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:

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")

The spectrum produced:

$2p$ to $3d$ excitations as one would measure using non-resonant inelastic x-ray scattering.

The output to standard out is:

nIXS_L23.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 
-1.655  1.989 11.988 16.766 -0.000  0.000 -0.861 -1.019 -0.877  3.008  4.992 
for q=	9	 per a0 (	17.007535121086	 per A) The ratio of k=1 and k=3 transition strength is:	0.081284239649905	0.04426369559805