FY $L_{2,3}M_{4,5}$

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====== FY $L_{2,3}M_{4,5}$ ======

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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.
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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.
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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). 
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The input is:
<code Quanty FY_L23M45.Quanty>
-- In this example we will calculate the fluorescence yield spectra 
-- One makes an excitation from 2p to 3d and then looks at a specific decay channel
-- Or at the sum over all channels
-- the spectra are integrated over the energy of the decay channel which allows for
-- extreme efficient calculation of these spectra. 
-- Note that most detectors will not be equally sensitive to all possible photon energies
-- and one thus would always measure some weighted sum over the different decay channels

-- this file calculates the Ni L23M45 spectra. 
-- (L23, i.e. we excite from 2p to 3d)
-- (M45, we decay from the 3d shell, into the 2p shell)

-- we minimize the output by setting the verbosity to 0
Verbosity(0)

-- In order to do crystal-field theory for NiO we need to define a Ni d-shell.
-- A d-shell has 10 elements and we label again the even spin-orbitals to be spin down
-- and the odd spin-orbitals to be spin up. In order to calculate 2p to 3d excitations we
-- also need a Ni 2p shell. We thus have a total of 10+6=16 fermions, 6 Ni-2p and 10 Ni-3d
-- spin-orbitals
NF=16
NB=0
IndexDn_2p={0,2,4}
IndexUp_2p={1,3,5}
IndexDn_3d={6,8,10,12,14}
IndexUp_3d={7,9,11,13,15}

-- 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)

-- new for core level spectroscopy are operators that define the interaction acting on the
-- Ni-2p shell. There is actually only one of these interactions, which is the Ni-2p
-- spin-orbit interaction

Oppcldots= NewOperator("ldots", NF, IndexUp_2p, IndexDn_2p)

-- we also need to define the Coulomb interaction between the Ni 2p- and Ni 3d-shell
-- Again the interaction (e^2/(|r_i-r_j|)) is expanded on spherical harmonics. For the interaction
-- between two shells we need to consider two cases. For the direct interaction a 2p electron
-- scatters of a 3d electron into a 2p and 3d electron. The radial integrals involve
-- the square of a 2p radial wave function at coordinate 1 and the square of a 3d radial
-- wave function at coordinate 2. The transfer of angular momentum can either be 0 or 2.
-- These processes are called direct and the resulting Slater integrals are F[0] and F[2].
-- The second proces involves a 2p electron scattering of a 3d electron into the 3d shell
-- and at the same time the 3d electron scattering into a 2p shell. These exchange processes
-- involve radial integrals over the product of a 2p and 3d radial wave function. The transfer
-- of angular momentum in this case can be 1 or 3 and the Slater integrals are called G1 and G3.

-- In Quanty you can enter these processes by labeling 4 indices for the orbitals, once
-- the 2p shell with spin up, 2p shell with spin down, 3d shell with spin up and 3d shell with
-- spin down. Followed by the direct Slater integrals (F0 and F2) and the exchange Slater 
-- integrals (G1 and G3)

-- Here we define the operators separately and later sum them with appropriate prefactors

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. 

-- 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)
-- 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)
-- 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)

-- 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()

-- the 3d to 2p dipole transition is the conjugate transpose of the 2p to 3d dipole transition
TXASxdag = ConjugateTranspose(TXASx)
TXASydag = ConjugateTranspose(TXASy)
TXASzdag = ConjugateTranspose(TXASz)
TXASldag = ConjugateTranspose(TXASl)
TXASrdag = ConjugateTranspose(TXASr)

-- 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
-- in order to make sure we have a filling of 8
-- electrons we need to define some restrictions
-- We need to restrict the occupation of the Ni-2p shell to 6 for the ground state and for
-- the Ni 3d-shell to 8.
StartRestrictions = {NF, NB, {"111111 0000000000",6,6}, {"000000 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","FYL23M45_XAS.dat"}});

-- and we calculate the FY spectra
FYSpectra = CreateFluorescenceYield(XASHamiltonian, {TXASz, TXASr, TXASl}, {TXASxdag, TXASydag, TXASzdag}, 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_crystal_field:fyl23m45.png?nolink}}
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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_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
</file>
###

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