FY $L_{2,3}M_{1}$

In Fluoressence Yield spectroscopy one can focus on different decay channels. Each of them will yield a different spectrum. For a $2p$ to $3d$ excitation one can look at the $3s$ to $2p$ decay. The edge measured is thus the $L_{2,3}M_{1}$ edge.

The input file is:

FY_L23M1.Quanty

-- This example calculates the fluorescence yield spectra for NiO (L23M45, i.e. 2p to
-- 3d excitation and decay from 3s 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
 
-- here we calculate the x-ray absorption spectra, not the main task of this file, but nice to do so we can compare
XASSpectra = CreateSpectra(XASHamiltonian, {T2p3dz, T2p3dr, T2p3dl}, psiList, {{"Emin",-10}, {"Emax",20}, {"NE",3500}, {"Gamma",1.0}});
XASSpectra.Print({{"file","FYL23M1_XAS.dat"}});
 
-- and we calculate the FY spectra
FYSpectra = CreateFluorescenceYield(XASHamiltonian, {T2p3dz, T2p3dr, T2p3dl}, {T3s2px, T3s2py, T3s2pz}, psiList, {{"Emin",-10}, {"Emax",20}, {"NE",3500}, {"Gamma",1.0}})
FYSpectra.Print({{"file","FYL23M1_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 'FYL23M1.ps'
set terminal postscript portrait enhanced color  "Times" 8 size 7.5,6
set yrange [0:0.6]
set size 1,1
 
set multiplot layout 3, 3
 
plot "FYL23M1_XAS.dat"  u 1:(-$3 )  title 'z-polarized Sz=-1' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
     "FYL23M1_Spec.dat" u 1:(4*$2)  title 'FY - x out' with lines ls 2,\
     "FYL23M1_Spec.dat" u 1:(4*$4)  title 'FY - y out' with lines ls 3,\
     "FYL23M1_Spec.dat" u 1:(4*$6)  title 'FY - z out' with lines ls 4
plot "FYL23M1_XAS.dat"  u 1:(-$5 )  title 'z-polarized Sz= 0' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
     "FYL23M1_Spec.dat" u 1:(4*$8)  title 'FY - x out' with lines ls 2,\
     "FYL23M1_Spec.dat" u 1:(4*$10) title 'FY - y out' with lines ls 3,\
     "FYL23M1_Spec.dat" u 1:(4*$12) title 'FY - z out' with lines ls 4
plot "FYL23M1_XAS.dat"  u 1:(-$7 )  title 'z-polarized Sz= 1' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
     "FYL23M1_Spec.dat" u 1:(4*$14) title 'FY - x out' with lines ls 2,\
     "FYL23M1_Spec.dat" u 1:(4*$16) title 'FY - y out' with lines ls 3,\
     "FYL23M1_Spec.dat" u 1:(4*$18) title 'FY - z out' with lines ls 4
 
plot "FYL23M1_XAS.dat"  u 1:(-$9 )  title 'r-polarized Sz=-1' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
     "FYL23M1_Spec.dat" u 1:(4*$20) title 'FY - x out' with lines ls 2,\
     "FYL23M1_Spec.dat" u 1:(4*$22) title 'FY - y out' with lines ls 3,\
     "FYL23M1_Spec.dat" u 1:(4*$24) title 'FY - z out' with lines ls 4
plot "FYL23M1_XAS.dat"  u 1:(-$11)  title 'r-polarized Sz= 0' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
     "FYL23M1_Spec.dat" u 1:(4*$26) title 'FY - x out' with lines ls 2,\
     "FYL23M1_Spec.dat" u 1:(4*$28) title 'FY - y out' with lines ls 3,\
     "FYL23M1_Spec.dat" u 1:(4*$30) title 'FY - z out' with lines ls 4
plot "FYL23M1_XAS.dat"  u 1:(-$13)  title 'r-polarized Sz= 1' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
     "FYL23M1_Spec.dat" u 1:(4*$32) title 'FY - x out' with lines ls 2,\
     "FYL23M1_Spec.dat" u 1:(4*$34) title 'FY - y out' with lines ls 3,\
     "FYL23M1_Spec.dat" u 1:(4*$36) title 'FY - z out' with lines ls 4
 
plot "FYL23M1_XAS.dat"  u 1:(-$15)  title 'l-polarized Sz=-1' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
     "FYL23M1_Spec.dat" u 1:(4*$38) title 'FY - x out' with lines ls 2,\
     "FYL23M1_Spec.dat" u 1:(4*$40) title 'FY - y out' with lines ls 3,\
     "FYL23M1_Spec.dat" u 1:(4*$42) title 'FY - z out' with lines ls 4
plot "FYL23M1_XAS.dat"  u 1:(-$17)  title 'l-polarized Sz= 0' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
     "FYL23M1_Spec.dat" u 1:(4*$44) title 'FY - x out' with lines ls 2,\
     "FYL23M1_Spec.dat" u 1:(4*$46) title 'FY - y out' with lines ls 3,\
     "FYL23M1_Spec.dat" u 1:(4*$48) title 'FY - z out' with lines ls 4
plot "FYL23M1_XAS.dat"  u 1:(-$19)  title 'l-polarized Sz= 1' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
     "FYL23M1_Spec.dat" u 1:(4*$50) title 'FY - x out' with lines ls 2,\
     "FYL23M1_Spec.dat" u 1:(4*$52) title 'FY - y out' with lines ls 3,\
     "FYL23M1_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("FYL23M1.gnuplot", "w")
file:write(gnuplotInput)
file:close()
 
-- call gnuplot to execute the script
os.execute("gnuplot FYL23M1.gnuplot")
-- and change the ps to pdf and eps

The resulting spectra are: (for a description see the text for the $L_{2,3}M_{45}$ spectra.

Fluorescence Yield spectra looking at the $3s$ to $2p$ decay channel after a $2p$ to $3d$ excitation for different incomming and outgoing polarizations compared to x-ray absorption.

And the text output is:

FY_L23M1.out

text produced as 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