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documentation:tutorials:nio_crystal_field:xas_l23_as_conductivity_tensor [2016/10/08 21:27] – created Maurits W. Haverkort | documentation:tutorials:nio_crystal_field:xas_l23_as_conductivity_tensor [2018/03/21 10:19] (current) – Stefano Agrestini | ||
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+ | {{indexmenu_n> | ||
+ | ====== XAS $L_{2,3}$ as conductivity tensor ====== | ||
+ | ### | ||
+ | Absorption spectra are polarization dependent. In principle one can choose an infinite different number of polarizations. Calculating for each different experimental geometry (or polarization) a new spectrum is cumbersome and not needed. The material properties are given by the conductivity tensor. For dipole transitions a 3 by 3 matrix. The absorption spectra for a given experiment are then found by the relation: | ||
+ | \begin{equation} | ||
+ | I(\omega, | ||
+ | \end{equation} | ||
+ | with $\epsilon$ the polarization vector, $\omega$ the photon energy, $\sigma(\omega)$ the energy dependent conductivity tensor, and $I$ the measured intensity. Quanty can calculate the conductivity tensor. This is an extra option given to the function CreateSpectra (\{" | ||
+ | ### | ||
+ | |||
+ | ### | ||
+ | The example below calculates the conductivity tensor at the Ni $L_{2,3}$ edge. We show two different methods. The first calculates 9 spectra and by linear combining them retrieves the tensor. Method two uses a Block algorithm. | ||
+ | <code Quanty XAS_tensor.Quanty> | ||
+ | -- This tutorial calculates the 2p to 3d x-ray absorption spectra of Ni in NiO using | ||
+ | -- crystal field theory | ||
+ | |||
+ | -- the spectra are represented as a 3 by 3 tensor, the conductivity tensor. We show two | ||
+ | -- different methods to calculate this tensor, once creating 9 spectra with different | ||
+ | -- polarizations, | ||
+ | |||
+ | -- Within crystal-field theory the solid is approximated by a single atom in an effective | ||
+ | -- potential. Although an extremely crude approximation it is useful for some cases. | ||
+ | -- For correlated transition metal insulators it captures the right symmetry of the | ||
+ | -- localized open d-shell. It is useful to determine magnetic g-factors, energies of d-d | ||
+ | -- excitations or core level x-ray absorption. (2p to 3d excitations L23 edges) | ||
+ | |||
+ | -- One should notice that the effective crystal-field potential is an affective potential | ||
+ | -- it is there to mimic the interaction with neighboring ligand atoms. In real materials | ||
+ | -- there do not exist such large electro static potentials. | ||
+ | |||
+ | -- 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, | ||
+ | IndexUp_2p={1, | ||
+ | IndexDn_3d={6, | ||
+ | IndexUp_3d={7, | ||
+ | |||
+ | -- We define several operators acting on the Ni -3d shell | ||
+ | |||
+ | OppSx | ||
+ | OppSy | ||
+ | OppSz | ||
+ | OppSsqr =NewOperator(" | ||
+ | OppSplus=NewOperator(" | ||
+ | OppSmin =NewOperator(" | ||
+ | |||
+ | OppLx | ||
+ | OppLy | ||
+ | OppLz | ||
+ | OppLsqr =NewOperator(" | ||
+ | OppLplus=NewOperator(" | ||
+ | OppLmin =NewOperator(" | ||
+ | |||
+ | OppJx | ||
+ | OppJy | ||
+ | OppJz | ||
+ | OppJsqr =NewOperator(" | ||
+ | OppJplus=NewOperator(" | ||
+ | OppJmin =NewOperator(" | ||
+ | |||
+ | Oppldots=NewOperator(" | ||
+ | |||
+ | -- The Coulomb interaction | ||
+ | |||
+ | OppF0 =NewOperator(" | ||
+ | OppF2 =NewOperator(" | ||
+ | OppF4 =NewOperator(" | ||
+ | |||
+ | -- The crystal-field operator | ||
+ | |||
+ | Akm = PotentialExpandedOnClm(" | ||
+ | OpptenDq = NewOperator(" | ||
+ | |||
+ | -- Count the number of eg and t2g electrons | ||
+ | |||
+ | Akm = PotentialExpandedOnClm(" | ||
+ | OppNeg = NewOperator(" | ||
+ | Akm = PotentialExpandedOnClm(" | ||
+ | OppNt2g = NewOperator(" | ||
+ | |||
+ | -- 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, | ||
+ | -- spin-orbit interaction | ||
+ | |||
+ | Oppcldots= NewOperator(" | ||
+ | |||
+ | -- we also need to define the Coulomb interaction between the Ni 2p- and Ni 3d-shell | ||
+ | -- Again the interaction (e^2/ | ||
+ | -- 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(" | ||
+ | OppUpdF2 = NewOperator(" | ||
+ | OppUpdG1 = NewOperator(" | ||
+ | OppUpdG3 = NewOperator(" | ||
+ | |||
+ | -- 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, | ||
+ | TXASx = NewOperator(" | ||
+ | -- y polarized light is defined as y = Sin[phi]Sin[theta] = sqrt(1/2) I ( C_1^{(-1)} + C_1^{(1)}) | ||
+ | Akm = {{1, | ||
+ | TXASy = Chop(NewOperator(" | ||
+ | -- z polarized light is defined as z = Cos[theta] = C_1^{(0)} | ||
+ | Akm = {{1,0,1}} | ||
+ | TXASz = NewOperator(" | ||
+ | |||
+ | |||
+ | -- 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 | ||
+ | -- 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:// | ||
+ | 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/ | ||
+ | -- in crystal field theory U drops out of the equation, also true for the interaction between the | ||
+ | -- Ni 2p and Ni 3d electrons | ||
+ | Upd | ||
+ | -- The Slater integrals between the 2p and 3d shell, again the numerical values can be found | ||
+ | -- in the back of my PhD. thesis. (http:// | ||
+ | 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 | ||
+ | -- 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 | ||
+ | -- In mean field theory the neighboring Ni sites give an effective potential acting on the | ||
+ | -- spin only when magnetically ordered. This exchange field in NiO is 6 J with J=27 meV. | ||
+ | Hex = 6*0.027 -- see Europhys. Lett., 32 259 (1995) [ and Phys. Rev. B 82, 094403 (2010) ] | ||
+ | |||
+ | -- the total Hamiltonian is the sum of the different operators multiplied with their prefactor | ||
+ | Hamiltonian = Chop(F0dd*OppF0 + F2dd*OppF2 + F4dd*OppF4 + tenDq*OpptenDq + zeta_3d*Oppldots + Bz*(2*OppSz+OppLz) + Hex*sqrt(1/ | ||
+ | |||
+ | -- 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 " | ||
+ | -- (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, {" | ||
+ | |||
+ | -- And calculate the lowest 3 eigenfunctions | ||
+ | psiList = Eigensystem(Hamiltonian, | ||
+ | |||
+ | -- 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, | ||
+ | |||
+ | -- next we loop over all operators and all states and print the expectation value | ||
+ | print(" | ||
+ | for i = 1,#psiList do | ||
+ | for j = 1,#oppList do | ||
+ | expectationvalue = Chop(psiList[i]*oppList[j]*psiList[i]) | ||
+ | io.write(string.format(" | ||
+ | end | ||
+ | io.write(" | ||
+ | end | ||
+ | |||
+ | |||
+ | |||
+ | -- calculating the spectra is simple and single line once all operators and wave-functions | ||
+ | -- are defined. | ||
+ | |||
+ | --------------------------- Method 1 ----------------------------- | ||
+ | -- in order to create the tensor we define 9 spectra using operators that are combinations | ||
+ | -- of x, y and z polarized light | ||
+ | |||
+ | TXASypz | ||
+ | TXASzpx | ||
+ | TXASxpy | ||
+ | TXASypiz = sqrt(1/ | ||
+ | TXASzpix = sqrt(1/ | ||
+ | TXASxpiy = sqrt(1/ | ||
+ | |||
+ | TimeStart(" | ||
+ | XASSpectra = CreateSpectra(XASHamiltonian, | ||
+ | TimeEnd(" | ||
+ | |||
+ | -- Broaden these 9 spectra | ||
+ | TimeStart(" | ||
+ | XASSpectra.Broaden(0.4, | ||
+ | TimeEnd(" | ||
+ | |||
+ | -- linear combine them into a tensor (note that the order here is given by the list of operators in the CreateSpectra function | ||
+ | |||
+ | XASSigma_method1 = Spectra.Sum(XASSpectra, | ||
+ | , | ||
+ | , | ||
+ | |||
+ | XASSigma_method1.Print({{" | ||
+ | |||
+ | -- prepare the gnuplot output for Sigma | ||
+ | gnuplotInput = [[ | ||
+ | set autoscale | ||
+ | set xtic auto # set xtics automatically | ||
+ | set ytic auto # set ytics automatically | ||
+ | set style line 1 lt 1 lw 2 lc 1 | ||
+ | set style line 2 lt 1 lw 2 lc 3 | ||
+ | |||
+ | set xlabel "E (eV)" font " | ||
+ | set ylabel " | ||
+ | |||
+ | set yrange [-0.3:0.3] | ||
+ | |||
+ | set out ' | ||
+ | set size 1.0, 1.0 | ||
+ | set terminal postscript portrait enhanced color " | ||
+ | |||
+ | set multiplot layout 6, 3 | ||
+ | |||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | |||
+ | unset multiplot | ||
+ | ]] | ||
+ | |||
+ | print(" | ||
+ | |||
+ | -- write the gnuplot script to a file | ||
+ | file = io.open(" | ||
+ | file: | ||
+ | file: | ||
+ | |||
+ | print("" | ||
+ | print(" | ||
+ | |||
+ | -- call gnuplot to execute the script | ||
+ | os.execute(" | ||
+ | |||
+ | --------------------------- Method 2 ----------------------------- | ||
+ | |||
+ | |||
+ | TimeStart(" | ||
+ | XASSigma_method2, | ||
+ | TimeEnd(" | ||
+ | |||
+ | -- Broaden these 9 spectra | ||
+ | TimeStart(" | ||
+ | XASSigma_method2.Broaden(0.4, | ||
+ | TimeEnd(" | ||
+ | |||
+ | XASSigma_method2.Print({{" | ||
+ | |||
+ | -- prepare the gnuplot output for Sigma | ||
+ | gnuplotInput = [[ | ||
+ | set autoscale | ||
+ | set xtic auto # set xtics automatically | ||
+ | set ytic auto # set ytics automatically | ||
+ | set style line 1 lt 1 lw 2 lc 1 | ||
+ | set style line 2 lt 1 lw 2 lc 3 | ||
+ | |||
+ | set xlabel "E (eV)" font " | ||
+ | set ylabel " | ||
+ | |||
+ | set yrange [-0.3:0.3] | ||
+ | |||
+ | set out ' | ||
+ | set size 1.0, 1.0 | ||
+ | set terminal postscript portrait enhanced color " | ||
+ | |||
+ | set multiplot layout 6, 3 | ||
+ | |||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | |||
+ | unset multiplot | ||
+ | ]] | ||
+ | |||
+ | print(" | ||
+ | |||
+ | -- write the gnuplot script to a file | ||
+ | file = io.open(" | ||
+ | file: | ||
+ | file: | ||
+ | |||
+ | print("" | ||
+ | print(" | ||
+ | |||
+ | -- call gnuplot to execute the script | ||
+ | os.execute(" | ||
+ | |||
+ | -------------------------- difference ------------------------ | ||
+ | |||
+ | XASSigma_diff = XASSigma_method2 - XASSigma_method1 | ||
+ | |||
+ | |||
+ | XASSigma_diff.Print({{" | ||
+ | |||
+ | -- prepare the gnuplot output for Sigma | ||
+ | gnuplotInput = [[ | ||
+ | set autoscale | ||
+ | set xtic auto # set xtics automatically | ||
+ | set ytic auto # set ytics automatically | ||
+ | set style line 1 lt 1 lw 2 lc 1 | ||
+ | set style line 2 lt 1 lw 2 lc 3 | ||
+ | |||
+ | set xlabel "E (eV)" font " | ||
+ | set ylabel " | ||
+ | |||
+ | set yrange [-0.3:0.3] | ||
+ | |||
+ | scale = 1000000000 | ||
+ | |||
+ | set out ' | ||
+ | set size 1.0, 1.0 | ||
+ | set terminal postscript portrait enhanced color " | ||
+ | |||
+ | set multiplot layout 6, 3 | ||
+ | |||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | plot " | ||
+ | " | ||
+ | |||
+ | unset multiplot | ||
+ | ]] | ||
+ | |||
+ | print(" | ||
+ | |||
+ | -- write the gnuplot script to a file | ||
+ | file = io.open(" | ||
+ | file: | ||
+ | file: | ||
+ | |||
+ | print("" | ||
+ | print(" | ||
+ | |||
+ | -- call gnuplot to execute the script | ||
+ | os.execute(" | ||
+ | |||
+ | |||
+ | ---------------- overview of timing ------------------- | ||
+ | TimePrint() | ||
+ | </ | ||
+ | ### | ||
+ | |||
+ | ### | ||
+ | The resulting spectra are for method 1 are: | ||
+ | ### | ||
+ | |{{: | ||
+ | ^ XAS spectra ($2p$ to $3d$) in form of a conductivity tensor ($\sigma$). For a particular polarization $\varepsilon$ the measured spectrum is $-\mathrm{Im}[\epsilon^* \cdot \sigma(\omega) \cdot \epsilon]$ ^ | ||
+ | |||
+ | ### | ||
+ | The resulting spectra are for method 2 are: | ||
+ | ### | ||
+ | |{{: | ||
+ | ^ XAS spectra ($2p$ to $3d$) in form of a conductivity tensor ($\sigma$). For a particular polarization $\varepsilon$ the measured spectrum is $-\mathrm{Im}[\epsilon^* \cdot \sigma(\omega) \cdot \epsilon]$ ^ | ||
+ | |||
+ | ### | ||
+ | The difference is: | ||
+ | ### | ||
+ | |{{ : | ||
+ | ^ Difference between calculation with method 1 and method 2 (should be zero) ^ | ||
+ | |||
+ | ### | ||
+ | The output of the script is: | ||
+ | <file Quanty_Output XAS_tensor.out> | ||
+ | text produced as output | ||
+ | Start of BlockGroundState. Converge 3 states to an energy with relative variance smaller than 1.490116119384766E-06 | ||
+ | |||
+ | Start of BlockOperatorPsiSerialRestricted | ||
+ | Outer loop 1, Number of Determinants: | ||
+ | Start of BlockOperatorPsiSerialRestricted | ||
+ | Start of BlockGroundState. Converge 3 states to an energy with relative variance smaller than 1.490116119384766E-06 | ||
+ | |||
+ | Start of BlockOperatorPsiSerial | ||
+ | < | ||
+ | -2.882 | ||
+ | -2.721 | ||
+ | -2.560 | ||
+ | Start of LanczosTriDiagonalizeMC | ||
+ | Start of LanczosTriDiagonalizeMC | ||
+ | Start of LanczosTriDiagonalizeMC | ||
+ | Start of LanczosTriDiagonalizeMC | ||
+ | Start of LanczosTriDiagonalizeMC | ||
+ | Start of LanczosTriDiagonalizeMC | ||
+ | Start of LanczosTriDiagonalizeMC | ||
+ | Start of LanczosTriDiagonalizeMC | ||
+ | Start of LanczosTriDiagonalizeMC | ||
+ | Spectra printed to file: XASSigma_method1.dat | ||
+ | Prepare gnuplot-file for Sigma | ||
+ | |||
+ | Execute the gnuplot to produce plots and convert the output into a pdf-file | ||
+ | Start of LanczosBlockTriDiagonalize | ||
+ | Start of LanczosBlockTriDiagonalizeMC | ||
+ | Spectra printed to file: XASSigma_method2.dat | ||
+ | Prepare gnuplot-file for Sigma | ||
+ | |||
+ | Execute the gnuplot to produce plots and convert the output into a pdf-file | ||
+ | Spectra printed to file: XASSigma_diff.dat | ||
+ | Prepare gnuplot-file for Sigma | ||
+ | |||
+ | Execute the gnuplot to produce plots and convert the output into a pdf-file | ||
+ | Timing results | ||
+ | | ||
+ | 0:00:04 | 1 | 0 | Mehtod1 | ||
+ | 0:00:16 | 2 | 0 | Broaden | ||
+ | 0:00:01 | 1 | 0 | Mehtod2 | ||
+ | ### | ||
+ | |||
+ | |||
+ | ===== Table of contents ===== | ||
+ | {{indexmenu> |