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For Review O

nly

Laser gain calculations for soft X-ray and XUV radiation

emitted from copper like ions by electron collisional pumping

Journal: Canadian Journal of Physics

Manuscript ID cjp-2016-0204.R1

Manuscript Type: Article

Date Submitted by the Author: 30-May-2016

Complete List of Authors: Saad, Mohamed; Misr University for Science and Technology, Basic Science

Allam, Sami; Cairo University, Physics El-Sherbini, Tharwat; Cairo University, Physics; Misr University for Science and Technology, Basic Science

Keyword: energy levels, oscillator strength, rate coefficients, soft X-ray, Laser gain

https://mc06.manuscriptcentral.com/cjp-pubs

Canadian Journal of Physics

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1

Laser gain calculations for soft X-ray and XUV radiation emitted

from copper like ions by electron collisional pumping

Mohamed A Sayed(1)*, Sami H Allam(2) and Tharwat M El-Sherbini(1), (2)

(1) Basic Science department, Faculty of Engineering, Misr University for Science and Technology, Giza, Egypt.

(2) Laboratory of Lasers and New Materials, Physics Department, Faculty of science, Cairo University, Giza, Egypt.

E-mail: [email protected]

Abstract. Electron impact excitation rate coefficients, Level population densities and gain coefficients for six excited ions with Z = 51, 52, 53, 54, 55, 56 in the copper isoelectronic sequence have been calculated. The electron collisional excitation rate coefficients are calculated according to the analytical formulas of Vriens and Smeets. Fine structure energy levels, transition probabilities and oscillator strengths needed in the calculations have been calculated using Cowan atomic structure code with relativistic corrections for [Ar]3d10nl with n = 4-7 and l = 0-6. The level population densities are calculated by solving the coupled rate equations involving 30 levels. Positive gain coefficients of the possible emitted lines are obtained at three selected electron temperatures namely 1/4, 1/2 and 3/4 of the ionization energy. The present calculated data show promising values for the production of soft X-ray and XUV laser by collisional pumping for the transitions 5p-5s and 6d-5f with wavelengths between 108 Å and 571 Å. The values of the maximum gain coefficient are found to increase with atomic number and their order of magnitude ranges from102 cm-1 to 104 cm−1.

Key words. energy levels, oscillator strength, rate coefficients, soft X-ray, Laser gain.

1. Introduction

Ions in the copper isoelectronic sequence have been extensively studied, because of their deceivingly simple electronic structure and their applications as sources of reference lines in spectroscopy. The study of spectra of highly excited ions of Cu-like ions has received a great deal of attention both experimentally and theoretically [1-10]. Laser produced plasma in the region of Soft X-ray (for wavelengths in the 2 ∼ 200 Å range) and XUV (for wavelengths in 300 ∼ 1000 Å range), have many uses in spectroscopy and is one mean for determining detailed electronic properties of materials [11]. X-ray lasers became reliable, efficient, economical and have several important applications [11]. The first demonstration of a soft x-ray laser has been done by Matthews et al in 1985 [12]. Many efforts to improve the features of soft x-ray lasers, such as the brightness, pulse duration, temporal and spatial coherence, spectral bandwidth and collimation, have been made [13, 14].

Shortly after the demonstration of the first soft X-ray amplification in neon-isoelectronic selenium by Mathews et al. [12], extensive work was done both theoretically and experimentally on other systems. Progress towards the development of soft X-ray lasers with several plasma-ion media of different isoelectronic sequences was achieved at many laboratories. Hagelstein et al. [15] observed stimulated emission in the soft X-ray spectral region in Ni-sequence by the collisional excitation scheme of elements of low atomic numbers. An extensive series of experiments for developing a soft X-ray laser using Li-sequence was performed by Jaeglé et al. [16] using a high power laser for pumping Al-targets. Silfvast and Wood [17] obtained a soft X-ray oscillator and amplifier from Na-sequence by photoionization pumping with laser produced plasmas.

The present paper includes calculations of electron impact excitation rate coefficients of the copper-like ions Sb XXIII, Te XXIV, I XXV, Xe XXVI, Cs XXVII and Ba XXVIII by calculating energy levels, the

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possible allowed transition probabilities and oscillator strengths upon using the Cowan atomic structure code with relativistic corrections. Life times, level populations and gain coefficients for the most promising soft X-ray transitions are also evaluated. Isoelectronic trends for the excitation rate and gain coefficients are also investigated.

2. Theoretical calculations of atomic Structure

The fine structure energy levels, oscillator strengths, and transition probabilities of Cu-like ions for elements from Z = 51 to Z = 56 are calculated by solving the Hartree-Fock equations with relativistic corrections (RHF) through the Cowan suite of atomic codes. [18, 19].

The energies of the various states of the field-free atom are given by the eigenvalues of matrices (one matrix for each possible value of total angular momentum) whose elements connecting states B and B' may be written in the form

( ) ( ) ( )k k

av BB k i j k i j i i

ijk i

B H B E f F g G dδ ζ′ ′ = + + + ∑ ∑l l l l l (1)

Here il and jl are orbital angular-momentum quantum numbers of electrons of a configuration

1 2

1 2 ...... qnn n

ql l l (2)

Where Eav is the central field energy which is the average energy of all states. In practice, modest ad hoc scaling adjustments are applied to the electrostatic and the spin-orbit interaction parameters Fk, Gk, and ζ but

are theoretically defined in terms of the radial portions ( )i

R rl

of the one-electron wavefunctions [18].

The parameter coefficients fk, gk, di in Eq. (1) depend on the angular quantum numbers of the basis

states B and B' of the configuration (2) in the chosen representation, but are independent of the radial wavefunctions and therefore of the particular atom or ion which exhibits this configuration. These coefficients are calculated in detail by R. D. Cowan [19].

The transition probability rate from a higher state JMγ ′ ′ ′ to all lower states M of the level Jγ is

given by [19]:

( )

4 2 2 3064

3 2 1

e aA S

h J

π σ=

′+ (3)

Where ( )1/ /J J J JE E hcσ λ ′ ′= = − is the wave number of transition and S the electric dipole line strength

is defined by:

( )

21

S J P Jγ γ ′ ′= (4)

This quantity is a measure of the total strength of the spectral line including all possible transition between

M, M′. The tensor operator P(1) (first order) in the reduced matrix element is the classical dipole moment for

the atom in units of ea0.

The absorption oscillator strength is related to S by [19]

( )

( )( )

2 208

3 2 1 3 2 1

j i

ij

E Emcaf S S

h J J

π σ −= =

+ + (5)

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In the present calculations the configuration 3d104s of Cu-like ions is taken as the ground state. The

energies of the higher electronic configurations 3d10nili are obtained by adjusting the scalling parameters Eav

and ζ(ri) and are shown in table (S1).

Our calculations of energy levels for the Cu-like ions Sb XXIII, Te XXIV and I XXV have been tabulated in table (S2) and Xe XXIV, Cs XXVII and Ba XXVIII in table (S3) and compared with the available experimental values in refs. [4, 5, 10]. The present calculations for energy levels have a good agreement with experiment within 0.0002 to 0.421%. Also tables (S4-S9) shows the calculated wavelength λ (Å), transition probability Aji (s

-1) and oscillator strength fij of the different allowed transitions for these ions. The present calculated data are compared with the available theoretical and experimental data [6, 7 ,9, 10]. Data in tables (S4 to S9) are used to calculate the total transition probabilities in (sec-1) of the excited states in Cu-like ions for Sb XXIII up to Ba XXVIII and are listed in table (S10).

we do not include any core polarization or core excited states in our calculation, we expect that the RHF oscillator strengths should become more reliable as the nuclear charge increases along the isoelectronic sequence because of the diminishing role of electron correlation which starts strong for neutral copper [20] and decreases as the nuclear charge increases along the copper isoelectronic sequence as indicated in reference [4]. For this reason the highly charged copper-like ions (Sb XXIII, Te XXIV, I XXV, Xe XXVI, Cs XXVII and Ba XXVIII) could be considered to have a simple structure (one electron outside a closed shell) where the energy levels approximately free from effects of configuration mixing.

Tables (S2 and S3) show that the energy levels increases with the nuclear charge. This increase due to the increase in the electrostatic interactions existing between the core charge and the excited electrons as the core charge increases in the various members of the sequence [18].

Tables (S4-S9) show that for most transitions the value of oscillator strength decreases with Z. This is because that the radial integrals for the dipole transition matrix element (represented in the line strength S in equation (5)) decrease when the nuclear charge Z increases. This result is explained by the contraction of the ionic orbitals toward the coordinate origin with increasing Z [21]. In some transitions, the f-value are found to increase with Z, this is due to the increase in transition energy ∆E and wave number σ in equation (5) is much faster than the decrease in radial integrals for the dipole transition matrix element in the same equation.

Studies of f-value regularities are of great practical importance for two reasons: First, they may be utilized to obtain, by graph interpolations, additional accurate oscillator strength values for ions not covered by existing experimental or theoretical data and, second, they may be utilized to evaluate the reliability of existing data from the degree of fit into an established systematic trend.

3- Gain Calculations

3.1 Rate Coefficients

Rate coefficients can be calculated by integrating the corresponding cross section over the Maxwellian energy distribution function. Different types of cross section approximations have been used by different authors to calculate the rate coefficients [22- 25].

Vriens and Smeets [24] constructed analytic semiempirical formulas for the excitation rate coefficient in cm-3s-1 from lower level having principal quantum number (p) to an upper level with quantum number (q) and is given by

( ) ( )0.571.6 10 exp / 0.32

lne pq ee e

pq pq pq pq

e pq pq

kT E kT kTRC f B

kT E R

−× − = + ∆ + + Γ

(6)

Where, both electron temperature, kTe and the Rydberg energy, R are in eV, Epq = Eq-Ep is the energy difference between the two levels and fpq is the absorption oscillator strength. The values of the parameters

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Γpq, ∆pq and Bpq are obtained in detail from [24, 25] and found to depend on the energy levels under consideration, ionization energy and oscillator strength. In addition, the de-excitation rate coefficient is given by

( )exp /pd e

qp pq pq e

q

gC C E kT

g= − (7)

Where gp and gq are the statistical weights of the lower and upper levels respectively.

For the hydrogen atom, p and q are the principal quantum numbers. For atoms or ions with a single electron outside closed shell (present work), the principle quantum numbers are replaced by effective quantum numbers in the above mentioned equations, i.e.

* eff

pi

Rp Z

E= and * eff

qi

Rq Z

E= (8)

Where Epi, Eqi are the ionization energies of the lower and upper levels respectively, R is the Rydberg constant and Zeff is the effective nuclear charge.

3.2 Level Populations

Level populations are calculated by Feldman et al [26-29] by solving the steady-state rate equations:

d e

j ji e ji ji

i j i j j i

d e

e i ij i ij i ij

j i i j j i

N A N C C

N N C N C N A

< < <

< < <

+ +

= + +

∑ ∑ ∑

∑ ∑ ∑ (9)

where Nj, is the level j population densities , Aji is the spontaneous decay rate from level j to level i in s-1, Ne is the electron density in cm-3, Ce

ji and Cdij are the electron collisional excitation and de-excitation rate

coefficient respectively in cm3s-1.

The population of the jth level is obtained from the identity [26-29],

j I T

j e

I T e

N N NN N

N N N

=

(10)

Where NI is the total number density of ions under consideration in all levels, NT is the total number density of ions in all ionization stages. Since the populations calculated from equation (9) are normalized such that

30

1

1n

j

j I

N

N

=

=

=∑ (11)

Where n =30 is the number of all the involved levels of the ion under consideration. The actual number density of any level is obtained from equation (10) where the normalized quantity Nj/NI is obtained from the solution of equation (9). NI/NT is assumed to be 1/4, i.e, 1/4 of the ions are in the considered degree of ionization in the Cu-like ions and the rest are in nearby degree of ionization [26,28]. NT/Ne = 1/κ where κ is the ionization stage of the particular ion under consideration [26, 11].

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Numerical calculations for the rate coefficients (equations 6 and 7) and solution of the steady state rate equations (9) in order to determine level population densities of the forty levels [Ar]3d10nl where n=4-7 and l=0,1,2,3,4,5,6 have been performed for Sb XXIII, Te XXIV, I XXV, Xe XXVI, Cs XXVII and Ba XXVIII using the computer code CRMO developed for collisional radiative model calculations [30]. The ionization energies needed in the calculations of the rate coefficients are obtained from NIST [10]. The most promising transitions that found to give population inversion are 3d105s (2S1/2) -3d105p (2P1/2), 3d105s (2S1/2) -3d105p (2P3/2), 3d105f (2

F5/2)-3d106d (2D3/2) and (2F7/2)-3d106d (2D5/2).

Figure (1) gives the schematic energy level diagram for such laser transitions showing the total transition probability of upper and lower lasing levels for Ba XXVIII. A demonstration of the ratio between the total excitation rate coefficients and the total de-excitation rate coefficients for the various lasing states as a function of electron temperature kT (eV) is shown in Figure (2) for Xe XXVI. To keep on the abundance of the copper like ions, the electron temperatures are chosen to be 1/4, 1/2 and 3/4 of the ionization energy. The Z-dependence of the collisional excitation rate coefficient between the lasing transitions at the three selected electron temperatures is shown in Figure (3) . It is clear that the excitation rates decrease with increasing the ionization stage in the Cu-isoelectronic sequence. This is mainly due to the increase in the energy gap Epq with Z as shown in tables (S2, S3) and also the decrease in the value of the oscillator strength fL with increasing the ionization stage as shown in tables (S4-S9). Figure (4) shows both of the variation of the energy gap Epq in eV and the oscillator strength fL with the ionization stage for the 3d105s (2S1/2)—3d105p(2P1/2) transition where the data of this plot is obtained from tables (S2-S9).

To illustrate the behavior of the reduced population of energy levels of the above mentioned transitions, an example of the calculated reduced populations (fractional level populations per unit statistical weight (Nj/gjNI)) as a function of electron densities at an electron temperature equals to 75% of the ionization energy are listed in Tables (1, 2) for Ba XXVIII and represented in figure (5).

The behavior of level populations shown in tables (1, 2) can be explained as follows: In general, The inversions occur because the higher lasing states 3d105p and 3d106d cannot decay fast to the allowed lower energy levels because they have total transition probabilities lower than that of the states 3d105s and 3d105f respectively as shown in table (S10) and Figure (1) for Ba XXVIII. Moreover, from figure (2) it is clear that the ratio between the total excitation and de-excitation rate coefficients is larger for the upper lasing levels than lower levels. This means that the lower lasing levels depopulate more rapidly than the higher lasing levels which enhances the population inversion between these levels.

In the case of collisional pumping, At low electron densities the reduced populations increase as functions of electron density, this is due to the increase in the collisional excitation rates with density [26, 29 & 31]. At high electron densities (Ne ≥ 1019 cm-3), where the collisional excitation rates exceed the radiative decay rates, the reduced populations are independent of electron density and are approximately equal (see Tables 1, 2). The population inversion is largest when the electron collisional de-excitation rate for the upper level is comparable to the radiative decay rate for this level [31].

3.3 Gain Coefficients

For a Doppler broadened spectral line, the gain coefficient α is given by [32]

1/23

8 2u l

ul u

i u l

N NMA g

kT g g

λα

π π

= −

(12)

λ is the wavelength of the transition between the upper and the lower levels, Aul is the rate of spontaneous emission from the upper level u to the lower levels l, M is the atomic mass of the ion, k is the Boltzmann constant, Ti is the ion temperature, Nu, Nl are the population densities of the two levels u and l and gu, gl are the statistical weights of the upper and lower levels respectively.

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In our calculations, for simplicity it is assumed that Ti in equation (12) equal to the electron temperature Te

which is larger than Ti. Since α ∝ T-1/2 the calculated gain coefficient is a lower limit and should be increased

by ( )1 2

e iT T . The electron temperature are assumed to be less than the ionization energy of the cu-like ions

and chosen to be 1/4, 1/2 and 3/4 the ionization energies of the ions under consideration.

The calculated results of gain coefficients are plotted against the electron density in Figures (6, 7) for different transitions in the copper like ions Sb XXIII, Te XXIV, I XXV, Xe XXVI, Cs XXVII and Ba XXVIII in cm-1. At low densities, the gain increases approximately as Ne. The maximum gain occurs at about the density where collisional depopulation of the upper level becomes comparable to the radiative decay. Above this density the upper level population increases slower than Ne while the lower level populations continue to increase as Ne and therefore, the difference in equation (12) begins to decrease. At very high densities, the population become almost statistically mixed and that difference goes to zero and then to slightly negative values for complete mixing. Also for increasing atomic number Z, the maximum gain occurs at higher electron densities; this is due to as Z increases, the radiative decay rate increases and the collisional excitation rate coefficient decreases and as mentioned in the previous section, the population inversions is largest when the electron collisional de-excitation rate for the upper level (increases with Ne) is comparable to the radiative decay rate for this level [31].

The maximum gain coefficients and the wavelengths for the most promising transitions in the copper like ions Sb XXIII, Te XXIV, I XXV, Xe XXVI, Cs XXVII and Ba XXVIII in cm-1, at different plasma electron temperatures Te (eV), are listed in Tables (3-8). In addition, the tables contains the corresponding life time of the upper and lower laser levels τ (ns) and the plasma electron density Ne (cm-3). The wavelengths of the various transitions are found to lie in the soft X-ray and XUV region of spectra. Unfortunately there are no previous experimental or theoretical data of laser gain for such ions to compare with. However, one of the aims of this article is to identify and predict new laser transitions in order to be a guide line to the experimental investigations.

4. Conclusion and outlook

Our calculations may give an attribute in the production of soft x-ray laser by collisional pumping under actual experimental conditions. In this case additional processes, such as radiative, dielectronic recombination, and perhaps resonances can significantly affect level populations which needs a future study. Experiments involving electron-collisional excitation pumping are explained in detail in reference [11], we suggest a direct electron excitation in a gaseous discharge may be used to produce the desired inversion. This method is used in some of the gaseous ion lasers. With this type of excitation the laser medium itself carries the discharge current. Under suitable conditions of pressure and current, the electrons in the discharge may directly excite the active ions to produce a higher population in certain levels compared to lower levels. In this case the relevant factors are the electron excitation rates and the life times of the various levels.

In the present work we presented the most promising excited states for the Cu-like ions Sb XXIII, Te XXIV, I XXV Xe XXVI, Cs XXVII and Ba XXVIII that can produce soft x-ray laser radiation. These transitions are 5p-5s and 6d-5f with wavelengths between 108 Å and 571 Å.. The value of the maximum gain coefficients ranges from 102 to 104 cm-1 and are found to increase with increasing the atomic number.

5. References

1. S. B. Utter, P. Beiersdorfer, E. Träbert, and E. J. Clothiaux, phys. REV. A 67, 032502 (2003). doi:10.1103/PhysRevA.67.032502.

2. J. F. Seely, J. O. Ekberg, C. M. Brown, U. Feldman, W. E. Behring, Joseph Reader, and M. C. Richardson, Phys. Rev. Lett. 57, 2924 (1986). doi:10.1103/PhysRevLett.57.2924.

3. Joseph Reader and Gabriel Luther Phys. Scr. 24 732 (1981). doi:10.1088/0031-8949/24/4/008. 4. Joseph Reader and Nicolo Acquista, J. Opt. Soc. Am. B 9, 347-349 (1992). doi:

10.1364/JOSAB.9.000347. 5. Lorenzo J. Curtis and Constantine E. Theodosiou, Phys. Rev. A 39, 605 (1989).

doi:10.1103/PhysRevA.39.605.

Page 6 of 15



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7

6. Martin, I., Lavín, C. and Barrientos, C., Int. J. Quantum Chem., 44: 465–474 (1992).

doi:10.1002/qua.560440842.

7. K. T. Cheng, Y. K. Kim, 22, 547-563 (1978). doi:10.1016/0092-640X(78)90023-2. 8. K.T. Cheng and R.A. Wagner, Phys. Rev. A 36, 5435 (1987). doi:10.1103/PhysRevA.36.5435. 9. B M Johnson, K W Jones, D C Gregory, J O Ekberg, L Engström, T H Kruse and J L Cecchi, Phys. Scr.,

32, 210-214, (1985). doi:10.1088/0031-8949/32/3/007.

10. Kramida, A., Ralchenko, Yu., Reader, J., and NIST ASD Team (2014). NIST Atomic Spectra

Database (ver. 5.2), [Online]. Available: http://www.nist.gov/pml/data/asd.cfm[2015, July 27]. National

Institute of Standards and Technology, Gaithersburg, MD.

11. R. C. Elton, "X-Ray Lasers"; Academic press, Inc. New York, ISBN: 0122380800 Lib. Congress Num:

TA1707.E37 (1990).

12. D. L. Matthews, P. L. Hagelstein, M. D. Rosen, M. J. Eckart, N. M. Ceglio, A. U. Hazi, H. Medecki, B.

J. MacGowan, J. E. Trebes, B. L. Whitten, E. M. Campbell, C. W. Hatcher, A. M. Hawryluk, R. L.

Kauffman, L. D. Pleasance, G. Rambach, J. H. Scofield, G. Stone, and T. A. Weaver, Phys. Rev.

Lett. 54, 110 (1985). doi:10.1103/PhysRevLett.54.110

13. Daido H, Reports on Progress in Physics, 65, 1513 (2002).

14. G J Tallents, J. Phys. D: Appl. Phys. , 36, R259 (2003). doi:10.1088/0022-3727/36/15/201.

15. P. L. Hagelstein, S. Basv, M. H. Muendel, J. P. Bravd, D. Tavber, S. Kavshik, J. Goodberlet, T. Y. Hung

and S. Maxon; Int. Colloquium of X-ray Lasers 116, 255 (1990).

16. Pierre Jaeglé, Gérard Jamelot, Antoine Carillon, Annie Klisnick, Alain Sureau, and Hélène Guennou, J.

Opt. Soc. Am. B 4, 563-574 (1987) doi:10.1364/JOSAB.4.000563.

17. W. T. Silfvast and O. R. Wood, J. Opt. Soc. Am. B 4, 609-618 (1987). doi:10.1364/JOSAB.4.000609.

18. R. D. Cowan, “the theory of atomic structure and spectra” Berkeley, Univ. of California Press (1981).

19. R. D. Cowan, J. Opt. Soc. Am.58, 808-818 (1968). doi:10.1364/JOSA.58.000808.

20. Jorgen Carlsson; Phys. Rev. A, 38 , 4, 1702-1710 (1988).

21. Jose Manuel, P. Serrao, J. Quant. Spectrosc. Radiat. Transfer Vol. 35, No. 4, pp. 265-276, (1986).

22. A. Hartgers, J. van Dijk, J. Jonkers and J. A. M. van der Mullen "CR-Model: A general collisional

radiative modeling code" Department of Physics, Eindhoven University of Technology, The

Netherlands, February 5, (2001); A. Hartgers, J. Jonkers and J. A. M. van der Mullen "Collisional

Radiative Models C++" Department of Physics, Eindhoven University of Technology, Eindhoven, The

Netherlands, June (1996).

23. H. W. Drawin. Collision and transport cross sections, eur-cea-fc-383 report. Technical report,

Association Euratom-CEA, Fontenay-aux-Roses (1966).

24. L. Vriens and A. H. M. Smeets, Phys. Rev. A 22, 940, (1980). doi:10.1103/PhysRevA.22.940

25. W.O. Younis, S.H. Allam, and Th.M. El-Sherbini. Can. J. Phys. 88, 257 (2010). doi:10.1139/P10-009.

26. U. Feldman, G. A. Doschek, and J. F. Seely, A. K. Bhatia; J. Appl. Phys. 58, 2909-2915 (1985).

doi:10.1063/1.335838

27. U. Feldman, J. F. Seely and G. A. Doschek; J. de Physique, C6-187 (1986). doi:

10.1051/jphyscol:1986626

28. U. Feldman, J. F. Seely and G. A. Doschek, A. K. Bhatia; J. Appl. Phys. 59, 3953 (1986). doi:

10.1063/1.336695

29. U. Feldman, J. F. Seely, A. K. Bhatia; J. Appl. Phys. J. Appl. Phys. 58, 3954 (1985). doi:

10.1063/1.335569

30. S. H. Allam, " CRMO – Collisional Radiative Model " (2003), private communication.

31. U. Feldman, J. F. Seely and A. K. Bhatia; J. Appl. Phys. 56, 2475 (1984). doi:10.1063/1.334308.

32. U. Feldman, A. K. Bhatia and S. Suckewer; J. Appl. Phys. 54, 2188 (1983). doi: 10.1063/1.332371.

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Figure 1. Energy level diagram of Ba XXVIII. The dashed lines show the most promising laser transitions

and the vertical arrows are labeled with the total radiative decay rates in units of 1011 s-1

Figure 2. Ratio of the total excitation and total de-excitation rate coefficients e d

ij jiC C∑ ∑ of the upper and

lower lasing levels as a function of the electron temperature kT for Xe XXVI.

λ = 334.5 Å

6s (2S1/2)

5s (2S1/2)

4s (2S1/2)

15.6

λ = 461 Å

10.6

2P3/2

2P1/2

4p

5p

2P3/2

2P1/2

6p

2P3/2

2P1/2

9.77

λ = 108 Åλ = 108.7 Å7.87

8.53

4d

2D5/2

2D3/2

2D5/2

2D3/2

5d

2D5/2

2D3/2

6d

13.5

13.9

2F7/2

2F5/2

4f

2F7/2

2F5/2

5f

2G9/2

2G7/2

5g

200 300 400 500 600 700

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

ΣC

e ij/Σ

Cd ji

kT (eV)

5s (2S

1/2)

5p (2P

1/2)

5p (2P

3/2)

5f (2F

5/2)

5f (2F

7/2)

6d (2D

3/2)

6d (2D

5/2)

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Figure 3. Z-dependence of the excitation rate coefficients at three different collisional electron temperatures

Te namely 1/4, 1/2 and 3/4 of the ionization potential of the ion.

Figure 4. The variation of the energy gap Epq and the oscillator strength fL with the ionization stage for the

3d105s (2S1/2) — 3d105p(2P1/2) transition.

Sb X

XIII

Te

XX

IV

I X

XV

Xe

XX

VI

Cs X

XV

II

Ba X

XV

III10

-10

10-9

10-8

10-7

Sb X

XIII

Te

XX

IV

I X

XV

Xe

XX

VI

Cs X

XV

II

Ba X

XV

III

Sb X

XIII

Te

XX

IV

I X

XV

Xe

XX

VI

Cs X

XV

II

Ba X

XV

III

Te=3/4 I.P.T

e=1/2 I.P.

Exci

tation rat

e coef

fici

ent (c

m3s-1

)

Te=1/4 I.P. 5s(

2S

1/2)--5p(

2P

1/2)

5s(2S

1/2)--5p(

2P

3/2)

5f(2F

5/2)--6d(

2D

3/2)

5f(2F

7/2)--6d(

2D

5/2)

Sb XXIII Te XXIV I XXV Xe XXVI Cs XXVII Ba XXVIII0.22

0.23

0.24

0.25

Ion

fl E

pq(eV)

Epq(eV)

fL

22

24

26

28

Page 9 of 15



For Review O

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Figure 5. The reduced population as a function of electron density Ne at 3/4 ionization potential for Ba

XXVIII of the levels (a) 3d105s(2S1/2) , 3d105p(2P1/2) and 3d105p(2P3/2) (b) 3d

105f(2F5/2), 3d105f(2F7/2),

3d106d(

2D3/2) and 3d

106d(

2D5/2).

Figure 6. Gain coefficient � as a function of electron density Ne at 3/4 ionization potential for the transitions

3d105p(2P1/2) —3d105s(2S1/2 and 3d105p(2P3/2) —3d105s(2S1/2) for Sb XXIII, Te XXIV, I XXV, Xe XXVI, Cs

XXVII and Ba XXVIII.

1014

1015

1016

1017

1018

1019

1020

1021

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

1014

1015

1016

1017

1018

1019

1020

1021

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

Reduced

Popula

tion

Ne (cm

-3)

(a)

5s (2S

1/2)

5p (2P

1/2)

5p (2P

3/2)

Reduced

Popula

tion

Ne (cm

-3)

(b)

5f (2F

5/2)

5f (2F

7/2)

6d (2D

3/2)

6d (2D

5/2)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Ion: Sb XXIII

Te = 519.9432 eV

3d105p(

2P

1/2)--3d

105s(

2S

1/2)

3d105p(

2P

3/2)--3d

105s(

2S

1/2)

Ion: Te XXIV

Te = 559.593375 eV 3d

105p(

2P

1/2)--3d

105s(

2S

1/2)

3d105p(

2P

3/2)--3d

105s(

2S

1/2)

Gai

n C

oef

fici

ent

α (×10

4cm

-1) Ion: I XXV

Te = 600.573225 eV

Gai

n C

oef

fici

ent

α (×10

4cm

-1) Ion: Xe XXVI

Te = 642.771225 eV

Ion: Cs XXVII

Te = 687.0522 eV

Ne×10

18(cm

-3)

Ion: Ba XXVIII

Te = 732.467625 eV

Ne×10

18(cm

-3)

Page 10 of 15



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Figure 7. Gain coefficient � as a function of electron density Ne at 3/4 ionization potential for the transitions 3d106d(2D3/2) —3d105f(2F5/2) and 3d106d(2D5/2) —3d105f(2F7/2) for Sb XXIII, Te XXIV, I XXV, Xe XXVI, Cs

XXVII and Ba XXVIII.

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Ion: Sb XXIII

Te = 519.9432 eV

3d106d(

2D

3/2)--3d

105f(

2F

5/2)

3d106d(

2D

5/2)--3d

105f(

2F

7/2)

Ion: Te XXIV

Te = 559.593375 eV

3d106d(

2D

3/2)--3d

105f(

2F

5/2)

3d106d(

2D

5/2)--3d

105f(

2F

7/2)

Ion: I XXV

Te = 600.573225 eV

Gai

n C

oef

fici

ent

α (×10

4cm

-1)

Gai

n C

oef

fici

ent

α (×10

4cm

-1)

Ion: Xe XXVI

Te = 642.771225 eV

Ion: Cs XXVII

Te = 687.0522 eV

Ne×10

18(cm

-3)

Ion: Ba XXVIII

Te = 732.467625 eV

Ne×10

18(cm

-3)

Page 11 of 15



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Table 1. Comparison between the reduced population of the lower energy level 3d105s(2S1/2), and other higher energy levels 3d105p(2P1/2), 3d

105p(2P3/2), as a function of electron density Ne (cm-3) at 3/4 ionization potential in Ba XXVIII.

Ne (cm-3)

Reduced Population Lower Level Upper levels 3d105s(2S1/2) 3d105p(2P1/2) 3d105p(2P3/2)

1.00E+13 1.75566E-11 1.67288E-10 1.50097E-10 1.00E+14 1.75627E-10 1.67286E-09 1.50094E-09 1.00E+15 1.76236E-09 1.67259E-08 1.50064E-08 1.00E+16 1.82201E-08 1.66915E-07 1.49801E-07 1.00E+17 2.33453E-07 1.63728E-06 1.47629E-06 1.00E+18 5.65717E-06 1.43699E-05 1.33384E-05 1.00E+19 1.40836E-04 1.00297E-04 9.64082E-05 1.00E+20 1.10812E-03 9.14775E-04 8.95634E-04 1.00E+21 4.41248E-03 4.11107E-03 4.04895E-03 1.00E+22 6.62241E-03 6.35279E-03 6.26409E-03 1.00E+23 6.98568E-03 6.73041E-03 6.63746E-03 1.00E+24 7.02446E-03 6.77087E-03 6.67747E-03 1.00E+25 7.02836E-03 6.77494E-03 6.6815E-03 1.00E+26 7.02875E-03 6.77535E-03 6.6819E-03 1.00E+27 7.02879E-03 6.77539E-03 6.68194E-03 1.00E+28 7.0288E-03 6.77539E-03 6.68195E-03 1.00E+29 7.0288E-03 6.77539E-03 6.68195E-03 1.00E+30 7.0288E-03 6.77539E-03 6.68195E-03

Table 2. Comparison between the reduced population of the lower energy levels 3d105f(2F5/2), 3d105f(

2F7/2)

and other higher energy levels 3d106d(2D3/2) and 3d106d(2D5/2) as a function of electron density Ne (cm-3) at 3/4 ionization potential in Ba XXVIII.

Ne (cm-3)

Reduced Population Lower Level Higher level Lower Level Higher level 3d105f(2F5/2) 3d106d(2D3/2) 3d105f(2F7/2) 3d106d(2D5/2)

1.00E+13 4.17278E-13 1.44171E-12 3.67409E-13 1.49700E-12 1.00E+14 3.97651E-12 1.44471E-11 3.83075E-12 1.49578E-11 1.00E+15 3.92347E-11 1.46498E-10 3.90276E-11 1.49065E-10 1.00E+16 3.98071E-10 1.53736E-09 3.97599E-10 1.53163E-09 1.00E+17 4.45149E-09 2.00592E-08 4.44812E-09 1.99613E-08 1.00E+18 1.08792E-07 4.96791E-07 1.08714E-07 4.95854E-07 1.00E+19 1.08468E-05 1.08610E-05 1.08390E-05 1.08431E-05 1.00E+20 4.44758E-04 1.34621E-04 4.44440E-04 1.34399E-04 1.00E+21 3.15999E-03 1.98037E-03 3.15773E-03 1.97711E-03 1.00E+22 5.45766E-03 4.49448E-03 5.45376E-03 4.48708E-03 1.00E+23 5.87496E-03 5.00771E-03 5.87076E-03 4.99947E-03 1.00E+24 5.92017E-03 5.06413E-03 5.91593E-03 5.05579E-03 1.00E+25 5.92473E-03 5.06982E-03 5.92049E-03 5.06148E-03 1.00E+26 5.92518E-03 5.07039E-03 5.92094E-03 5.06205E-03 1.00E+27 5.92523E-03 5.07045E-03 5.92099E-03 5.06211E-03 1.00E+28 5.92523E-03 5.07046E-03 5.92099E-03 5.06211E-03 1.00E+29 5.92523E-03 5.07046E-03 5.92099E-03 5.06211E-03 1.00E+30 5.92523E-03 5.07046E-03 5.92099E-03 5.06211E-03

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Table 3. Wavelength λ (Å), life time τu (ps) and maximum gain coefficients αmax for Sb XXIII as a function of electron density Ne at three selected temperatures Te namely quarter, half and three quarter of the ionization energy.

Transition λ (Å) τu (Ps) τL (Ps) Te (eV) Ne (cm-3) αmax (cm-1)

5p(2P1/2)--5s(2S1/2) 571.25 2.04 1.26 173.3144 6.527E+17 3.458E+02

346.6288 9.556E+17 1.473E+03

519.9432 1.051E+18 2.525E+03

5p(2P3`2)--5s(2S1/2) 451.13 1.91 1.26 173.3144 6.527E+17 5.347E+02

346.6288 8.687E+17 2.497E+03

519.9432 1.051E+18 4.378E+03

6d(2D3/2)--5f(2F5/2) 167.39 2.73 1.79 173.3144 2.479E+18 5.502E+02

346.6288 3.299E+18 3.658E+03

519.9432 3.629E+18 6.742E+03

6d(2D5/2)--5f(2F7/2) 166.46 2.91 1.84 173.3144 2.479E+18 7.776E+02

346.6288 2.999E+18 5.217E+03

519.9432 3.629E+18 9.643E+03

Table 4. Wavelength λ (Å), life time τu (ps) and maximum gain coefficients αmax for Te XXIV as a function of electron density Ne at three selected temperatures Te namely quarter, half and three quarter of the ionization energy.


5p(2P1/2)--5s(2S1/2) 545.03 1.76 1.09 186.53112 8.687E+17 5.068E+02

373.06224 1.156E+18 2.130E+03

559.59336 1.399E+18 3.588E+03

5p(2P3/2)--5s(2S1/2) 423.75 1.64 1.09 186.53112 7.897E+17 7.618E+02

373.06224 1.051E+18 3.473E+03

559.59336 1.272E+18 6.110E+03

6d(2D3/2)--5f(2F5/2) 152.29 2.27 1.46 186.53112 2.726E+18 5.840E+02

373.06224 3.629E+18 4.060E+03

559.59336 4.391E+18 7.412E+03

6d(2D5/2)--5f(2F7/2) 151.44 2.43 1.51 186.53112 2.726E+18 8.183E+02

373.06224 3.629E+18 5.733E+03

559.59336 4.391E+18 1.048E+04

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Table 5. Wavelength λ (Å), life time τu (ps) and maximum gain coefficients αmax for I XXV as a function of electron density Ne at three selected temperatures Te namely quarter, half and three quarter of the ionization energy.


5p(2P1/2)--5s(2S1/2) 521.88 1.52 0.94 200.191075 1.051E+18 7.122E+02

400.38215 1.399E+18 2.947E+03

600.573225 1.693E+18 5.000E+03

5p(2P3/2)--5s(2S1/2) 399.02 1.42 0.94 200.191075 9.556E+17 1.032E+03

400.38215 1.399E+18 4.685E+03

600.573225 1.539E+18 8.153E+03

6d(2D3/2)--5f(2F5/2) 139.17 1.90 1.21 200.191075 3.299E+18 5.941E+02

400.38215 4.391E+18 4.284E+03

600.573225 4.830E+18 7.988E+03

6d(2D5/2)--5f(2F7/2) 138.38 2.04 1.24 200.191075 3.299E+18 8.324E+02

400.38215 4.391E+18 6.061E+03

600.573225 4.830E+18 1.134E+04

Table 6. Wavelength λ (Å), life time τu (ps) and maximum gain coefficients αmax for Xe XXVI as a function of electron density Ne at three selected temperatures Te namely quarter, half and three quarter of the ionization energy.


5p(2P1/2)--5s(2S1/2) 499.77 1.33 0.83 214.257075 1.272E+18 9.900E+02

428.51415 1.693E+18 4.046E+03

642.771225 2.048E+18 6.882E+03

5p(2P3/2)--5s(2S1/2) 375.72 1.23 0.83 214.257075 1.156E+18 1.378E+03

428.51415 1.693E+18 6.236E+03

642.771225 1.862E+18 1.074E+04

6d(2D3/2)--5f(2F5/2) 127.69 1.61 1.01 214.257075 3.629E+18 6.171E+02

428.51415 4.830E+18 4.527E+03

642.771225 5.844E+18 8.678E+03

6d(2D5/2)--5f(2F7/2) 126.96 1.73 1.04 214.257075 3.629E+18 8.684E+02

428.51415 4.830E+18 6.428E+03

642.771225 5.844E+18 1.234E+04

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Table 7. Wavelength λ (Å), life time τu (ps) and maximum gain coefficients αmax for Cs XXVII as a function of electron density Ne at three selected temperatures Te namely quarter, half and three quarter of the ionization energy.

Transition λ (Å) τu (ps) τL (ps) Te (eV) Ne (cm-3) αmax (cm-1)

5p(2P1/2)--5s(2S1/2) 480.01 1.16 0.73 229.0174 1.539E+18 1.357E+03

458.0348 2.253E+18 5.552E+03

687.0522 2.479E+18 9.354E+03

5p(2P3/2)--5s(2S1/2) 354.53 1.07 0.73 229.0174 1.399E+18 1.830E+03

458.0348 2.048E+18 8.256E+03

687.0522 2.479E+18 1.422E+04

6d(2D3/2)--5f(2F5/2) 117.59 1.37 0.85 229.0174 3.992E+18 6.272E+02

458.0348 5.844E+18 4.855E+03

687.0522 6.429E+18 9.209E+03

6d(2D5/2)--5f(2F7/2) 116.90 1.48 0.88 229.0174 3.992E+18 8.789E+02

458.0348 5.844E+18 6.848E+03

687.0522 6.429E+18 1.304E+04

Table 8. Wavelength λ (Å), life time τu (ps) and maximum gain coefficients αmax for Ba XXVIII as a function of electron density Ne at three selected temperatures Te namely quarter, half and three quarter of the ionization energy.


5p(2P1/2)--5s(2S1/2) 461.00 1.02 0.64 244.155875 2.048E+18 1.872E+03

488.31175 2.726E+18 7.666E+03

732.467625 2.999E+18 1.279E+04

5p(2P3/2)--5s(2S1/2) 334.48 0.94 0.64 244.155875 1.693E+18 2.403E+03

488.31175 2.479E+18 1.079E+04

732.467625 2.999E+18 1.857E+04

6d(2D3/2)--5f(2F5/2) 108.65 1.17 0.72 244.155875 4.830E+18 6.640E+02

488.31175 6.429E+18 5.133E+03

732.467625 7.779E+18 1.009E+04

6d(2D5/2)--5f(2F7/2) 108.01 1.27 0.74 244.155875 4.830E+18 9.294E+02

488.31175 6.429E+18 7.262E+03

732.467625 7.779E+18 1.431E+04

Page 15 of 15



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For Review Only - University of Toronto T-Space · Mathews et al. [12], ... obtained a soft X-ray oscillator and amplifier from Na-sequence by photoionization pumping with laser produced