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====== nIXS $M_{4,5}$ ($d$-$d$ excitations) ======

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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}.
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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. 
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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>
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The spectrum produced:

{{:documentation:tutorials:nio_crystal_field:nionixs_dd.png?nolink}}
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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.
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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>
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