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documentation:tutorials:nio_crystal_field:nixs_l23 [2016/10/08 21:25] – created Maurits W. Haverkortdocumentation:tutorials:nio_crystal_field:nixs_l23 [2019/02/21 08:25] (current) – Maurits W. Haverkort
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 +{{indexmenu_n>10}}
 +====== nIXS $L_{2,3}$ ======
  
 +###
 +Besides low energy transitions nIXS can be used as a core level spectroscopy technique. One then measures resonances with non-resonant inelastic x-ray scattering :-).
 +###
 +
 +###
 +The input script:
 +<code Quanty nIXS_L23.Quanty>
 +-- using inelastic x-ray scattering one can not only measure low energy excitations,
 +-- but equally well core to core transitions. This allows one to probe for example
 +-- 3p to 3d transitions using octupole operators. 
 +
 +-- We set the output of the program to a minimum
 +Verbosity(0)
 +
 +-- we need a 2p and 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}
 +
 +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})
 +
 +Akm = PotentialExpandedOnClm("Oh", 2, {0.6,-0.4})
 +OpptenDq = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm)
 +
 +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)
 +
 +Oppcldots= NewOperator("ldots", NF, IndexUp_2p, IndexDn_2p)
 +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})
 +
 +-- in crystal field theory U drops out of the equation
 +U       =  0.000 
 +F2dd    = 11.142 
 +F4dd    =  6.874
 +F0dd    = U+(F2dd+F4dd)*2/63
 +-- in crystal field theory U drops out of the equation
 +Upd     =  0.000 
 +F2pd    =  6.667
 +G1pd    =  4.922
 +G3pd    =  2.796
 +F0pd    =  Upd + G1pd*1/15 + G3pd*3/70
 +tenDq   =  1.100
 +zeta_3d =  0.081
 +zeta_2p = 11.498
 +Bz      = 0.000001
 +
 +Hamiltonian =  F0dd*OppF0 + F2dd*OppF2 + F4dd*OppF4 + tenDq*OpptenDq + zeta_3d*Oppldots + Bz*(2*OppSz + OppLz)
 +
 +XASHamiltonian = Hamiltonian + zeta_2p * Oppcldots + F2pd * OppUpdF2 + G1pd * OppUpdG1 + G3pd * OppUpdG3
 +
 +-- we now can create the lowest Npsi eigenstates:
 +Npsi=3
 +-- in order to make sure we have a filling of 2 electrons we need to define some restrictions
 +StartRestrictions = {NF, NB, {"111111 0000000000",6,6}, {"000000 1111111111",8,8}}
 +
 +psiList = Eigensystem(Hamiltonian, StartRestrictions, Npsi)
 +
 +oppList={Hamiltonian, OppSsqr, OppLsqr, OppJsqr, OppSz, OppLz, Oppldots, OppF2, OppF4, OppNeg, OppNt2g}
 +
 +print(" <E>    <S^2>  <L^2>  <J^2>  <S_z>  <L_z>  <l.s>  <F[2]> <F[4]> <Neg>  <Nt2g>");
 +for key,psi in pairs(psiList) do
 +  expvalue = psi * oppList * psi
 +  for k,v in pairs(expvalue) do
 +    io.write(string.format("%6.3f ",v))
 +  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
 +
 +-- pd transitions from 2p (index 4 in Rnl) to 3d (index 7 in Rnl)
 +-- <R(r) | j_k(q r) | R(r)>
 +function RjRpd (q)
 +  Rj1R = 0
 +  Rj3R = 0
 +  dr = Rnl[3][1]-Rnl[2][1]
 +  r0 = Rnl[2][1]-2*dr
 +  for ir = 2, #Rnl, 1 do
 +    r = r0 + ir * dr
 +    Rj1R = Rj1R + Rnl[ir][4] * math.SphericalBesselJ(1,q*r) * Rnl[ir][7] * dr
 +    Rj3R = Rj3R + Rnl[ir][4] * math.SphericalBesselJ(3,q*r) * Rnl[ir][7] * dr
 +  end
 +  return Rj1R, Rj3R
 +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 * math.SphericalHarmonicC(k,m,theta,phi)})
 +  end
 +  return ret
 +end
 +
 +-- define nIXS transition operators
 +function TnIXS_pd(q, theta, phi)
 +  Rj1R, Rj3R = RjRpd(q)
 +  k=1
 +  A1 = ExpandOnClm(k, theta, phi, I^k*(2*k+1)*Rj1R)
 +  T1 = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, A1)
 +  k=3
 +  A3 = ExpandOnClm(k, theta, phi, I^k*(2*k+1)*Rj3R)
 +  T3 = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, A3)
 +  T = T1+T3
 +  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=9.0
 +
 +print("for q=",q," per a0 (",q / a0," per A) The ratio of k=1 and k=3 transition strength is:", RjRpd(q))
 +
 +-- define some transition operators
 +qtheta=0
 +qphi=0
 +Tq001 = TnIXS_pd(q,qtheta,qphi)
 +
 +qtheta=Pi/2
 +qphi=Pi/4
 +Tq110 = TnIXS_pd(q,qtheta,qphi)
 +
 +qtheta=math.acos(math.sqrt(1/3))
 +qphi=Pi/4
 +Tq111 = TnIXS_pd(q,qtheta,qphi)
 +
 +qtheta=math.acos(math.sqrt(9/14))
 +qphi=math.acos(math.sqrt(1/5))
 +Tq123 = TnIXS_pd(q,qtheta,qphi)
 +
 +-- calculate the spectra
 +nIXSSpectra = CreateSpectra(XASHamiltonian, {Tq001, Tq110, Tq111, Tq123}, psiList, {{"Emin",-10}, {"Emax",20}, {"NE",6000}, {"Gamma",1.0}})
 +
 +-- print the spectra to a file
 +nIXSSpectra.Print({{"file","NiOnIXS_L23.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 "#000000"
 +set style line  5 lt 1 lw 3 lc rgb "#808080"
 +
 +set xlabel "E (eV)" font "Times,12"
 +set ylabel "Intensity (arb. units)" font "Times,12"
 +
 +set out 'NiOnIXS_L23.ps'
 +set size 1.0, 0.3
 +set terminal postscript portrait enhanced color  "Times" 8
 +
 +energyshift=857.6
 +
 +plot "NiOnIXS_L23.dat" using ($1+energyshift):(-$9  -$11 -$13 +0.16) title '011' with lines ls  2,\
 +     "NiOnIXS_L23.dat" using ($1+energyshift):(-$15 -$17 -$19 +0.11) title '111' with lines ls  3,\
 +     "NiOnIXS_L23.dat" using ($1+energyshift):(-$21 -$23 -$25 +0.06) title '123' with lines ls  4,\
 +     "NiOnIXS_L23.dat" using ($1+energyshift):(-$3   -$5  -$7 +0.01) title '001' with lines ls  1
 +
 +]]
 +
 +-- write the gnuplot script to a file
 +file = io.open("NiOnIXS_L23.gnuplot", "w")
 +file:write(gnuplotInput)
 +file:close()
 +
 +-- call gnuplot to execute the script
 +os.execute("gnuplot NiOnIXS_L23.gnuplot")
 +-- transform to pdf and eps
 +os.execute("ps2pdf NiOnIXS_L23.ps  ; ps2eps NiOnIXS_L23.ps  ;  mv NiOnIXS_L23.eps temp.eps  ; eps2eps temp.eps NiOnIXS_L23.eps  ; rm temp.eps")
 +</code>
 +###
 +
 +###
 +The spectrum produced:
 +###
 +|{{:documentation:tutorials:nio_crystal_field:nionixs_l23.png?nolink |}}|
 +^ $2p$ to $3d$ excitations as one would measure using non-resonant inelastic x-ray scattering.  ^
 +
 +
 +###
 +The output to standard out is:
 +<file Quanty_Output nIXS_L23.out>
 +-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= 9 per a0 ( 17.007535121086 per A) The ratio of k=1 and k=3 transition strength is: 0.081284239649905 0.04426369559805
 +</file>
 +###
 +
 +
 +===== Table of contents =====
 +{{indexmenu>.#1|msort}}

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