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documentation:tutorials:nio_crystal_field:nixs_m45 [2016/10/08 21:23] – created Maurits W. Haverkortdocumentation:tutorials:nio_crystal_field:nixs_m45 [2019/02/21 08:22] (current) – Maurits W. Haverkort
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 +{{indexmenu_n>9}}
 +====== nIXS $M_{4,5}$ ($d$-$d$ excitations) ======
  
 +###
 +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. 
 +###
 +
 +###
 +This example shows low energy $d$-$d$ transitions in NiO. The input is:
 +<code Quanty nIXS_M45.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 set the output to a minimum
 +Verbosity(0)
 +
 +-- define the basis of one particle spin-orbitals
 +-- we only need the d orbitals in this case
 +NF=10
 +NB=0
 +IndexDn_3d={0,2,4,6,8}
 +IndexUp_3d={1,3,5,7,9}
 +
 +-- define operators on this basis
 +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)
 +
 +-- define the coulomb operator
 +-- we here define the part depending on F0 seperately from the part depending on F2
 +-- when summing we can put in the numerical values of the slater integrals
 +
 +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})
 +
 +-- define the crystal-field operator
 +
 +Akm = PotentialExpandedOnClm("Oh", 2, {0.6,-0.4})
 +OpptenDq = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm)
 +
 +-- define number operators counting 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)
 +
 +-- set some parameters (see PRB 85, 165113 (2012) for more information)
 +beta    =  0.8
 +U       =  0.000
 +F2dd    = 11.142 * beta
 +F4dd    =  6.874 * beta
 +F0dd    = U+(F2dd+F4dd)*2/63
 +tenDq   = 1.100
 +zeta_3d = 0.081
 +Bz      = 0.000001
 +
 +-- create a parameter dependent Hamiltonian
 +Hamiltonian =  F0dd*OppF0 + F2dd*OppF2 + F4dd*OppF4 + tenDq*OpptenDq + zeta_3d*Oppldots + Bz*(2*OppSz + OppLz)
 +
 +
 +-- 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 8 electrons in the 3d shell
 +StartRestrictions = {NF, NB, {"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
 +
 +-- 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_crystal_field:nionixs_dd.png?nolink}}
 +###
 +
 +###
 +We calculate the spectrum in 4 different directions of momentum transfer. The experimental spectra (by Verbeni (2009) and Huotari (2008) //et al.//) are measured on a powder sample and thus do not show the strong momentum direction dependence. In previous measurements (Larson //et al.// (2007) and later in great detail by Hiroaka //et al.// (2009) this angular dependence has been observed, which agrees well with the theoretical predictions.
 +###
 +
 +###
 +For completeness the output of the script is:
 +<file Quanty_Output nIXS_M45.out>
 + <E>    <S^2>  <L^2>  <J^2>  <S_z>  <L_z>  <l.s>  <F[2]> <F[4]> <Neg>  <Nt2g>
 +-2.444  1.999 12.000 15.118 -0.994 -0.285 -0.331 -1.020 -0.878  2.011  5.989 
 +-2.444  1.999 12.000 15.118 -0.000 -0.000 -0.331 -1.020 -0.878  2.011  5.989 
 +-2.444  1.999 12.000 15.118  0.994  0.285 -0.331 -1.020 -0.878  2.011  5.989 
 +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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