# 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 ",Complex.Re(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
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