This is an old revision of the document!


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 corresponding input file is:

FY_L23M1.Quanty
-- in this example we calculate the fluorescence yield spectra for the L23M1 edge.
-- i.e. we make an 2p to 3d excitation followed by a 3s to 2p decay. 
 
-- we minimize the output by setting the verbosity to 0
Verbosity(0)
 
-- in order to do this calculation we need a Ni 2p shell a Ni 3s shell and a Ni 3d shell
NF=18
NB=0
IndexDn_2p={0,2,4}
IndexUp_2p={1,3,5}
IndexDn_3s={6}
IndexUp_3s={7}
IndexDn_3d={8,10,12,14,16}
IndexUp_3d={9,11,13,15,17}
 
-- just like in the previous example we define several operators acting on the Ni -3d shell
 
OppSx   =NewOperator("Sx"   ,NF, IndexUp_3d, IndexDn_3d)
OppSy   =NewOperator("Sy"   ,NF, IndexUp_3d, IndexDn_3d)
OppSz   =NewOperator("Sz"   ,NF, IndexUp_3d, IndexDn_3d)
OppSsqr =NewOperator("Ssqr" ,NF, IndexUp_3d, IndexDn_3d)
OppSplus=NewOperator("Splus",NF, IndexUp_3d, IndexDn_3d)
OppSmin =NewOperator("Smin" ,NF, IndexUp_3d, IndexDn_3d)
 
OppLx   =NewOperator("Lx"   ,NF, IndexUp_3d, IndexDn_3d)
OppLy   =NewOperator("Ly"   ,NF, IndexUp_3d, IndexDn_3d)
OppLz   =NewOperator("Lz"   ,NF, IndexUp_3d, IndexDn_3d)
OppLsqr =NewOperator("Lsqr" ,NF, IndexUp_3d, IndexDn_3d)
OppLplus=NewOperator("Lplus",NF, IndexUp_3d, IndexDn_3d)
OppLmin =NewOperator("Lmin" ,NF, IndexUp_3d, IndexDn_3d)
 
OppJx   =NewOperator("Jx"   ,NF, IndexUp_3d, IndexDn_3d)
OppJy   =NewOperator("Jy"   ,NF, IndexUp_3d, IndexDn_3d)
OppJz   =NewOperator("Jz"   ,NF, IndexUp_3d, IndexDn_3d)
OppJsqr =NewOperator("Jsqr" ,NF, IndexUp_3d, IndexDn_3d)
OppJplus=NewOperator("Jplus",NF, IndexUp_3d, IndexDn_3d)
OppJmin =NewOperator("Jmin" ,NF, IndexUp_3d, IndexDn_3d)
 
Oppldots=NewOperator("ldots",NF, IndexUp_3d, IndexDn_3d)
 
-- as in the previous example we define the Coulomb interaction
 
OppF0 =NewOperator("U", NF, IndexUp_3d, IndexDn_3d, {1,0,0})
OppF2 =NewOperator("U", NF, IndexUp_3d, IndexDn_3d, {0,1,0})
OppF4 =NewOperator("U", NF, IndexUp_3d, IndexDn_3d, {0,0,1})
 
-- as in the previous example we define the crystal-field operator
 
Akm = PotentialExpandedOnClm("Oh", 2, {0.6,-0.4})
OpptenDq = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm)
 
-- and as in the previous example we define operators that count the number of eg and t2g
-- electrons
 
Akm = PotentialExpandedOnClm("Oh", 2, {1,0})
OppNeg = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm)
Akm = PotentialExpandedOnClm("Oh", 2, {0,1})
OppNt2g = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm)
 
-- we need to define the core level spin-orbit interaction on the Ni 2p shell
 
Oppcldots= NewOperator("ldots", NF, IndexUp_2p, IndexDn_2p)
 
-- and the interaction between the Ni 2p shell and the Ni 3d shell
 
OppUpdF0 = NewOperator("U", NF, IndexUp_2p, IndexDn_2p, IndexUp_3d, IndexDn_3d, {1,0}, {0,0})
OppUpdF2 = NewOperator("U", NF, IndexUp_2p, IndexDn_2p, IndexUp_3d, IndexDn_3d, {0,1}, {0,0})
OppUpdG1 = NewOperator("U", NF, IndexUp_2p, IndexDn_2p, IndexUp_3d, IndexDn_3d, {0,0}, {1,0})
OppUpdG3 = NewOperator("U", NF, IndexUp_2p, IndexDn_2p, IndexUp_3d, IndexDn_3d, {0,0}, {0,1})
 
-- next we define the dipole operator. The dipole operator is given as epsilon.r
-- with epsilon the polarization vector of the light and r the unit position vector
-- We can expand the position vector on (renormalized) spherical harmonics and use
-- the crystal-field operator to create the dipole operator. 
 
-- we both define the dipole operator between the 2p and 3d shell as well as the dipole 
-- operator between the 3s and 2p shell
 
-- x polarized light is defined as x = Cos[phi]Sin[theta] = sqrt(1/2) ( C_1^{(-1)} - C_1^{(1)})
Akm = {{1,-1,sqrt(1/2)},{1, 1,-sqrt(1/2)}}
TXASx  = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm)
T3s2px = NewOperator("CF", NF, IndexUp_2p, IndexDn_2p, IndexUp_3s, IndexDn_3s, Akm)
-- y polarized light is defined as y = Sin[phi]Sin[theta] = sqrt(1/2) I ( C_1^{(-1)} + C_1^{(1)})
Akm = {{1,-1,sqrt(1/2)*I},{1, 1,sqrt(1/2)*I}}
TXASy  = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm)
T3s2py = NewOperator("CF", NF, IndexUp_2p, IndexDn_2p, IndexUp_3s, IndexDn_3s, Akm)
-- z polarized light is defined as z = Cos[theta] = C_1^{(0)}
Akm = {{1,0,1}}
TXASz  = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm)
T3s2pz = NewOperator("CF", NF, IndexUp_2p, IndexDn_2p, IndexUp_3s, IndexDn_3s, Akm)
 
-- besides linear polarized light one can define circular polarized light as the sum of 
-- x and y polarizations with complex prefactors
TXASr = sqrt(1/2)*(TXASx - I * TXASy)
TXASl =-sqrt(1/2)*(TXASx + I * TXASy)
 
-- we can remove zero's from the dipole operator by chopping it.
TXASr.Chop()
TXASl.Chop()
 
-- once all operators are defined we can set some parameter values.
 
-- the value of U drops out of a crystal-field calculation as the total number of electrons
-- is always the same
U       =  0.000 
-- F2 and F4 are often referred to in the literature as J_{Hund}. They represent the energy
-- differences between different multiplets. Numerical values can be found in the back of
-- my PhD. thesis for example. http://arxiv.org/abs/cond-mat/0505214 
F2dd    = 11.142 
F4dd    =  6.874
-- F0 is not the same as U, although they are related. Unimportant in crystal-field theory
-- the difference between U and F0 is so important that I do include it here. (U=0 so F0 is not)
F0dd    = U+(F2dd+F4dd)*2/63
-- in crystal field theory U drops out of the equation, also true for the interaction between the 
-- Ni 2p and Ni 3d electrons
Upd     =  0.000 
-- The Slater integrals between the 2p and 3d shell, again the numerical values can be found
-- in the back of my PhD. thesis. (http://arxiv.org/abs/cond-mat/0505214)
F2pd    =  6.667
G1pd    =  4.922
G3pd    =  2.796
-- F0 is not the same as U, although they are related. Unimportant in crystal-field theory
-- the difference between U and F0 is so important that I do include it here. (U=0 so F0 is not)
F0pd    =  Upd + G1pd*1/15 + G3pd*3/70
-- tenDq in NiO is 1.1 eV as can be seen in optics or using IXS to measure d-d excitations
tenDq   =  1.100
-- the Ni 3d spin-orbit is small but finite
zeta_3d =  0.081
-- the Ni 2p spin-orbit is very large and should not be scaled as theory is quite accurate here
zeta_2p = 11.498
-- we can add a small magnetic field, just to get nice expectation values. (units in eV... )
Bz      = 0.000001
 
-- the total Hamiltonian is the sum of the different operators multiplied with their prefactor
Hamiltonian = F0dd*OppF0 + F2dd*OppF2 + F4dd*OppF4 + tenDq*OpptenDq + zeta_3d*Oppldots + Bz*(2*OppSz+OppLz)
 
-- We normally do not include core-valence interactions between filed and partially filled 
-- shells as it drops out anyhow. For the XAS we thus need to define a "different" 
-- (more complete) Hamiltonian.
XASHamiltonian = Hamiltonian + zeta_2p * Oppcldots + F0pd * OppUpdF0 + F2pd * OppUpdF2 + G1pd * OppUpdG1 + G3pd * OppUpdG3
 
-- We saw in the previous example that NiO has a ground-state doublet with occupation 
-- t2g^6 eg^2 and S=1 (S^2=S(S+1)=2). The next state is 1.1 eV higher in energy and thus
-- unimportant for the ground-state upto temperatures of 10 000 Kelvin. We thus restrict 
-- the calculation to the lowest 3 eigenstates.
Npsi=3
-- We need a filling of 6 electrons in the 2p shell
-- 2 electrons in the 3s shell
-- and 8 electrons in the 3d shell
StartRestrictions = {NF, NB, {"111111 11 0000000000",8,8}, {"000000 00 1111111111",8,8}}
 
-- And calculate the lowest 3 eigenfunctions
psiList = Eigensystem(Hamiltonian, StartRestrictions, Npsi)
 
-- In order to get some information on these eigenstates it is good to plot expectation values
-- We first define a list of all the operators we would like to calculate the expectation value of
oppList={Hamiltonian, OppSsqr, OppLsqr, OppJsqr, OppSz, OppLz, Oppldots, OppF2, OppF4, OppNeg, OppNt2g};
 
-- next we loop over all operators and all states and print the expectation value
print(" <E>    <S^2>  <L^2>  <J^2>  <S_z>  <L_z>  <l.s>  <F[2]> <F[4]> <Neg>  <Nt2g>");
for i = 1,#psiList do
  for j = 1,#oppList do
    expectationvalue = Chop(psiList[i]*oppList[j]*psiList[i])
    io.write(string.format("%6.3f ",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, {TXASz, TXASr, TXASl}, 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, {TXASz, TXASr, TXASl}, {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
os.execute("ps2pdf FYL23M1.ps ; ps2eps FYL23M1.ps ;  mv FYL23M1.eps temp.eps ; eps2eps temp.eps FYL23M1.eps ; rm temp.eps")

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

And the text output is:

FY_L23M1.out
 <E>    <S^2>  <L^2>  <J^2>  <S_z>  <L_z>  <l.s>  <F[2]> <F[4]> <Neg>  <Nt2g>
-2.721  1.999 12.000 15.120 -0.994 -0.286 -0.324 -1.020 -0.878  2.011  5.989 
-2.721  1.999 12.000 15.120 -0.000 -0.000 -0.324 -1.020 -0.878  2.011  5.989 
-2.721  1.999 12.000 15.120  0.994  0.286 -0.324 -1.020 -0.878  2.011  5.989 
Start of LanczosTriDiagonalizeKrylovRR
Start of LanczosTriDiagonalizeKrylovRR
Start of LanczosTriDiagonalizeKrylovRR
Start of LanczosTriDiagonalizeKrylovRR
Start of LanczosTriDiagonalizeKrylovRR
Start of LanczosTriDiagonalizeKrylovRR
Start of LanczosTriDiagonalizeKrylovRR
Start of LanczosTriDiagonalizeKrylovRR
Start of LanczosTriDiagonalizeKrylovRR

Table of contents

Print/export