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

¡¤ *p*^{2}
is replaced by [** p** – (

¡¤ (*e*/*c*) ** p**∙

¡¤ *A*^{2} is ignored for
one-photon interaction

¡¤ ** p**∙

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

[** r**,

¡¤ 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 = Einstein¡¯s *A*
coefficient in the dipole (*E*1) approx.

*A*_{nn¡Ç}
= (4*e*^{2}*¥ø*^{3}/3*©¤c*^{3})|<*¥×** _{n}*|¥Ò

*¥ø** =* (*E _{n}* –

*E _{n}* = Total energy
of the upper state

*E _{n}*

¡¤ 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:

*f _{n}*

¡¤ Thomas-Kuhn-Reich sum
rule:

¥Ò_{n}*f _{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:

*M*1, *E*2, *M*2, *E*3, etc.

¡¤ The MCDF2002 code can calculate all types of
multipole transitions

¡¤ *E*1 transition is the strongest. All other transitions are much weaker
than *E*1

¡¤ When a local potential [example, *V*(*r*),
*e*^{2}/*r _{ij}*] 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

¡¤ ¡¾10% for strong *E*1 transitions is
considered excellent. Only rarely
an experimental *f* value is known to better than ¡¾5%

¡¤ 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 ¥Ä*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* = *E*_{2}*Z*^{2} + *E*_{1}*Z*
+ *E*_{0} + *E*_{–1}/*Z* + *E*_{–2}/*Z*^{2}
+ ...

¡¤ For hydrogen-like ions

*E*_{2} = − (2*n*^{2})^{−1
}∙ ¨ö*mc*^{2}*¥á*^{2}

¡¤ *E _{¥ë}* is a function of
quantum numbers of the state of interest, but independent of

¡¤ In principle,
*E _{¥ë}* can be calculated from eigenfunctions
of the hydrogen atom using perturbation theory

¡¤ In practice,
only *E*_{1}, *E*_{0}, and *E*_{–1}
are known for some states

¡¤ Hartree-Fock
solutions have the correct *E*_{2}, but only parts of the rest of *E _{¥ë}*

¡¤ MCHF solutions containing all allowed
configurations of a Coulomb complex have the correct *E*_{1}, and *E*_{0}

¡¤ MCHF solutions have only parts of *E*_{–1}
and beyond

¡¤ *Z* dependence of *f* value:

In atomic units,

*f _{n}*

¡¤ *¥ø** *~* Z*^{2}, < *r* > ~ 1/*Z*, hence *f*
~ *Z*^{0}

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

¡¤ In reality, relativity makes *¥ø *~* Z*^{2}
+ *Z*^{4} + ¡¦

and *f *value exhibits a strong *Z* dependence at
high *Z*

¡¤ Relativity introduces *Z¥á* dependence
in *E _{¥ë}*

¡¤ For highly-charged ions, using all allowed
configurations in a relativistic Coulomb complex will generate correct *E*_{1}
and *E*_{0}

¡¤ 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:

< *r ^{n}*
>

For heavy atoms,
< *r ^{n}* >

¡¤ Binding energy:

| *¥å*_{R}
| > | *¥å*_{NR} | for most cases

| *¥å*_{R}
| < | *¥å*_{NR} | for some outer shells

¡¤ *f*
values: Since | *¥å*_{R}
| > | *¥å*_{NR} | for most cases,

*f*_{R} > *f*_{NR} for most transitions in heavy atoms

¡¤ Intercombination
lines: *E*1 allowed, spin
forbidden transitions

Example:
1s2p ^{3}P_{1} ¡æ 1s^{2} ^{1}S_{0}
of He

Spin
selection rules are valid only in the *LS* coupling

Example:
3p^{5}4s ^{3}P_{1} ¡æ 3p^{6} ^{1}S_{0}
of Ar

Caution: 4p^{5}5s
^{3}P_{1} ¡æ 4p^{6} ^{1}S_{0} of Kr

5P^{5}6s
^{3}P_{1} ¡æ 5p^{6} ^{1}S_{0} of Xe

Above transitions are
comparable to *E*1 allowed transitions

4p^{5}5s ^{1}P_{1}
¡æ 4p^{6} ^{1}S_{0} of Kr

5P^{5}6s
^{1}P_{1} ¡æ 5p^{6} ^{1}S_{0} of Xe

¡¤ Magnitudes of *A* values for neutral
atoms:

Strong *E*1
transitions: > 10^{7}
sec^{−1}

Weak *E*1
transitions: ~ 10^{6} sec^{−1}

*M*1, *E*2
transitions: < 10^{4}
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* ¡æ ¡Ä

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

¡¤ Example: F atom

F^{+}: 2s^{2} 2p^{4} leads to ^{3}P,
^{1}D, and ^{1}S

*c* ¡æ ¡Ä 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 2p_{1/2} and 2p_{3/2} orbitals
of the DF wave function for the F atom in the limit *c* ¡æ ¡Ä

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 *E*2 transitions are weak, NR
offset cannot be used for *E*2 transitions because *E*2 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