​Emin",​-10},​ {"​Emax",​20},​ {"​NE",​3500},​ {"​Gamma",​1.0}}) + XASSpectra.Print({{"​file","​FYL23M45_XAS.dat"​}});​ + + -- and we calculate the FY spectra + FYSpectra = CreateFluorescenceYield(XASHamiltonian,​ {T2p3dz, T2p3dr, T2p3dl}, {T3d2px, T3d2py, T3d2pz}, psiList, {{"​Emin",​-10},​ {"​Emax",​20},​ {"​NE",​3500},​ {"​Gamma",​1.0}});​ + FYSpectra.Print({{"​file","​FYL23M45_Spec.dat"​}});​ + + -- in order to plot both the XAS and FY spectra we can define a gnuplot script + gnuplotInput = [[ + set autoscale ​ + set xtic auto  ​ + set ytic auto + set style line  1 lt 1 lw 1 lc rgb "#​000000"​ + set style line  2 lt 1 lw 1 lc rgb "#​FF0000"​ + set style line  3 lt 1 lw 1 lc rgb "#​00FF00"​ + set style line  4 lt 1 lw 1 lc rgb "#​0000FF"​ + + set xlabel "E (eV)" font "​Times,​10"​ + set ylabel "​Intensity (arb. units)"​ font "​Times,​10"​ + + set out '​FYL23M45.ps'​ + set terminal postscript portrait enhanced color  "​Times"​ 8 size 7.5,6 + set yrange [0:0.6] + + set multiplot layout 3, 3 + + plot "​FYL23M45_XAS.dat" ​ u 1:(-$3 ) title '​z-polarized Sz=-1' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ + "​FYL23M45_Spec.dat"​ u 1:​(4*$2) ​ title 'FY - x out' with lines ls 2,\ + "​FYL23M45_Spec.dat"​ u 1:​(4*$4) ​ title 'FY - y out' with lines ls 3,\ + "​FYL23M45_Spec.dat"​ u 1:​(4*$6) ​ title 'FY - z out' with lines ls 4 + plot "​FYL23M45_XAS.dat" ​ u 1:(-$5 ) title '​z-polarized Sz= 0' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ + "​FYL23M45_Spec.dat"​ u 1:​(4*$8) ​ title 'FY - x out' with lines ls 2,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$10) title 'FY - y out' with lines ls 3,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$12) title 'FY - z out' with lines ls 4 + plot "​FYL23M45_XAS.dat" ​ u 1:(-$7 ) title '​z-polarized Sz= 1' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$14) title 'FY - x out' with lines ls 2,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$16) title 'FY - y out' with lines ls 3,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$18) title 'FY - z out' with lines ls 4 + + plot "​FYL23M45_XAS.dat" ​ u 1:(-$9 ) title '​r-polarized Sz=-1' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$20) title 'FY - x out' with lines ls 2,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$22) title 'FY - y out' with lines ls 3,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$24) title 'FY - z out' with lines ls 4 + plot "​FYL23M45_XAS.dat" ​ u 1:​(-$11) ​ title '​r-polarized Sz= 0' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$26) title 'FY - x out' with lines ls 2,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$28) title 'FY - y out' with lines ls 3,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$30) title 'FY - z out' with lines ls 4 + plot "​FYL23M45_XAS.dat" ​ u 1:​(-$13) ​ title '​r-polarized Sz= 1' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$32) title 'FY - x out' with lines ls 2,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$34) title 'FY - y out' with lines ls 3,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$36) title 'FY - z out' with lines ls 4 + + plot "​FYL23M45_XAS.dat" ​ u 1:​(-$15) ​ title '​l-polarized Sz=-1' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$38) title 'FY - x out' with lines ls 2,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$40) title 'FY - y out' with lines ls 3,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$42) title 'FY - z out' with lines ls 4 + plot "​FYL23M45_XAS.dat" ​ u 1:​(-$17) ​ title '​l-polarized Sz= 0' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$44) title 'FY - x out' with lines ls 2,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$46) title 'FY - y out' with lines ls 3,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$48) title 'FY - z out' with lines ls 4 + plot "​FYL23M45_XAS.dat" ​ u 1:​(-$19) ​ title '​l-polarized Sz= 1' with filledcurves y1=0 ls 1 fs transparent solid 0.5,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$50) title 'FY - x out' with lines ls 2,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$52) title 'FY - y out' with lines ls 3,\ + "​FYL23M45_Spec.dat"​ u 1:(4*$54) title 'FY - z out' with lines ls 4 + + unset multiplot + ]] + + + -- write the gnuplot script to a file + file = io.open("​FYL23M45.gnuplot",​ "​w"​) + file:​write(gnuplotInput) + file:​close() + + -- call gnuplot to execute the script + os.execute("​gnuplot FYL23M45.gnuplot"​) + -- and transform the ps to pdf + os.execute("​ps2pdf FYL23M45.ps ; ps2eps FYL23M45.ps ;  mv FYL23M45.eps temp.eps ; eps2eps temp.eps FYL23M45.eps ; rm temp.eps"​) + ​ + ### + + ### + The script returns 9 plots with each 4 curves. The local ground-state of Ni in NiO is 3-fold degenerate in the paramagnetic phase ($S=1$) The different columns show the spectra for the states with different $S_z$. In the paramagnetic phase one should summ these 3 spectra, in a full magnetized sample one measurers either the left or the right column. The different rows the different incoming polarization. Top row z-polarized,​ middle right bottom left polarized light. The black filed curve shows the absorption cross section. The red, green and blue curve show the spectra for different outgoing polarization. ​ + ### + + | {{:​documentation:​tutorials:​nio_ligand_field:​fyl23m45.png?​nolink |}} | + ^ Fluorescence yield spectra for different geometries of both incoming and outgoing polarization compared to x-ray absorbtion ^ + + ### + The script shows some information on the ground-state,​ here the text output. + ​ + #    <​E> ​     <​S^2> ​   <​L^2> ​   <​J^2> ​   <​S_x^3d>​ <​L_x^3d>​ <​S_y^3d>​ <​L_y^3d>​ <​S_z^3d>​ <​L_z^3d>​ <​l.s> ​   <​F[2]> ​  <​F[4]> ​  <​Neg^3d>​ <​Nt2g^3d><​Neg^Ld>​ <​Nt2g^Ld><​N^3d>​ + 1   ​-3.395 ​   1.999   ​12.000 ​  ​15.147 ​   0.000    0.000    0.000    0.000   ​-0.905 ​  ​-0.280 ​  ​-0.319 ​  ​-1.043 ​  ​-0.925 ​   2.189    5.989    3.823    6.000    8.178 + 2   ​-3.395 ​   1.999   ​12.000 ​  ​15.147 ​   0.000    0.000    0.000    0.000   ​-0.000 ​  ​-0.000 ​  ​-0.319 ​  ​-1.043 ​  ​-0.925 ​   2.189    5.989    3.823    6.000    8.178 + 3   ​-3.395 ​   1.999   ​12.000 ​  ​15.147 ​   0.000    0.000    0.000    0.000    0.905    0.280   ​-0.319 ​  ​-1.043 ​  ​-0.925 ​   2.189    5.989    3.823    6.000    8.178 + Start of LanczosTriDiagonalizeKrylovMC + Start of LanczosTriDiagonalizeKrylovMC + Start of LanczosTriDiagonalizeKrylovMC + Start of LanczosTriDiagonalizeKrylovMC + Start of LanczosTriDiagonalizeKrylovMC + Start of LanczosTriDiagonalizeKrylovMC + Start of LanczosTriDiagonalizeKrylovMC + Start of LanczosTriDiagonalizeKrylovMC + Start of LanczosTriDiagonalizeKrylovMC + ​ + ### + + ===== Table of contents ===== + {{indexmenu>​.#​1|msort}}