The effect of temperature can be added by calculating excited states and expectation values of excited states. The temperature dependent expectation value is then created using Boltzmann statistics.
A small example:
-- Sofar we calculated eigenstates and expectation values (or spectra) of these
-- eigenstates. At 0 K one would measure the expectation value of the lowest eigenstate
-- at finite temperature one would measure an average over several states weighted by
-- Boltzmann statistics. In this example we calculate the temperature dependent
-- x-ray absorption spectra of NiO. (Ni L23 edge 2p to 3d)
-- we set the verbosity to 0 in order to minimize the output
Verbosity(0)
-- the beginning of this file is the same as example 21 where x-ray absorption is calculated
-- 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)
-- x 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)
-- 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... )
-- we define a magnetic field in units of tesla EnergyUnits.Tesla.value is a constant
-- expressing Tesla in units of eV
B = 10*EnergyUnits.Tesla.value
-- once all parameters are set we can define the Hamiltonian for both the ground-state
-- and the excited state as a sum of operators multiplied with the numerical interaction strength
Hamiltonian = F0dd*OppF0 + F2dd*OppF2 + F4dd*OppF4 + tenDq*OpptenDq + zeta_3d*Oppldots + B*(2*OppSz + OppLz)
XASHamiltonian = Hamiltonian + zeta_2p * Oppcldots + F2pd * OppUpdF2 + G1pd * OppUpdG1 + G3pd * OppUpdG3
-- we set restrictions to have 6 electrons in the p-shell and 8 electrons in the d-shell
StartRestrictions = {NF, NB, {"111111 0000000000",6,6}, {"000000 1111111111",8,8}}
-- and we calculate all 45 eigenstates
Npsi=45
psiList = Eigensystem(Hamiltonian, StartRestrictions, Npsi)
-- Boltzmann statistics contains the exponent of the eigen energy. In order to prevent
-- number overflow we set later the ground-state energy to zero. Here we calculate
-- the ground state energy
Egrd = psiList[1] * Hamiltonian * psiList[1]
-- 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
-- now we can calculate temperature averaged expectation values
-- The temperature we will take here is 10 Kelvin (again we enter it in units of eV)
T = 10 * EnergyUnits.Kelvin.value
-- we will calculate the partition function Z
Z=0
-- the total magnetic moment M
M=0
-- the total spin moment MS
MS=0
-- the total angular moment ML
ML=0
-- and temperature averaged spectra for z, r and l polarized light.
Spectra_z=0
Spectra_r=0
Spectra_l=0
-- the temperature averaged spectra are calculated as sums over the different states
-- weighted by the Boltzmann occupation. In order to make these sums we set them first to
-- zero (done above)
-- and now we can make the sums
for j=1, 3 do
Z = Z + exp(-(psiList[j] * Hamiltonian * psiList[j] - Egrd)/T)
M = M + psiList[j] * (2 * OppSz + OppLz) * psiList[j] * exp(-(psiList[j] * Hamiltonian * psiList[j] - Egrd)/T)
MS = MS + psiList[j] * (OppSz) * psiList[j] * exp(-(psiList[j] * Hamiltonian * psiList[j] - Egrd)/T)
ML = ML + psiList[j] * (OppLz) * psiList[j] * exp(-(psiList[j] * Hamiltonian * psiList[j] - Egrd)/T)
Spectra_z = Spectra_z + CreateSpectra(XASHamiltonian, TXASz,psiList[j], {{"Emin",-10}, {"Emax",20}, {"NE",3500}, {"Gamma",1.0}}) * exp(-(psiList[j] * Hamiltonian * psiList[j] - Egrd)/T)
Spectra_r = Spectra_r + CreateSpectra(XASHamiltonian, TXASr,psiList[j], {{"Emin",-10}, {"Emax",20}, {"NE",3500}, {"Gamma",1.0}}) * exp(-(psiList[j] * Hamiltonian * psiList[j] - Egrd)/T)
Spectra_l = Spectra_l + CreateSpectra(XASHamiltonian, TXASl,psiList[j], {{"Emin",-10}, {"Emax",20}, {"NE",3500}, {"Gamma",1.0}}) * exp(-(psiList[j] * Hamiltonian * psiList[j] - Egrd)/T)
end
-- In order to normalize we should device by the partition function Z
M = M / Z
MS = MS / Z
ML = ML / Z
Spectra_z = Spectra_z/Z
Spectra_r = Spectra_r/Z
Spectra_l = Spectra_l/Z
-- and we can print the results to the screen
print("For a magnetic field of ",B/EnergyUnits.Tesla.value,"Tesla")
print("At temperature ",T/EnergyUnits.Kelvin.value," Kelvin the magnetic moment is",M)
print("The spin contribution is",MS)
print("The angular contribution is",ML)
-- we can calculate the isotropic spectra and the magnetic circular dichroism
Spectra_iso = (Spectra_z + Spectra_l + Spectra_r)/3
Spectra_XMCD = (Spectra_r - Spectra_l)
-- and print them to file
Spectra_iso.Print({{"file", "TemperatureXASSpecIso.dat"}});
Spectra_XMCD.Print({{"file", "TemperatureXASSpecXMCD.dat"}});
-- from here on you can use your favorite program to plot these spectra
-- I include a gnuplot script to make these plots
-- 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 xlabel "E (eV)" font "Times,12"
set ylabel "Intensity (arb. units)" font "Times,12"
set out 'Temperature.ps'
set size 1.0, 1.0
set terminal postscript portrait enhanced color "Times" 8
plot "TemperatureXASSpecIso.dat" u 1:(-$3) title 'Iso ' with lines ls 1,\
"TemperatureXASSpecXMCD.dat" u 1:(-$3) title 'XMCD' with lines ls 2
]]
-- write the gnuplot script to a file
file = io.open("Temperature.gnuplot", "w")
file:write(gnuplotInput)
file:close()
-- call gnuplot to execute the script
os.execute("gnuplot Temperature.gnuplot ")
-- change the postscript file to pdf or eps
os.execute("ps2pdf Temperature.ps ; ps2eps Temperature.ps ; mv Temperature.eps temp.eps ; eps2eps temp.eps Temperature.eps ; rm temp.eps")
The output is:
<E> <S^2> <L^2> <J^2> <S_z> <L_z> <l.s> <F[2]> <F[4]> <Neg> <Nt2g> -2.722 1.999 12.000 15.127 -0.994 -0.290 -0.326 -1.020 -0.878 2.011 5.989 -2.721 1.999 12.000 15.120 -0.000 -0.004 -0.324 -1.020 -0.878 2.011 5.989 -2.720 1.999 12.000 15.113 0.994 0.281 -0.323 -1.020 -0.878 2.010 5.990 -1.655 1.989 11.989 16.760 -0.178 -0.160 -0.859 -1.019 -0.877 3.008 4.992 -1.655 1.989 11.988 16.766 -0.010 0.003 -0.861 -1.019 -0.877 3.008 4.992 -1.637 1.999 12.000 15.802 -0.496 -0.052 -0.477 -1.020 -0.878 3.007 4.993 -1.637 1.999 12.000 15.815 0.178 0.148 -0.482 -1.020 -0.878 3.007 4.993 -1.636 1.999 12.000 15.815 0.482 0.062 -0.481 -1.020 -0.878 3.007 4.993 -1.587 1.997 11.989 12.560 -0.488 -0.151 0.266 -1.020 -0.877 2.996 5.004 -1.587 1.997 11.989 12.563 -0.060 0.006 0.265 -1.020 -0.877 2.996 5.004 -1.586 1.997 11.989 12.566 0.503 0.139 0.264 -1.020 -0.877 2.996 5.004 -1.567 2.000 12.000 12.000 0.070 -0.015 0.500 -1.020 -0.878 3.000 5.000 -0.956 1.995 11.496 18.956 -0.036 -0.038 -1.473 -1.004 -0.886 3.588 4.412 -0.890 1.999 11.439 15.343 -0.498 0.859 -0.513 -1.003 -0.887 3.573 4.427 -0.890 1.999 11.439 15.351 0.025 0.039 -0.515 -1.003 -0.887 3.573 4.427 -0.890 1.999 11.440 15.357 0.480 -0.877 -0.516 -1.003 -0.887 3.573 4.427 -0.809 1.764 10.850 9.064 -0.014 0.026 0.004 -0.984 -0.879 3.367 4.633 -0.809 1.764 10.850 9.065 0.012 0.010 0.003 -0.984 -0.879 3.367 4.633 -0.782 1.997 11.307 9.989 -0.482 0.501 0.789 -0.999 -0.889 3.550 4.450 -0.782 1.997 11.307 9.985 0.014 -0.019 0.790 -0.999 -0.889 3.550 4.450 -0.782 1.997 11.308 9.981 0.500 -0.473 0.791 -0.999 -0.889 3.550 4.450 -0.490 0.251 9.749 9.156 0.000 -0.004 1.055 -0.900 -0.806 2.269 5.731 -0.490 0.251 9.749 9.156 0.000 -0.001 1.055 -0.900 -0.806 2.269 5.731 0.464 0.138 7.384 7.649 -0.033 -0.555 -0.681 -0.889 -0.803 3.132 4.868 0.464 0.137 7.388 7.653 -0.000 0.004 -0.680 -0.889 -0.803 3.132 4.868 0.465 0.137 7.393 7.657 0.033 0.553 -0.680 -0.889 -0.803 3.132 4.868 0.822 0.030 18.844 18.867 -0.000 -0.016 0.039 -0.767 -0.843 2.422 5.578 0.939 1.992 2.753 6.346 -0.109 -0.091 -0.310 -0.736 -1.035 3.441 4.559 0.939 1.992 2.751 6.345 -0.007 0.006 -0.313 -0.736 -1.035 3.441 4.559 0.966 1.865 3.179 6.537 -0.475 -0.392 0.365 -0.747 -1.019 3.426 4.574 0.967 1.866 3.177 6.537 0.109 0.092 0.361 -0.747 -1.019 3.426 4.574 0.967 1.866 3.178 6.537 0.458 0.407 0.362 -0.747 -1.019 3.425 4.575 1.001 1.994 2.614 2.880 -0.490 -0.433 0.413 -0.732 -1.038 3.417 4.583 1.002 1.994 2.613 2.879 -0.021 0.018 0.413 -0.732 -1.038 3.417 4.583 1.003 1.994 2.613 2.879 0.507 0.419 0.413 -0.732 -1.038 3.417 4.583 1.033 1.972 2.737 1.255 0.028 -0.024 0.745 -0.730 -1.037 3.398 4.602 1.280 0.008 19.948 19.962 -0.002 -0.501 0.081 -0.776 -0.855 3.003 4.997 1.280 0.008 19.948 19.961 0.000 0.007 0.081 -0.776 -0.855 3.003 4.997 1.281 0.008 19.948 19.961 0.002 0.497 0.081 -0.776 -0.855 3.003 4.997 2.042 0.004 16.663 16.669 -0.000 -0.059 0.115 -0.809 -0.836 3.915 4.085 2.042 0.004 16.662 16.669 0.000 0.011 0.115 -0.809 -0.836 3.915 4.085 2.118 0.004 18.133 18.138 0.000 -2.033 0.087 -0.794 -0.845 3.886 4.114 2.119 0.004 18.137 18.142 0.000 0.059 0.087 -0.794 -0.845 3.886 4.114 2.120 0.004 18.141 18.147 -0.000 2.039 0.086 -0.794 -0.845 3.886 4.114 5.964 0.003 0.923 0.920 0.000 0.000 0.190 -0.581 -0.585 3.592 4.408 For a magnetic field of 10 Tesla At temperature 10 Kelvin the magnetic moment is -1.7185713390306 The spin contribution is -0.74960257265827 The angular contribution is -0.21936619371406