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documentation:tutorials:nio_ligand_field:nixs_m45 [2016/10/09 15:42] – created Maurits W. Haverkortdocumentation:tutorials:nio_ligand_field:nixs_m45 [2018/03/20 11:06] (current) – Maurits W. Haverkort
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 +{{indexmenu_n>10}}
 +====== nIXS $M_{4,5}$ ======
  
 +###
 +Inelastic x-ray scattering IXS (non-resonant) nIXS or x-ray Raman scattering allows one to measure non-dipolar allowed transitions. A powerful technique to look at even $d$-$d$ transitions with well defined selection rules \cite{Haverkort:2007bv, vanVeenendaal:2008kv, Hiraoka:2011cq}, but can also be used to determine orbital occupations of rare-earth ions that are fundamentally not possible to determine using dipolar spectroscopy \cite{Willers:2012bz}.
 +###
 +
 +###
 +This tutorial compares calculated spectra to experiment. In order to make the plots you need to download the experimental data. You can download them in a zip file here {{ :documentation:tutorials:nio_crystal_field:nio_data.zip |}}. Please unpack this file and make sure to have the folders NiO_Experiment and NiO_Radial in the same folder as you do the calculations. And as always, if used in a publication, please cite the original papers that published the data. 
 +###
 +
 +###
 +The first example shows low energy $d$-$d$ transitions in NiO. The input is:
 +<code Quanty NIXS_dd.Quanty>
 +-- This example calculates the d-d excitations in NiO using non-resonant Inelastic X-ray
 +-- Scattering. This is one of the most beautiful spectroscopy techniques as the selection
 +-- rules are very "simple" and straight forward.
 +
 +-- We use the A^2 term of the interaction to make transitions between states with photons
 +-- of much higher energy. These photons now cary non negligible momentum and one can make
 +-- transitions beyond the dipole limit.
 +
 +-- Here we look at k=2 and k=4 transitions between the Ni 3d orbitals
 +
 +-- We use the definitions of all operators and basis orbitals as defined in the file
 +-- include and can afterwards directly continue by creating the Hamiltonian
 +-- and calculating the spectra
 +
 +dofile("Include.Quanty")
 +
 +-- The parameters and scheme needed only includes the ground-state (d^8) configuration
 +
 +-- 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)
 +-- 
 +-- 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
 +Delta   =  4.7
 +-- parameters obtained from DFT (PRB 85, 165113 (2012))
 +F2dd    = 11.14 
 +F4dd    =  6.87
 +F2pd    =  6.67
 +tenDq   =  0.56
 +tenDqL  =  1.44
 +Veg     =  2.06
 +Vt2g    =  1.21
 +zeta_3d =  0.081
 +Bz      =  0.000001
 +H112    =  0.120
 +
 +ed      = (10*Delta-nd*(19+nd)*Udd/2)/(10+nd)
 +eL      = nd*((1+nd)*Udd/2-Delta)/(10+nd)
 +
 +F0dd    = Udd + (F2dd+F4dd) * 2/63
 +
 +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
 +              
 +-- 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
 +
 +
 +-- in order to calculate nIXS we need to determine the intensity ratio for the different multipole intensities
 +-- ( see PRL 99, 257401 (2007) for the formalism )
 +-- in short the A^2 interaction is expanded on spherical harmonics and Bessel functions
 +-- The 3d Wannier functions are expanded on spherical harmonics and a radial wave function
 +-- For the radial wave-function we calculate <R(r) | j_k(q r) | R(r)>
 +-- which defines the transition strength for the multipole of order k
 +
 +-- The radial functions here are calculated for a Ni 2+ atom and stored in the folder NiO_Radial
 +-- more sophisticated methods can be used
 +
 +-- read the radial wave functions
 +-- order of functions
 +-- r 1S 2S 2P 3S 3P 3D
 +file = io.open( "NiO_Radial/RnlNi_Atomic_Hartree_Fock", "r")
 +Rnl = {}
 +for line in file:lines() do
 +  RnlLine={}
 +  for i in string.gmatch(line, "%S+") do
 +    table.insert(RnlLine,i)
 +  end
 +  table.insert(Rnl,RnlLine)
 +end
 +
 +-- some constants
 +a0      =  0.52917721092
 +Rydberg = 13.60569253
 +Hartree = 2*Rydberg
 +
 +-- dd transitions from 3d (index 7 in Rnl) to 3d (index 7 in Rnl)
 +-- <R(r) | j_k(q r) | R(r)>
 +function RjRdd (q)
 +  Rj0R = 0
 +  Rj2R = 0
 +  Rj4R = 0
 +  dr = Rnl[3][1]-Rnl[2][1]
 +  r0 = Rnl[2][1]-2*dr
 +  for ir = 2, #Rnl, 1 do
 +    r = r0 + ir * dr
 +    Rj0R = Rj0R + Rnl[ir][7] * SphericalBesselJ(0,q*r) * Rnl[ir][7] * dr
 +    Rj2R = Rj2R + Rnl[ir][7] * SphericalBesselJ(2,q*r) * Rnl[ir][7] * dr
 +    Rj4R = Rj4R + Rnl[ir][7] * SphericalBesselJ(4,q*r) * Rnl[ir][7] * dr
 +  end
 +  return Rj0R, Rj2R, Rj4R
 +end
 +
 +-- the angular part is given as C(theta_q, phi_q)^* C(theta_r, phi_r)
 +-- which is a potential expanded on spherical harmonics
 +function ExpandOnClm(k,theta,phi,scale)
 +  ret={}
 +  for m=-k, k, 1 do
 +    table.insert(ret,{k,m,scale * SphericalHarmonicC(k,m,theta,phi)})
 +  end
 +  return ret
 +end
 +
 +-- define nIXS transition operators
 +function TnIXS_dd(q, theta, phi)
 +  Rj0R, Rj2R, Rj4R = RjRdd(q)
 +  k=0
 +  A0 = ExpandOnClm(k, theta, phi, I^k*(2*k+1)*Rj0R)
 +  T0 = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, A0)
 +  k=2
 +  A2 = ExpandOnClm(k, theta, phi, I^k*(2*k+1)*Rj2R)
 +  T2 = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, A2)
 +  k=4
 +  A4 = ExpandOnClm(k, theta, phi, I^k*(2*k+1)*Rj4R)
 +  T4 = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, A4)
 +  T = T0+T2+T4
 +  T.Chop()
 +  return T
 +end
 +
 +-- q in units per a0 (if you want in units per A take 5*a0 to have a q of 5 per A)
 +q=4.5
 +
 +print("for q=",q," per a0 (",q / a0," per A) The ratio of k=0, k=2 and k=4 transition strength is:",RjRdd(q))
 +
 +-- define some transition operators
 +qtheta=0
 +qphi=0
 +Tq001 = TnIXS_dd(q,qtheta,qphi)
 +
 +qtheta=Pi/2
 +qphi=Pi/4
 +Tq110 = TnIXS_dd(q,qtheta,qphi)
 +
 +qtheta=acos(sqrt(1/3))
 +qphi=Pi/4
 +Tq111 = TnIXS_dd(q,qtheta,qphi)
 +
 +qtheta=acos(sqrt(9/14))
 +qphi=acos(sqrt(1/5))
 +Tq123 = TnIXS_dd(q,qtheta,qphi)
 +
 +-- calculate the spectra
 +nIXSSpectra = CreateSpectra(Hamiltonian, {Tq001, Tq110, Tq111, Tq123}, psiList, {{"Emin",-1}, {"Emax",6}, {"NE",3000}, {"Gamma",0.1}})
 +
 +-- print the spectra to a file
 +nIXSSpectra.Print({{"file","NiOnIXS_dd.dat"}})
 +
 +-- a gnuplot script to make the plots
 +gnuplotInput = [[
 +set autoscale  
 +set xtic auto   
 +set ytic auto    
 +set style line  1 lt 1 lw 1 lc rgb "#FF0000"
 +set style line  2 lt 1 lw 1 lc rgb "#0000FF"
 +set style line  3 lt 1 lw 1 lc rgb "#00C000"
 +set style line  4 lt 1 lw 1 lc rgb "#800080"
 +set style line  5 lt 1 lw 3 lc rgb "#000000"
 +
 +set xlabel "E (eV)" font "Times,12"
 +set ylabel "Intensity (arb. units)" font "Times,12"
 +
 +set out 'NiOnIXS_dd.ps'
 +set size 1.0, 0.3
 +set terminal postscript portrait enhanced color  "Times" 12
 +
 +set yrange [0:6.5]
 +
 +plot "NiO_Experiment/NIXS_dd_JSR_16_469_2009" using 1:($2*0.01) title 'experiment' with filledcurves y1=0 ls 5 fs transparent solid 0.5,\
 +     "NiOnIXS_dd.dat" using 1:(-$15 -$17 -$19 +3.25) title 'q // 111' with lines ls  3,\
 +     "NiOnIXS_dd.dat" using 1:(-$21 -$23 -$25 +2.50) title 'q // 123' with lines ls  4,\
 +     "NiOnIXS_dd.dat" using 1:(-$9  -$11 -$13 +1.75) title 'q // 011' with lines ls  2,\
 +     "NiOnIXS_dd.dat" using 1:(-$3   -$5  -$7 +1.00) title 'q // 001' with lines ls  1
 +]]
 +
 +-- write the gnuplot script to a file
 +file = io.open("NiOnIXS_dd.gnuplot", "w")
 +file:write(gnuplotInput)
 +file:close()
 +
 +-- call gnuplot to execute the script
 +os.execute("gnuplot NiOnIXS_dd.gnuplot")
 +-- transform to pdf and eps
 +os.execute("ps2pdf NiOnIXS_dd.ps  ; ps2eps NiOnIXS_dd.ps  ;  mv NiOnIXS_dd.eps temp.eps  ; eps2eps temp.eps NiOnIXS_dd.eps  ; rm temp.eps")
 +</code>
 +###
 +
 +The spectrum produced:
 +| {{ :documentation:tutorials:nio_ligand_field:nionixs_dd.png?nolink |}} |
 +^ nonresonant inelastic x-ray scattering spectra orientations of the momentum compared to the experimental spectra of a powder. ^
 +
 +###
 +We calculate the spectrum in 4 different directions of momentum transfer. The experimental spectra \cite{Verbeni:2009dx, Huotari:2008kx} are measured on a powder sample and thus do not show the strong momentum direction dependence. In previous \cite{Larson:2007ev} and subsequent \cite{Hiraoka:2009fa} this angular dependence has been observed.
 +###
 +
 +###
 +For completeness the output of the script is:
 +<file Quanty_Output NIXS_dd.out>
 +  #    <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   -3.503    1.999   12.000   15.095   -0.370   -0.115   -0.370   -0.115   -0.741   -0.230   -0.305   -1.042   -0.924    2.186    5.990    3.825    6.000    8.175 
 +  2   -3.395    1.999   12.000   15.160   -0.002   -0.000   -0.002   -0.000   -0.003   -0.001   -0.322   -1.043   -0.925    2.189    5.988    3.823    6.000    8.178 
 +  3   -3.286    1.999   12.000   15.211    0.369    0.113    0.369    0.113    0.737    0.227   -0.336   -1.043   -0.925    2.193    5.987    3.820    6.000    8.180 
 +for q= 4.5 per a0 ( 8.5037675605428 per A) The ratio of k=0, k=2 and k=4 transition strength is: 0.069703673179605 0.1609791731565 0.086144672158063
 +</file>
 +###
 +
 +
 +===== Table of contents =====
 +{{indexmenu>.#1|msort}}

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