Measurement of total, radiative and radiationless (Auger) vacancy transfer probabilities from K to L_i sub-shells of Cs, Ba and La

Abstract

The vacancy transfer probabilities of K to L₁, L₂ and L₃ shells were measured with a new approach. The measured L X-ray yields from targets excited by 59.5 keV incident photons, i.e., above the K edge of elements, were detected with a high-resolution Si(Li) detector. For comparison with experimental results, theoretical calculations were made using radiative and radiationless transitions. The radiative transitions of these elements were taken from the relativistic Hartree–Slater model. The radiationless transitions were taken from the Dirac–Hartree–Slater model. The measured results were found to be in good agreement with theoretical values.

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Introduction

When an atom's K or L shell/sub-shell has been ionised, it is de-excited through radiative or radiationless (Auger) transitions. Knowledge of average vacancy distributions is important for the study of processes such as nuclear electron capture, internal conversion of γ-rays, the photoelectric effect and, in general, whenever primary vacancies produced in the shell must be distinguished from multiple ionization due to the decay of the inner vacancy.

The L X-ray and M X-ray production cross-sections and average L and M shell fluorescence yields were measured experimentally in recent years.^1–7 Ertugrul et al.⁸ measured K to L shell radiative vacancy transfer probabilities for elements in the atomic number range 72 ≤ Z ≤ 92 using a cobalt-57 radioisotope source.

The enhancement effect of K to L shell vacancy transfer on the Kα to Lα intensity ratio was measured for lanthanides.⁹ The enhancement of Coster–Kronig transitions on L sub-shell X-rays was investigated for heavy elements in the atomic number range 79 ≤ Z ≤ 92.¹⁰ K to L shell^11,12 and L to M shell¹³ vacancy transfer probabilities have been measured experimentally using two radioisotope sources. In addition, Puri et al.¹⁴ calculated vacancy transfer probabilities of K to L₁, K to L₂, K to L₃, K to L, L₁ to M₁, L₂ to M, L₃ to M shells. The values of η_KLi (i = 1, 2, 3) for elements in the atomic number range 20 ≤ Z ≤ 94 have also been calculated.¹ In these calculations, the contributions due to Auger and radiative transitions were derived using the best fitted experimental data on the fluorescence yields, and the intensity ratios of different components of K–LX (X = L, M, N…) Auger electrons to K X-rays are currently available.

In the earlier measurements^11–13 two radioisotope sources for excitation of targets were used.^11,12 The method was based on the number of L X-rays produced at the photon excitation energy below the K edge and at the excitation energy above the K edge, where the major contribution to L shell vacancies comes from the decay of K shell vacancies. In the present work, the vacancy probabilities of K to L₁, K to L₂ and K to L₃ were measured using: L_i sub-shell X-ray production cross-sections (σ_L1x, σ_L2x and σ_L3x); theoretical values of sub-shell photoionization cross-sections; sub-shell fluorescence yields; and Coster–Kronig transition probabilities. The measured values were compared with the calculated values deduced using Auger transitions¹⁵ and radiative X-ray emissions,¹⁶ and found to be in agreement.

Method and experimental measurements

Measurements of cross-sections for the production of L_i sub-shell X-rays of Cs, Ba and La were made using the ²⁴¹Am radioisotope and the experimental set-up shown in Fig. 1. In this set-up, 59.5 keV gamma rays emitted from a 100 mCi americium-241 point source were used for excitation of the target. The radioisotope of ²⁴¹Am was completely (∼99.99%) filtered out with the help of graded filters of Pb, Cu and Al of thickness 0.1, 0.1 and 1 mm, respectively. The emitted L X-rays from the target were detected by a calibrated Si(Li) detector (active area = 12 mm², sensitive depth = 3 mm, Be window thickness = 12.5 µm) with an energy resolution of approximately 160 eV at 5.96 keV. The measurement of L X-rays at the emission angle 90° accounts for the anisotropic emission of Ll, Lα and L₃β lines because the effects of anisotropy at angles smaller and greater than 90° compensate one another.^17,18

Fig. 1 Experimental set-up.

High purity (99.9%) samples of thickness from 5 to 20 µg were used for the preparation of the targets. Each target was excited and the data were recorded for time intervals ranging from 3 to 12 h. Also, to minimize any systematic errors, the data for each target were taken on different occasions. A typical X-ray spectrum for La is shown in Fig. 2.

Fig. 2 K and L X-ray spectra of La.

Data analysis

The experimental L_i XRF cross-sections σ_Lix were evaluated using the relationwhere N_Lix is the number of counts per unit time under the photopeak corresponding to L X-rays of the elements, the product I₀G is the intensity of the exciting radiation falling on the area of the target sample visible to the detector, ε_Lix is the efficiency of the detector at the L_ix X-ray energy of the element, t is the mass per area of the element in g cm⁻², M is the atomic weight, N_A is Avogadro's number and β_Lix is the self absorption factor for the incident and emitted L_ix X-ray photons. The values of β_Lix have been calculated by using the following expression obtained by assuming that the fluorescent X-rays are incident normally at the detector:where μ_inc and μ_emt are the absorption coefficients at the incident and emitted X-ray photon energy, respectively (values are taken from the tables of Hubbell and Seltzer¹⁹), and θ₁ are θ₂ are the angles of incident photons and emitted X-rays with respect to the normal at the surface of the sample. In the present work, the values of I₀Gε_Lix were determined by collecting the K X-ray spectra of thin samples, Ti (99.9%), V (99%), Fe (99.9%), Ni (99%), Zn (99.9%), As₂O₃ (99.9%), SeO₂ (99.9%), KBr (99%) and SrCl₂·6H₂O (99.9%), in the same geometry in which the L XRP cross sections were measured using the equation:where N_Kα, β_Kα and ε_Kα have the same meaning as in eqn. (1) except that they correspond to K X-rays instead of the ith group of L X-rays. In the calculations, the theoretical values of Kα XRP cross sections were calculated using the equationwhere σ_K(E) is the K shell photoionization cross section²⁰ for the elements at the excitation energy E, ω_K is the K shell fluorescence yield from the tables of Krause²¹ and f_Kα is the fractional X-ray emission rate for Kα X-rays and is defined as:where I_Kβ/I_Kα is the Kβ to Kα X-ray intensity ratio.¹⁶

In eqn. (1), the counts per unit time under the photopeak corresponding to L_ix X-rays were evaluated according to the following.

(i) Because the L₃l, L₃α, Lβ and Lγ X-rays are well resolved the net counts were taken under the photopeak.

(ii) Because the L₃β X-rays cannot be resolved they are found using N_L3α, the net intensity with the following equation:

(iii) From this point, we can write

and

From eqns. (7)–(9)

where ε_Lix (i = 1, 2, 3 and x = α, β, γ) and β_Lix are described in eqn. (1), the I_L3β/I_L3α, I_L1β/I_L1γ and I_L2β/I_L2γ relative emission rates are given by the relationships:where Γ(L_i–X_i) are the radiative emission rates, which are taken from the table of Scofield¹⁶ and tabulated as relative radiative emission rates. The values of N_L3β, N_L2β, N_L2γ, N_L1β and N_L2γ involve the total L₃β, L₂β, L₂γ, L₁β and L₁γ X-rays originating from L₃, L₂ and L₁ sub-shells, for example, N_L1β = N_Lβ3 + N_Lβ4 + N_Lβ9 + N_Lβ10.

Table 1 L sub-shell X-ray production cross-sections with theoretical values

	σ L3l/b atom^−1^a		σ L3α/b atom^−1		σ L3β/b atom^−1		σ L2β/b atom^−1		σ L2γ/b atom^−1		σ L1β/b atom^−1		σ L1γ/b atom^−1
	Exp.	Theo.	Exp.	Theo.	Exp.	Theo.	Exp.	Theo.	Exp.	Theo.	Exp.	Theo.	Exp.	Theo.
a 1 b (barn) = 10^−28 m^−2.
55Cs	2.8 ± 0.3	2.8	73.6 ± 3	74	12.4 ± 0.8	12	40.7 ± 2	39	6.7 ± 0.8	6.3	8.3 ± 1	9.1	2.0 ± 0.2	2.3
56Ba	3.2 ± 0.3	3.1	81.0 ± 5	79	13.5 ± 0.8	13	45.4 ± 3	44	7.7 ± 1	7.4	8.5 ± 1	8.7	2.3 ± 0.2	2.2
57La	3.8 ± 0.4	3.7	94.5 ± 6	97	17.1 ± 1	17	52.2 ± 3	49	9.0 ± 1	8.7	9.6 ± 1	9.6	2.6 ± 0.2	2.5

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Table 2 σ _L1x, σ_L2x and σ_L3x X-ray production cross-sections with theoretical values

	σ L1x/b atom^−1		σ L2x/b atom^−1		σ L3x/b atom^−1
	Exp.	Theo.	Exp.	Theo.	Exp.	Theo.
55Cs	11.3 ± 1	11.4	47.4 ± 3	45.3	88.8 ± 4	88.8
56Ba	10.8 ± 1	10.9	53.1 ± 4	51.4	98.7 ± 6	95.1
57La	12.5 ± 1	12.1	61.2 ± 4	57.7	115.4 ± 7	117.7

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Table 3 Total vacancy transfer probabilities η_KLi

	η KL1		η KL2		η KL3
	Exp.	Theo.	Exp.	Theo.	Exp.	Theo.
55Cs	0.051 ± 0.002	0.047	0.271 ± 0.01	0.302	0.545 ± 0.02	0.545
56Ba	0.041 ± 0.002	0.045	0.304 ± 0.01	0.301	0.527 ± 0.02	0.542
57La	0.045 ± 0.002	0.043	0.306 ± 0.01	0.300	0.540 ± 0.02	0.539

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Table 4 Intensity ratios of K X-rays

	I(Kβ)/I(Kα)		I(Kα2)/I(Kα1)	I(K′β1)/I(Kα1)
	Exp.	Ref. 16	Ref. 16	Ref. 16
55Cs	0.239 ± 0.014	0.2240	0.542	0.2807
56Ba	0.243 ± 0.007	0.2273	0.543	0.2832
57La	0.247 ± 0.010	0.2304	0.545	0.2857

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Table 5 Radiative vacancy transfer probabilities η_KLi(R)

	η KL2(R)		η KL3(R)
	Exp.	Theo.	Exp.	Theo.
55Cs	0.254 ± 0.01	0.255	0.469 ± 0.02	0.473
56Ba	0.255 ± 0.01	0.257	0.470 ± 0.02	0.473
57La	0.256 ± 0.01	0.258	0.470 ± 0.02	0.474

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Table 6 Radiationless (Auger) vacancy transfer probabilities η_KLi(A)

	η KL1(A)		η KL2(A)		η KL3(A)
	Exp.	Theo.	Exp.	Theo.	Exp.	Theo.
55Cs	0.051 ± 0.002	0.047	0.047 ± 0.001	0.047	0.076 ± 0.003	0.072
56Ba	0.041 ± 0.002	0.045	0.049 ± 0.002	0.044	0.058 ± 0.003	0.069
57La	0.045 ± 0.002	0.043	0.048 ± 0.002	0.042	0.070 ± 0.003	0.065

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Table 7 , , , ,

	Exp.	Theo.	Exp.	Theo.	Exp.	Theo.	Exp.	Theo.	Exp.	Theo.
55Cs	1	1	0.17	0.15	0.83	0.84	0.14	0.13	0.86	0.86
56Ba	1	1	0.16	0.14	0.83	0.85	0.11	0.12	0.89	0.87
57La	1	1	0.15	0.14	0.83	0.86	0.13	0.12	0.87	0.88

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The η_KLi values were evaluated by the relations:

The present values of η_KL1 (for calculation of η_KL2) and of η_KL1 and η_KL2 (for calculation of η_KL3) were used in eqns. (17) and (18), respectively. The experimental radiative transition probabilities were measured using the Kβ/Kα intensity ratio of these elements. For this the following equations were used^1,8where ω_K is the fluorescence yield of the K shell,²¹I(Kα₂)/I(Kα₁), and I(Kβ₁)/I(Kα₁) and I(Kβ)/I(Kα) are the intensity ratios of K X-rays. I(Kα₂)/I(Kα₁) and I(Kβ₁)/I(Kα₁) intensity ratios were taken from Scofield.¹⁶ The I(Kβ)/I(Kα) intensity ratio is observed by using the following relation:where N(Kβ) and N(Kα) are the net counts observed under the photopeaks corresponding to I(Kβ) and I(Kα) X-rays , respectively.

The radiationless (Auger) transition probabilities were found by using the equations:

Theoretical calculations

In this work we calculated the theoretical L X-ray production cross-sections for the elements at the 59.5 keV incident photon energy using the following equations:where σ_i (i = 1, 2, 3) is the L sub-shell photoionization cross section,²⁰ω_i (i = 1, 2, 3) is the L sub-shell fluorescence yield,²¹f_ij (i = 1, 2 and j = 2, 3) is the Coster–Kronig transition probability²¹ and F_ij (F_3l, F_3α, F_3β,…) is the fraction of the radiative transition of the sub-shell L_i (i = 1, 2, 3) contained in the jth spectral line.

The F_ij values are given by the following:

where Γ_i (i = 1, 2, 3) is the total radiative width of the L_i sub-shell¹⁶ and Γ (X_i–Y_j) is the partial width.

From eqns. (25)–(32), we can obtain the following equations:

where the quantities are described earlier.

The K to L_i sub-shell vacancy transfer probability is defined as the number of L_i sub-shell vacancies produced in the decay of one K shell vacancy through radiative K–L_i transitions or through Auger K–L_iL_j and K–L_iX (X = M, N, O...) transitions. The average value of η_KLi is given by:

where η_KLi(R) and η_KLi(A) are the radiative and Auger transition probabilities of the K to L_i sub-shells.

The η_KLi values have been calculated using the following equations:

where X = M, N, O andwhere Γ_R(KL_i) is the radiative K shell partial width, Γ_A(K–L_iL_j) and Γ_A(K–L_iX) are the radiationless partial widths and Γ(K) is the total K level width. The radiative transition rates and Auger transition rates have been tabulated by Scofield¹⁶ and Chen et al.,¹⁵ respectively. In eqns. (42)–(44), the first parts are the radiative transition probabilities and the second parts are the radiationless (Auger) transition probabilities.

Results and discussion

L sub-shell X-ray production cross-sections, total vacancy transfer probabilities, intensity ratios of K X-rays, radiative and radiationless (Auger) vacancy transfer probabilities are shown in Tables 1–6, respectively. The measured values of the K to L_i sub-shell vacancy transfer probability for the elements Cs, Ba and La are listed in Table 3. The overall error in the measured values is estimated to be 3–7%. This error is attributed to the uncertainties in the different parameters used to deduce η_KLi values, namely, the error in the evaluation of the area under the K shell and L sub-shell X-ray peak (4%), the error in the absorption correction factor ratio (<1%) and other systematic errors (2–3%).

In earlier measurements, Puri et al.¹¹ and Ertugrul et al.¹² measured K to L shell vacancy transfer probabilities for elements in the atomic number range 37 ≤ Z ≤ 42 and 73 ≤ Z ≤ 92, respectively. Furthermore, from L to M shell vacancy transfer probabilities were measured for elements in the atomic number region 70 ≤ Z ≤ 92.¹³ In these measurements, the targets were excited by two energies that were below and above the K edge of the elements. In addition, the radiative vacancy transfer probabilities for K to L₂, K to L₃ and K to M shells were measured for elements in the atomic number range 69 ≤ Z ≤ 92 by using a Co-57 radioisotope.⁸ In the present study, an Am-241 radioisotope source was used for excitation of targets. The experimental values, η_KL, were compared with the calculated values and found to be in good agreement with the calculated values for the elements. In the calculations, the K shell radiative rates, based on the relativistic Hartree–Slater theory, were taken from the tabulations of Scofield.¹⁶ The radiationless (Auger) transition rates, based on the relativistic Dirac–Hartree–Slater model were taken from Chen et al.¹⁵

When the excitation energy is above the K edge of an atom, a significant contribution to the L shell vacancy comes from the decay of K shell vacancies. To estimate this contribution, knowledge of K shell to L_i sub-shell vacancy transfer probabilities, η_KLi, is required. While the Kα X-rays (Kα₁ and Kα₂) occur as a result of the transition of vacancy transfer from the L₂ to K (Kα₂) and L₃ to K (Kα₁) shells, the Kβ X-rays are formed as a result of the transition of vacancy transfer from the K shell to M, N and other higher shells. The fractional emission rate for Kα X-rays (f_Kα) gives information on the K to L₂ and L₃ radiative vacancy transfer. The f_Kα values for Cs, Ba and La were calculated as 0.816, 0.814 and 0.812, respectively. So, it can be said that the K shell vacancies decay to L₂, L₃ sub-shells and enhance L₂, L₃ sub-shell vacancies. A similar process occurs with radiationless transition. The K to L₁ vacancy transfer is radiationless only. The K to L₂ and L₃ vacancy transfer is the sum of the radiative and radiationless transfer. As shown in Table 7, while the percentages of radiative transition from the L₂ sub-shell into the K shell are 84%, 85% and 86% for Cs, Ba and La, respectively, the same quantities for the L₃ sub-shell are 86%, 87% and 88%, respectively.

Because the total vacancy transfer from the K shell to L₃ sub-shell is higher than the K shell to L₁ sub-shell and K shell to L₂ sub-shell, respectively, the L₃ X-rays (L_l, Lα, some of Lβ) are the most enhanced. The counts of vacancy in the L₃ sub-shell after photoionization are formed by primary ionization, K shell to L₃ sub-shell vacancy transfer and L₁ to L₃ and L₂ to L₃ Coster–Kronig vacancy transfer. The percentages of the vacancies in the L₃ sub-shell for Cs at 59.5 keV are calculated as 66%, 23%, 6% and 3.6% for K shell to L₃ sub-shell vacancy transfer, L₁ to L₃ sub-shell Coster–Kronig vacancy transfer, L₂ to L₃ sub-shell Coster–Kronig transfer and primary ionization, respectively. Considering that the vacancies from K to L₁ and K to L₂ vacancy transfer are carried out by the L₁ to L₃ , L₂ to L₃ and the L₁ to L₂ and L₂ to L₃ Coster–Kronig transitions (together), the enhancement of L₃ X-rays by the K to L_i sub-shell vacancy transfer are very important for L₃ X-ray production cross sections. Similar calculations were made for L₁ and L₂ sub-shells of Cs. It is seen that the percentage of K to L₁ vacancies in the L₁ sub-shell vacancy is 34% and the percentage of K shell to L₂ sub-shell vacancy and L₁ to L₂ Coster–Kronig vacancy in the L₂ sub-shell are 86% and 7.5%, respectively. According to these results, when the K and L shell are excited together, the vacancy transfer of K to L_i sub-shells should have a very important role. For this reason, knowledge of the vacancy transfer is necessary. In this study, K to L₁, K to L₂ and K to L₃ vacancy transfers (total, radiative and radiationless) were evaluated. Since a lot of physical parameters are calculated theoretically, this method can be called semi-empirical. However, K to L₁, K to L₂ and K to L₃ vacancy transfer probabilities for total, radiative and radiationless transitions have been evaluated in this study for the first time.

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