Lecture Note for Oct.16, 2002 (Wed.)

 

 

V.  INTERACTION OF PHOTONS WITH ATOMS

 

 

V-A.  Electric and Magnetic Multipole Transitions

 

     Interaction of one photon with an atom

 

     Photon is introduced into the Hamiltonian through a vector potential A

 

     p2 is replaced by [p – (e/c)A]2

 

     (e/c) pA is treated as a perturbation to p2

 

     A2 is ignored for one-photon interaction

 

     pA leads to the velocity form of the transition operator

 

     The velocity form of the transition operator is converted to the length form of the operator using the commutation relation

 

              [r, H] = (i/m)p

 

     This conversion between the velocity and length forms is valid only for nonrelativistic H

     There is also the accelerator form, but it leads to unreliable results, and should not be used

 

     Transition probability = Einsteins A coefficient in the dipole (E1) approx.

 

              Ann = (4e23/3c3)|<n|j rj| n>|2

 

              = (EnEn)/

 

En = Total energy of the upper state

 

En = Total energy of the lower state

 

     Transition probability (dimension sec–1) = total # of photons emitted per second

 

     For photoabsorption, it is customary to use the dipole oscillator strength, or the f value:

 

              fnn = (mc3gn/2e23gn) Ann

 

     Thomas-Kuhn-Reich sum rule:

 

              n fnn = N

 

              N = total # of electrons in the target atom

 

     The TKR sum rule is for the entire atom, not for each orbital

 

     However, an approx. sum rule for f values is satisfied for orbitals with the same principal quantum number

 

     Electron correlation makes the f values for different orbitals to be shared

 

     Relativistic form of the transition operator includes the photon energy.  For high-energy photons, it is important to use the relativistic formula.  The MCDF2002 code uses relativistic formulas for all electromagnetic transitions

 

     Formulas for other electromagnetic transitions are derived by expanding the electromagnetic field representing the photon into multipoles:

 

       M1, E2, M2, E3, etc.

 

     The MCDF2002 code can calculate all types of multipole transitions

 

     E1 transition is the strongest.  All other transitions are much weaker than E1

 

     When a local potential [example, V(r), e2/rij] is used, the three forms of the transition operator should lead to the same answer

 

     Since the NR Hamiltonian contains only local potentials, the length and velocity forms lead to the same result if exact solutions of the NR Hamiltonian are used

 

     The two forms of solutions may accidentally agree even when poor wave functions or wave functions based on local potentials are used [example: Herman-Skillman wave functions, scaled hydrogenic wave functions]

 

     Length and velocity results obtained from Hartree-Fock wave functions do not agree in principle because the potential used in the HF equation is non-local

 

     Accuracy of calculated transition probabilities depends on the accuracy of the wave functions used

 

     Accuracy of experimental transition probabilities is very poor compared to the accuracy of experimental transition wavelengths

 

     10% for strong E1 transitions is considered excellent.  Only rarely an experimental f value is known to better than 5%

 

     Non-dipole transitions are known as the forbidden transitions

 

     In general, theory is more reliable than experiments for forbidden transitions

 

     Since accurate wavelengths are often known from experiments, it is better to use experimental wavelengths with theoretical transition matrix elements in the length form to obtain the best results

 

     The velocity form often leads to less accurate results than the length form

 

     When results from the length and velocity forms are compared, theoretical wavelengths should be used, because the eigenvalues of the Hamiltonian are used in converting the velocity form to the length form


    For applications in which many transition probabilities are used, such as plasma modeling, consistency in the wave functions used is important.  Do not mix wave functions of different types or qualities in calculating transition probabilities

 

     For mass production of transition probabilities, configuration average wave functions are recommended

 

     For individual f transition probabilities, use correlated wave functions, such as multi-configuration Hartree-Fock or Dirac-Fock wave functions

 

 


V-B.  Selection Rules for Electromagnetic Transitions

 

     Selection rules arise from the conservation of quantum numbers for the combined system of the incident photon and the target atom

 

     Note that the # of photon is different before and after a transition

 

     Each multipole transition has its own set of selection rules in terms of J and the parity of the target atom

 

     Strict selection rules are in terms of the jj coupling

 

     There are approx. selection rules in terms of the LS coupling

 

     See Eq. (54), Table III, and Table IV of the lecture note for the selection rules for common types of electromagnetic multipole transitions

 

     For applications to laser spectroscopy and plasma diagnostics, one must be  familiar with the selection rules in Tables III and IV

 

 

 

V-C.  Charge Expansion of Atomic Properties

 

     Many atomic properties are smooth functions of Z (= nuclear charge) along an isoelectronic sequence

 

     Nonrelativistic total energy of atoms :

 

       E = E2Z2 + E1Z + E0 + E–1/Z + E–2/Z2 + ...

 

     For hydrogen-like ions

 

       E2 = − (2n2)−1 mc22

 

     E is a function of quantum numbers of the state of interest, but independent of Z

 

     In principle, E can be calculated from eigenfunctions of the hydrogen atom using perturbation theory

 

     In practice, only E1,  E0, and E–1 are known for some states

 

     Hartree-Fock solutions have the correct E2, but only parts of the rest of E

 

     MCHF solutions containing all allowed configurations of a Coulomb complex have the correct E1, and E0

 

     MCHF solutions have only parts of E–1 and beyond

 

     Z dependence of f value:

 

In atomic units,

 

fnn = (2gn/3gn) | < n | j rj | n> |2

 

     ~ Z2,  < r > ~ 1/Z, hence f ~ Z0

       i.e., nonrelativistic f  for a transition along an isoelectronic sequence is independent of Z

 

     In reality, relativity makes ~ Z2 + Z4 +

       and f value exhibits a strong Z dependence at high Z

 

     Relativity introduces Z dependence in E

 

     For highly-charged ions, using all allowed configurations in a relativistic Coulomb complex will generate correct E1 and E0

 

     Using all allowed configurations in a relativistic or nonrelativistic Coulomb complex will represent valence correlation well

 

 


V-D.  Differences in Hartree-Fock and Dirac-Fock Calculations

 

     Orbital size:

 

< rn >R  <  < rn >NR

 

       For heavy atoms, < rn >R  >  < rn >NR for outer shells

 

     Binding energy:

 

| R | > | NR | for most cases

 

| R | < | NR | for some outer shells

 

     f values:  Since | R | > | NR | for most cases,

      

       fR > fNR for most transitions in heavy atoms

 

     Intercombination lines:  E1 allowed, spin forbidden transitions

 

       Example:  1s2p 3P1 1s2 1S0 of He

 

       Spin selection rules are valid only in the LS coupling

 

       Example:  3p54s 3P1 3p6 1S0 of Ar

 

       Caution: 4p55s 3P1 4p6 1S0 of Kr

 

              5P56s 3P1 5p6 1S0 of Xe

 

Above transitions are comparable to E1 allowed transitions

 

4p55s 1P1 4p6 1S0 of Kr

      

              5P56s 1P1 5p6 1S0 of Xe

 

     Magnitudes of A values for neutral atoms:

 

Strong E1 transitions:  > 107 sec−1

 

Weak E1 transitions:  ~ 106 sec−1

 

M1, E2 transitions:  < 104 sec−1

 

 

 

 


V-E.  Nonrelativistic Limit of DF wave functions and Nonrelativistic Offsets

 

     Some DF wave functions do not reduce to the correct nonrelativistic HF wave functions in the limit c

 

     Rule of thumb:  If the ion core has non-unique LS terms, c will not produce the correct HF limit

 

     Example:  F atom

 

       F+:  2s2 2p4 leads to 3P, 1D, and 1S

 

       c limit of the DF wave function for the F atom will not lead to the correct HF wave function

 

       This can be checked by comparing 2p1/2 and 2p3/2 orbitals of the DF wave function for the F atom in the limit c

 

       The correct HF wave function should have

2p* = 2p  in the NR limit

 

     If configuration average DF wave functions are used, they reduce to the correct HF wave functions

 

     DF wave functions with incorrect NR limits will also produce incorrect fine-structure splittings

 

       Example:  Fine structure splitting in F

 

       Experiment:  404.10 cm−1

 

       Dirac-Fock:  371.057 cm−1

 

       NR offset:     −23.771 cm−1 (2p* lower than 2p)

 

       DF+NR offset:  371.057 − 23.771 = 394.828 cm−1

 

       DF(config. ave.) = 404.404 cm−1

 

     Similar NR offset exists for transition probabilities

 

     If the NR limits of the DF wave functions used to calculate transition probabilities are incorrect, the matching nonrelativistic matrix element with signs must be subtracted from the relativistic matrix element before it is squared

 

     The MCDF2002 code can accept NR offset for transition probabilities and correct the relativistic result accordingly

     NR offsets in energy and transition matrix elements are both small, and will not affect results if excitation energies and transition probabilities are large

 

     They are important only when excitation energy is small, such as the fine-structure splitting, or transition probability is small, such as forbidden transitions in neutral atoms

 

     Subtracting NR offsets works only when the quantity being calculated vanishes in the NR limit

 

     Although E2 transitions are weak, NR offset cannot be used for E2 transitions because E2 transitions are allowed in the NR limit

 

 

 

 

 

Homework: 

 

Download mcdf2002.exe, mdfgme.dat, dff.f05, and dff.out from the KAERI website, http://amods.kaeri.re.kr/mcdf/MCDF.html

 

Run mcdf2002.exe by typing the binary file name, mcdf2002.exe

 

The code looks for an input file named dff.f05 because the second line in mdfgme.dat indicates the generic file name, dff

Output will come out in dff.f06, while binary file for the wave function will be stored in dff.f09

 

Compare dff.out and dff.f06 and confirm that the contents of the two files are the same except for the execution time.  Execution time will depend on the speed of your PC

 

This homework will confirm that the MCDF2002 code is running correctly on your PC

 

The output will be used when practice on the MCDF2002 code itself starts on Nov. 6