Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Next revision		Previous revision
documentation:tutorials:nio_crystal_field:rixs_l23m45 [2016/10/08 21:18] – created Maurits W. Haverkortdocumentation:tutorials:nio_crystal_field:rixs_l23m45 [2018/03/21 10:12] (current) – Stefano Agrestini
Line 1: Line 1:
 +{{indexmenu_n>7}}
 +====== RIXS $L_{2,3}M_{4,5}$ ======
  
 +###
 +Using the function ResonantSpectra we can calculate inelastic x-ray scattering. (or other second order processess) Here we show the example of $d$-$d$ excitations in NiO using RIXS. Due to an ever increasing resolution this method gains rapidly in popularity and impact. See for example the combined work of Ghiringhelli \textit{et al.} For NiO see \cite{Ghiringhelli:2005kp}.
 +###
 +
 +###
 +A script to calculate these spectra is:
 +<code Quanty RIXS_L23M45.Quanty>
 +-- this example calculates the resonant inelastic x-ray scattering in NiO we look at the
 +-- Ni L23M45 edge, i.e. we make an excitation from 2p to 3d (L23) and decay from the
 +-- 3d shell back to the 2p shell (final "hole" in the 3d shell M45 edge). These spectra
 +-- measure d-d excitations and or magnons.
 +
 +-- we minimize the output by setting the verbosity to 0
 +Verbosity(0)
 +
 +-- in order to do this calculation we need a Ni 2p shell and a Ni 3d shell
 +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)
 +
 +-- In order te describe the resonance we need the interaction on the 2p shell (spin-orbit)
 +Oppcldots= NewOperator("ldots", NF, IndexUp_2p, IndexDn_2p)
 +
 +-- and the Coulomb interaction between the 2p and 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. 
 +
 +-- 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("%8.3f ",Complex.Re(expectationvalue)))
 +  end
 +  io.write("\n")
 +end
 +
 +-- Not the main task of this example, but in order to know where the resonance sits it is
 +-- nice to calculate the x-ray absorption spectra
 +XASSpectra = CreateSpectra(XASHamiltonian, TXASx, psiList[1], {{"Emin",-10}, {"Emax",20}, {"NE",3500}, {"Gamma",1.0}})
 +XASSpectra.Print({{"file", "RIXSL23M45_XAS.dat"}})
 +
 +-- We also calculate the fluorescence yield spectra which are equal to the integral over the
 +-- RIXS spectra
 +FYSpectra = CreateFluorescenceYield(XASHamiltonian, TXASx, TXASydag, psiList[1], {{"Emin",-10}, {"Emax",20}, {"NE",3500}, {"Gamma",1.0}})
 +FYSpectra.Print({{"file","RIXSL23M45_FY.dat"}})
 +
 +-- and we calculate the RIXS spectra. Note that in order to calculate RIXS you need to
 +-- specify two Hamiltonians.
 +-- These calculations can be lengthy and one should think that for each incoming energy
 +-- (E1) one needs to a new calculation. In this case there are thus 120 calculations
 +RIXSSpectra = CreateResonantSpectra(XASHamiltonian, Hamiltonian, TXASx, TXASydag, psiList[1], {{"Emin1",-10}, {"Emax1",20}, {"NE1",120}, {"Gamma1",1.0}, {"Emin2",-0.5}, {"Emax2",7.5}, {"NE2",8000}, {"Gamma2",0.05}})
 +RIXSSpectra.Print({{"file", "RIXSL23M45.dat"}})
 +
 +print("Finished calculating the spectra now start plotting.\nThis might take more time than the calculation");
 +
 +-- Once finished we can make some nice plots. The spectra are saved to disk in ASCII format
 +-- but I like to add gnuplot scripts so you can look at pictures imeidately
 +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 out 'RIXSL23M45_Map.ps'
 +set terminal postscript portrait enhanced color  "Times" 8 size 7.5,6
 +
 +unset colorbox
 +
 +energyshift=857.6
 +
 +set multiplot 
 +set size 0.5,0.55
 +set origin 0,0
 +
 +set ylabel "resonant energy (eV)" font "Times,10"
 +set xlabel "energy loss (eV)" font "Times,10"
 +
 +set yrange [852:860]
 +set xrange [-0.5:7.5]
 +
 +plot "<(awk '((NR>5)&&(NR<8007)){for(i=3;i<=NF;i=i+2){printf \"%s \", $i}{printf \"%s\", RS}}' RIXSL23M45.dat)" matrix using ($2/1000-0.5):($1/4+energyshift-10):(-$3) with image notitle 
 +
 +set origin 0.5,0
 +
 +set yrange [869:877]
 +set xrange [-0.5:7.5]
 +
 +plot "<(awk '((NR>5)&&(NR<8007)){for(i=3;i<=NF;i=i+2){printf \"%s \", $i}{printf \"%s\", RS}}' RIXSL23M45.dat)" matrix using ($2/1000-0.5):($1/4+energyshift-10):(-$3) with image notitle 
 +
 +unset multiplot
 +
 +set out 'RIXSL23M45_Spec.ps'
 +set terminal postscript portrait enhanced color  "Times" 8 size 7.5,5
 +
 +set multiplot 
 +set size 0.25,1.0
 +set origin 0,0
 +
 +set ylabel "E (eV)" font "Times,10"
 +set xlabel "XAS" font "Times,10"
 +set yrange [energyshift-10:energyshift+20]
 +set xrange [-0.3:0]
 +plot "RIXSL23M45_XAS.dat"  using       3:($1+energyshift) notitle with lines ls  1,\
 +     "RIXSL23M45_FY.dat"   using (-3*$2):($1+energyshift) notitle with lines ls  4
 +
 +set size 0.8,1.0
 +set origin 0.2,0.0
 +
 +set xlabel "Energy loss (eV)" font "Times,10"
 +unset ylabel 
 +unset ytics
 +set xrange [-0.5:7.5]
 +
 +ofset = 0.25 
 +scale=3
 +
 +plot for [i=0:120] "RIXSL23M45.dat"  using 1:(-column(3+2*i)*scale+ofset*i-10 + energyshift) notitle with lines ls  4
 +
 +unset multiplot
 +]]
 +
 +-- write the gnuplot script to a file
 +file = io.open("RIXSL23M45.gnuplot", "w")
 +file:write(gnuplotInput)
 +file:close()
 +
 +-- call gnuplot to execute the script
 +os.execute("gnuplot RIXSL23M45.gnuplot")
 +-- transform to pdf and eps
 +os.execute("ps2pdf RIXSL23M45_Map.ps  ; ps2eps RIXSL23M45_Map.ps  ;  mv RIXSL23M45_Map.eps temp.eps  ; eps2eps temp.eps RIXSL23M45_Map.eps  ; rm temp.eps")
 +os.execute("ps2pdf RIXSL23M45_Spec.ps ; ps2eps RIXSL23M45_Spec.ps ;  mv RIXSL23M45_Spec.eps temp.eps ; eps2eps temp.eps RIXSL23M45_Spec.eps ; rm temp.eps")
 +
 +</code>
 +###
 +
 +###
 +It produces two plots. 
 +
 +{{:documentation:tutorials:nio_crystal_field:rixsl23m45_spec.png?nolink |}}
 + 
 +The first shows on the left the XAS spectra in black and the integrated RIXS spectra (FY) in blue. The resonant enhanced $d$-$d$ excitations are shown in the right plot. 
 +###
 +
 +###
 +{{:documentation:tutorials:nio_crystal_field:rixsl23m45_map.png?nolink |}}
 +
 +The second plot shows the same RIXS spectra, but now as an intensity map.
 +###
 +
 +###
 +The text output of the script is:
 +<file Quanty_Output RIXS_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
 +Finished calculating the spectra now start plotting.
 +This might take more time than the calculation
 +</file>
 +###
 +
 +
 +===== Table of contents =====
 +{{indexmenu>.#1|msort}}

You are here: Quanty - a quantum many body script language » Documentation » Tutorials » NiO crystal field » RIXS $L_{2,3}M_{4,5}$

Print/export