10. Bound-bound oscillator strengths (data_BBT, data_SRBBT)

Bound-bound oscillator strengths are needed in our evaluation of the anomalous scattering factors. As noted earlier, bound-bound transitions contribute as much as 30% or more to the Thomas-Reiche-Kuhn sum rule. Our evaluation of relativistic multipole bound-bound oscillator strengths follows the formulation of Scofield (1975).

The bound-bound oscillator strengths needed for our anomalous scattering factors connect occupied orbitals of the atom with unoccupied orbitals. These values are stored in the RTAB database in files with the filename 'ZZZ_bbtIM', where 'ZZZ', 'I' and 'M' indicate the atomic number, ioniticity, and model as noted earlier, and '_bbt' indicates that the file contains occupied-to-unoccupied bound-bound oscillator strengths. We consider all transitions down to 10^-6 of the largest oscillator strength found. In order to satisfy this requirement, we often need to include transitions to states with principal quantum numbers up to 100 or more.

For example, the file 'data_BBT/082_bbt0sl' contains our occupied-to-unoccupied bound-bound oscillator strenghts for Pb. Selected bound-bound oscillator strengths are listed in Table 8. The file contains a total of 1670 bound-bound transitions for a total oscillator strength of about 3.98; the maximum oscillator strength of about 1.06 is for the P1 -> P3 transition.

We also have need for oscillator strengths for transitions between occupied states of the atom. These transitions are needed to subtract spurious resonances from our S-matrix amplitudes. Recall that our SM prescription mixes predictions of S-matrix amplitudes for inner-shell electrons with modified relativistic form-factor predictions for outer-shell electrons. The S-matrix amplitudes, for a single electron transistion in a potential, include contributions for transitions to all bound and continuum states of the potential, including the occupied orbitals of the atom. If the total scattering amplitude for all occupied orbitals were computed solely via the S matrix, the contributions from transitions between occupied orbitals would vanish. If the transitions between occupied orbitals were not subtracted from the S-matrix amplitudes, however, spurious resonances connecting orbitals computed via the S matrix with orbitals computed via modified relativistic form factor would survive under our SM prescription. Consequently, we routinely subtract the contribution for spurious resonances (bound-bound transitions between occupied orbitals of the atom) from our S-matrix amplitudes.

The spurious-resonance bound-bound oscillator strengths for transitions between occupied orbitals of the atom are stored in the RTAB database in files with the filename 'ZZZ_srbbtIM'. We consider all transitions that connect occupied orbitals of the atom, and sample values from 'data_SRBBT/082_srbbt0sl' are listed in Table 9. This file contains 276 transitions between the occupied orbitals of Pb, for a total oscillator strength of about 39.6; the maximum contribution is about 3.83 for the N5 -> N7 transition.

Table 8. Selected occupied-to-unoccupied bound-bound oscillator strengths for Pb are extracted from the file ‘data_BBT/082_bbt0sl’ in the RTAB database.
return to tables

Transition	Transition energy (keV)	Initial state, occupied (n₁, κ₁)	Final state, unoccupied (n₂, κ₂)	Δℓ = ℓ₂-ℓ₁	ƒ_total (per initial electron)	ƒ_electric (per initial electron & final hole)	ƒ_magnetic (per initial electron & final hole)
K → Q1	8.83511E+01	(1,-1)	(7,-1)	0	3.55161E-08	0.00E+00	1.78E-08
K → P3	8.83486E+01	(1,-1)	(6,-2)	1	1.08147E-04	4.06E-05	1.89E-09
K → P2	8.83469E+01	(1,-1)	(6, 1)	1	6.95786E-05	5.22E-05	0.00E+00
K → P5	8.83524E+01	(1,-1)	(6,-3)	2	7.32820E-08	1.22E-08	1.88E-12
L3 → Q1	1.30365E+01	(2,-2)	(7,-1)	-1	2.71613E-05	1.36E-05	7.60E-12
P1 → P3	9.80446E-03	(6,-1)	(6,-2)	1	1.06036E+00	3.98E-01	3.19E-17

Table 9. Selected spurious-resonance bound-bound oscillator strengths for Pb are extracted from the file ‘data_SRBBT/082_srbbt0sl’ in the RTAB database.
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Transition	Transition energy (keV)	Initial state, occupied (n₁, κ₁)	Final state, occupied (n₂, κ₂)	Δℓ = ℓ₂-ℓ₁	ƒ_total (per initial electron)	ƒ_electric (per initial electron & final hole)	ƒ_magnetic (per initial electron & final hole)
K → L1	7.24965E+01	(1,–1)	(2,-1)	0	3.68471E-04	0.00E+00	1.84E-04
K → L2	7.31016E+01	(1,–1)	(2, 1)	1	2.16403E-01	1.08E-01	0.00E+00
K → L3	7.53146E+01	(1,–1)	(2,–2)	1	3.48130E-01	8.70E-02	3.71E-06
L1 → L2	6.05025E-01	(2,-1)	(2, 1)	1	3.25986E-02	1.63E-02	0.00E+00
L3 → M5	1.05507E+01	(2,–2)	(3,-3)	1	1.81068E+00	3.02E-01	5.31E-08
M1 → N7	3.67986E+00	(3,-1)	(4,-4)	3	1.12378E-05	1.40E-06	1.37E-14
N5 → N7	2.64735E-01	(4,-3)	(4,-4)	1	3.82834E+00	4.79E-01	9.97E-13

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Information date: Sep. 2, 2000 lk