Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Next revision		Previous revision
documentation:tutorials:nio_crystal_field:fy_l23m45 [2016/10/08 19:14] – created Maurits W. Haverkortdocumentation:tutorials:nio_crystal_field:fy_l23m45 [2018/04/26 15:21] (current) – Maurits W. Haverkort
Line 1: Line 1:
 +{{indexmenu_n>5}}
 +====== FY $L_{2,3}M_{4,5}$ ======
  
 +###
 +The absorption cross section is in principle measured using transmission. Transmission experiments in the soft-x-ray regime can be difficult as the absorption is quite high. Alternatively one can measure the reflectivity, which allows one to retrive the complete conductivity tensor using ellipsometry. As the x-ray wave-length is not large compared to the sample thickness this does not return the average sample absorption, but gives spatial information as well. Known as resonant scattering or reflectometry a beautiful technique, but might be overkill in some situations. A simple but effective way to measure the absorption cross section is to use the total electron yield, which can be measured by grounding the sample via an Ampare meter.
 +###
 +
 +###
 +An alternative is to measure the fluorescence yield. Although not proportional to the absorption cross section \cite{Kurian:2012de,vanVeenendaal:1996tb,deGroot:1994tz} an extremely useful technique that contains similar information as absorption. Actually it is often more sensitive to differences in the ground-state and shows more detail in the spectral features. The calculation of fluorescence yield is similar to the calculation of absorption.
 +###
 +
 +###
 +In the following example we calculate the excitation of a $2p$ electron into the $3d$ shell of Ni in NiO. ($L_{2,3}$ edge). We integrate over the decay of a $3d$ electron into the $2p$ orbital (removing an electron from the $3d$-shell i.e. $M_{4,5}$) We thus look at the $L_{2,3}$-$M_{4,5}$ FY spectra. (note that one should always list both the excitation as well as the decay channel as the spectra change between different channels). 
 +###
 +
 +###
 +The input is:
 +<code Quanty FY_L23M45.Quanty>
 +-- In this example we will calculate the fluorescence yield spectra 
 +-- One makes an excitation from 2p to 3d and then looks at a specific decay channel
 +-- Or at the sum over all channels
 +-- the spectra are integrated over the energy of the decay channel which allows for
 +-- extreme efficient calculation of these spectra. 
 +-- Note that most detectors will not be equally sensitive to all possible photon energies
 +-- and one thus would always measure some weighted sum over the different decay channels
 +
 +-- this file calculates the Ni L23M45 spectra. 
 +-- (L23, i.e. we excite from 2p to 3d)
 +-- (M45, we decay from the 3d shell, into the 2p shell)
 +
 +-- we minimize the output by setting the verbosity to 0
 +Verbosity(0)
 +
 +-- In order to do crystal-field theory for NiO we need to define a Ni d-shell.
 +-- A d-shell has 10 elements and we label again the even spin-orbitals to be spin down
 +-- and the odd spin-orbitals to be spin up. In order to calculate 2p to 3d excitations we
 +-- also need a Ni 2p shell. We thus have a total of 10+6=16 fermions, 6 Ni-2p and 10 Ni-3d
 +-- spin-orbitals
 +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}
 +
 +-- just like in the previous example we define several operators acting on the Ni -3d shell
 +
 +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)
 +
 +-- as in the previous example we define the Coulomb interaction
 +
 +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})
 +
 +-- as in the previous example we define the crystal-field operator
 +
 +Akm = PotentialExpandedOnClm("Oh",2,{0.6,-0.4})
 +OpptenDq = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, Akm)
 +
 +-- and as in the previous example we define operators that count 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)
 +
 +-- new for core level spectroscopy are operators that define the interaction acting on the
 +-- Ni-2p shell. There is actually only one of these interactions, which is the Ni-2p
 +-- spin-orbit interaction
 +
 +Oppcldots= NewOperator("ldots", NF, IndexUp_2p, IndexDn_2p)
 +
 +-- we also need to define the Coulomb interaction between the Ni 2p- and Ni 3d-shell
 +-- Again the interaction (e^2/(|r_i-r_j|)) is expanded on spherical harmonics. For the interaction
 +-- between two shells we need to consider two cases. For the direct interaction a 2p electron
 +-- scatters of a 3d electron into a 2p and 3d electron. The radial integrals involve
 +-- the square of a 2p radial wave function at coordinate 1 and the square of a 3d radial
 +-- wave function at coordinate 2. The transfer of angular momentum can either be 0 or 2.
 +-- These processes are called direct and the resulting Slater integrals are F[0] and F[2].
 +-- The second proces involves a 2p electron scattering of a 3d electron into the 3d shell
 +-- and at the same time the 3d electron scattering into a 2p shell. These exchange processes
 +-- involve radial integrals over the product of a 2p and 3d radial wave function. The transfer
 +-- of angular momentum in this case can be 1 or 3 and the Slater integrals are called G1 and G3.
 +
 +-- In Quanty you can enter these processes by labeling 4 indices for the orbitals, once
 +-- the 2p shell with spin up, 2p shell with spin down, 3d shell with spin up and 3d shell with
 +-- spin down. Followed by the direct Slater integrals (F0 and F2) and the exchange Slater 
 +-- integrals (G1 and G3)
 +
 +-- Here we define the operators separately and later sum them with appropriate prefactors
 +
 +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})
 +
 +-- next we define the dipole operator. The dipole operator is given as epsilon.r
 +-- with epsilon the polarization vector of the light and r the unit position vector
 +-- We can expand the position vector on (renormalized) spherical harmonics and use
 +-- the crystal-field operator to create the dipole operator. 
 +
 +-- x polarized light is defined as x = Cos[phi]Sin[theta] = sqrt(1/2) ( C_1^{(-1)} - C_1^{(1)})
 +Akm = {{1,-1,sqrt(1/2)},{1, 1,-sqrt(1/2)}}
 +TXASx = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm)
 +-- y polarized light is defined as y = Sin[phi]Sin[theta] = sqrt(1/2) I ( C_1^{(-1)} + C_1^{(1)})
 +Akm = {{1,-1,sqrt(1/2)*I},{1, 1,sqrt(1/2)*I}}
 +TXASy = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm)
 +-- z polarized light is defined as z = Cos[theta] = C_1^{(0)}
 +Akm = {{1,0,1}}
 +TXASz = NewOperator("CF", NF, IndexUp_3d, IndexDn_3d, IndexUp_2p, IndexDn_2p, Akm)
 +
 +-- besides linear polarized light one can define circular polarized light as the sum of 
 +-- x and y polarizations with complex prefactors
 +TXASr = sqrt(1/2)*(TXASx - I * TXASy)
 +TXASl =-sqrt(1/2)*(TXASx + I * TXASy)
 +
 +-- we can remove zero's from the dipole operator by chopping it.
 +TXASr.Chop()
 +TXASl.Chop()
 +
 +-- the 3d to 2p dipole transition is the conjugate transpose of the 2p to 3d dipole transition
 +TXASxdag = ConjugateTranspose(TXASx)
 +TXASydag = ConjugateTranspose(TXASy)
 +TXASzdag = ConjugateTranspose(TXASz)
 +TXASldag = ConjugateTranspose(TXASl)
 +TXASrdag = ConjugateTranspose(TXASr)
 +
 +-- once all operators are defined we can set some parameter values.
 +
 +-- the value of U drops out of a crystal-field calculation as the total number of electrons
 +-- is always the same
 +U       =  0.000 
 +-- F2 and F4 are often referred to in the literature as J_{Hund}. They represent the energy
 +-- differences between different multiplets. Numerical values can be found in the back of
 +-- my PhD. thesis for example. http://arxiv.org/abs/cond-mat/0505214 
 +F2dd    = 11.142 
 +F4dd    =  6.874
 +-- F0 is not the same as U, although they are related. Unimportant in crystal-field theory
 +-- the difference between U and F0 is so important that I do include it here. (U=0 so F0 is not)
 +F0dd    = U+(F2dd+F4dd)*2/63
 +-- in crystal field theory U drops out of the equation, also true for the interaction between the 
 +-- Ni 2p and Ni 3d electrons
 +Upd     =  0.000 
 +-- The Slater integrals between the 2p and 3d shell, again the numerical values can be found
 +-- in the back of my PhD. thesis. (http://arxiv.org/abs/cond-mat/0505214)
 +F2pd    =  6.667
 +G1pd    =  4.922
 +G3pd    =  2.796
 +-- F0 is not the same as U, although they are related. Unimportant in crystal-field theory
 +-- the difference between U and F0 is so important that I do include it here. (U=0 so F0 is not)
 +F0pd    =  Upd + G1pd*1/15 + G3pd*3/70
 +-- tenDq in NiO is 1.1 eV as can be seen in optics or using IXS to measure d-d excitations
 +tenDq   =  1.100
 +-- the Ni 3d spin-orbit is small but finite
 +zeta_3d =  0.081
 +-- the Ni 2p spin-orbit is very large and should not be scaled as theory is quite accurate here
 +zeta_2p = 11.498
 +-- we can add a small magnetic field, just to get nice expectation values. (units in eV... )
 +Bz      = 0.000001
 +
 +-- the total Hamiltonian is the sum of the different operators multiplied with their prefactor
 +Hamiltonian = F0dd*OppF0 + F2dd*OppF2 + F4dd*OppF4 + tenDq*OpptenDq + zeta_3d*Oppldots + Bz*(2*OppSz+OppLz)
 +
 +-- We normally do not include core-valence interactions between filed and partially filled 
 +-- shells as it drops out anyhow. For the XAS we thus need to define a "different" 
 +-- (more complete) Hamiltonian.
 +XASHamiltonian = Hamiltonian + zeta_2p * Oppcldots + F0pd * OppUpdF0 + F2pd * OppUpdF2 + G1pd * OppUpdG1 + G3pd * OppUpdG3
 +
 +-- 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
 +-- in order to make sure we have a filling of 8
 +-- electrons we need to define some restrictions
 +-- We need to restrict the occupation of the Ni-2p shell to 6 for the ground state and for
 +-- the Ni 3d-shell to 8.
 +StartRestrictions = {NF, NB, {"111111 0000000000",6,6}, {"000000 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 ",expectationvalue))
 +  end
 +  io.write("\n")
 +end
 +
 +-- here we calculate the x-ray absorption spectra, not the main task of this file, but nice to do so we can compare
 +XASSpectra = CreateSpectra(XASHamiltonian, {TXASz, TXASr, TXASl}, psiList, {{"Emin",-10}, {"Emax",20}, {"NE",3500}, {"Gamma",1.0}});
 +XASSpectra.Print({{"file","FYL23M45_XAS.dat"}});
 +
 +-- and we calculate the FY spectra
 +FYSpectra = CreateFluorescenceYield(XASHamiltonian, {TXASz, TXASr, TXASl}, {TXASxdag, TXASydag, TXASzdag}, psiList, {{"Emin",-10}, {"Emax",20}, {"NE",3500}, {"Gamma",1.0}});
 +FYSpectra.Print({{"file","FYL23M45_Spec.dat"}});
 +
 +-- in order to plot both the XAS and FY spectra we can define a gnuplot script
 +gnuplotInput = [[
 +set autoscale 
 +set xtic auto  
 +set ytic auto 
 +set style line  1 lt 1 lw 1 lc rgb "#000000"
 +set style line  2 lt 1 lw 1 lc rgb "#FF0000"
 +set style line  3 lt 1 lw 1 lc rgb "#00FF00"
 +set style line  4 lt 1 lw 1 lc rgb "#0000FF"
 +
 +set xlabel "E (eV)" font "Times,10"
 +set ylabel "Intensity (arb. units)" font "Times,10"
 +
 +set out 'FYL23M45.ps'
 +set terminal postscript portrait enhanced color  "Times" 8 size 7.5,6
 +set yrange [0:0.6]
 +
 +set multiplot layout 3, 3
 +
 +plot "FYL23M45_XAS.dat"  u 1:(-$3 )  title 'z-polarized Sz=-1' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
 +     "FYL23M45_Spec.dat" u 1:(4*$2)  title 'FY - x out' with lines ls 2,\
 +     "FYL23M45_Spec.dat" u 1:(4*$4)  title 'FY - y out' with lines ls 3,\
 +     "FYL23M45_Spec.dat" u 1:(4*$6)  title 'FY - z out' with lines ls 4
 +plot "FYL23M45_XAS.dat"  u 1:(-$5 )  title 'z-polarized Sz= 0' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
 +     "FYL23M45_Spec.dat" u 1:(4*$8)  title 'FY - x out' with lines ls 2,\
 +     "FYL23M45_Spec.dat" u 1:(4*$10) title 'FY - y out' with lines ls 3,\
 +     "FYL23M45_Spec.dat" u 1:(4*$12) title 'FY - z out' with lines ls 4
 +plot "FYL23M45_XAS.dat"  u 1:(-$7 )  title 'z-polarized Sz= 1' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
 +     "FYL23M45_Spec.dat" u 1:(4*$14) title 'FY - x out' with lines ls 2,\
 +     "FYL23M45_Spec.dat" u 1:(4*$16) title 'FY - y out' with lines ls 3,\
 +     "FYL23M45_Spec.dat" u 1:(4*$18) title 'FY - z out' with lines ls 4
 +
 +plot "FYL23M45_XAS.dat"  u 1:(-$9 )  title 'r-polarized Sz=-1' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
 +     "FYL23M45_Spec.dat" u 1:(4*$20) title 'FY - x out' with lines ls 2,\
 +     "FYL23M45_Spec.dat" u 1:(4*$22) title 'FY - y out' with lines ls 3,\
 +     "FYL23M45_Spec.dat" u 1:(4*$24) title 'FY - z out' with lines ls 4
 +plot "FYL23M45_XAS.dat"  u 1:(-$11)  title 'r-polarized Sz= 0' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
 +     "FYL23M45_Spec.dat" u 1:(4*$26) title 'FY - x out' with lines ls 2,\
 +     "FYL23M45_Spec.dat" u 1:(4*$28) title 'FY - y out' with lines ls 3,\
 +     "FYL23M45_Spec.dat" u 1:(4*$30) title 'FY - z out' with lines ls 4
 +plot "FYL23M45_XAS.dat"  u 1:(-$13)  title 'r-polarized Sz= 1' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
 +     "FYL23M45_Spec.dat" u 1:(4*$32) title 'FY - x out' with lines ls 2,\
 +     "FYL23M45_Spec.dat" u 1:(4*$34) title 'FY - y out' with lines ls 3,\
 +     "FYL23M45_Spec.dat" u 1:(4*$36) title 'FY - z out' with lines ls 4
 +
 +plot "FYL23M45_XAS.dat"  u 1:(-$15)  title 'l-polarized Sz=-1' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
 +     "FYL23M45_Spec.dat" u 1:(4*$38) title 'FY - x out' with lines ls 2,\
 +     "FYL23M45_Spec.dat" u 1:(4*$40) title 'FY - y out' with lines ls 3,\
 +     "FYL23M45_Spec.dat" u 1:(4*$42) title 'FY - z out' with lines ls 4
 +plot "FYL23M45_XAS.dat"  u 1:(-$17)  title 'l-polarized Sz= 0' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
 +     "FYL23M45_Spec.dat" u 1:(4*$44) title 'FY - x out' with lines ls 2,\
 +     "FYL23M45_Spec.dat" u 1:(4*$46) title 'FY - y out' with lines ls 3,\
 +     "FYL23M45_Spec.dat" u 1:(4*$48) title 'FY - z out' with lines ls 4
 +plot "FYL23M45_XAS.dat"  u 1:(-$19)  title 'l-polarized Sz= 1' with filledcurves y1=0 ls  1 fs transparent solid 0.5,\
 +     "FYL23M45_Spec.dat" u 1:(4*$50) title 'FY - x out' with lines ls 2,\
 +     "FYL23M45_Spec.dat" u 1:(4*$52) title 'FY - y out' with lines ls 3,\
 +     "FYL23M45_Spec.dat" u 1:(4*$54) title 'FY - z out' with lines ls 4
 +
 +unset multiplot
 +]]
 +
 +
 +-- write the gnuplot script to a file
 +file = io.open("FYL23M45.gnuplot", "w")
 +file:write(gnuplotInput)
 +file:close()
 +
 +-- call gnuplot to execute the script
 +os.execute("gnuplot FYL23M45.gnuplot")
 +-- and transform the ps to pdf
 +os.execute("ps2pdf FYL23M45.ps ; ps2eps FYL23M45.ps ;  mv FYL23M45.eps temp.eps ; eps2eps temp.eps FYL23M45.eps ; rm temp.eps")
 +</code>
 +###
 +
 +###
 +The script returns 9 plots with each 4 curves. The local ground-state of Ni in NiO is 3-fold degenerate in the paramagnetic phase ($S=1$) The different columns show the spectra for the states with different $S_z$. In the paramagnetic phase one should summ these 3 spectra, in a full magnetized sample one measurers either the left or the right column. The different rows the different incoming polarization. Top row z-polarized, middle right bottom left polarized light. The black filed curve shows the absorption cross section. The red, green and blue curve show the spectra for different outgoing polarization. 
 +
 +{{:documentation:tutorials:nio_crystal_field:fyl23m45.png?nolink}}
 +###
 +
 +###
 +The script shows some information on the ground-state, here the text output.
 +<file Quanty_Output FY_L23M45.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 
 +-2.721  1.999 12.000 15.120  0.994  0.286 -0.324 -1.020 -0.878  2.011  5.989 
 +Start of LanczosTriDiagonalizeKrylovRR
 +Start of LanczosTriDiagonalizeKrylovRR
 +Start of LanczosTriDiagonalizeKrylovRR
 +Start of LanczosTriDiagonalizeKrylovRR
 +Start of LanczosTriDiagonalizeKrylovRR
 +Start of LanczosTriDiagonalizeKrylovRR
 +Start of LanczosTriDiagonalizeKrylovRR
 +Start of LanczosTriDiagonalizeKrylovRR
 +Start of LanczosTriDiagonalizeKrylovRR
 +</file>
 +###
 +
 +===== Table of contents =====
 +{{indexmenu>.#1|msort}}

You are here: Quanty - a quantum many body script language » Documentation » Tutorials » NiO crystal field » FY $L_{2,3}M_{4,5}$

Print/export