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documentation:tutorials:nio_ligand_field:fy_l23m45 [2016/10/10 09:41] (current)
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 +{{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:
 +<code Quanty FY_L23M45.Quanty>​
 +-- 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*$54) title 'FY - z out' with lines ls 4
 +
 +unset multiplot
 +]]
 +
 +
 +-- write the gnuplot script to a file
 +file = io.open("​FYL23M45.gnuplot",​ "​w"​)
 +file:​write(gnuplotInput)
 +file:​close()
 +
 +-- call gnuplot to execute the script
 +os.execute("​gnuplot FYL23M45.gnuplot"​)
 +-- and transform the ps to pdf
 +os.execute("​ps2pdf FYL23M45.ps ; ps2eps FYL23M45.ps ;  mv FYL23M45.eps temp.eps ; eps2eps temp.eps FYL23M45.eps ; rm temp.eps"​)
 +</​code>​
 +###
 +
 +###
 +The script returns 9 plots with each 4 curves. The local ground-state of Ni in NiO is 3-fold degenerate in the paramagnetic phase ($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.
 +<file Quanty_Output FY_L23M45.out>​
 +  #    <​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
 +</​file>​
 +###
 +
 +===== Table of contents =====
 +{{indexmenu>​.#​1|msort}}
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