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Thermal ionization of Cs Rydberg states
Nuclear Instruments and Methods in Physics Research B 267 (2009) 310–312
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Thermal ionization of Cs Rydberg states I.L. Glukhov *, V.D. Ovsiannikov Department of Physics, Voronezh State University, University Square 1, Voronezh 394006, Russian Federation
a r t i c l e
i n f o
Available online 21 October 2008 PACS: 32.80.Fb 44.40.+a Keywords: Rydberg states Black-body radiation Thermal photoionization
a b s t r a c t Rates Pnl of photoionization from Rydberg ns-, np-, nd-states of a valence electron in Cs, induced by blackbody radiation, were calculated on the basis of the modified Fues model potential method. The numerical data were approximated with a three-term expression which reproduces in a simple analytical form the dependence of Pnl on the ambient temperature T and on the principal quantum number n. The comparison between approximate and exactly calculated values of the thermal ionization rate demonstrates the applicability of the proposed approximation for highly excited states with n from 20 to 100 in a wide temperature range of T from 100 to 10,000 K. We present coefficients of this approximation for the s-, p- and d-series of Rydberg states. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Black-body radiation (BBR) is a ubiquitous perturbation factor affecting every quantum system unless environmental temperature is absolute zero. BBR photons cause redistribution of state populations by means of induced upward and downward electron transitions, shifts of energy levels and sublevels, induce thermal photoionization of neutral atoms. In recent years access to fast computers and suitable software have made possible the study of photoionization of Rydberg states with n = 20–50. These states are of interest at normal temperatures where the maximum of the Planck distribution for BBR coincides with their ionization potentials. On the one hand, photoionization of Rydberg atoms induced by BBR is a very convenient method for ultracold plasma formation, while, on the other hand, the liberated photoelectrons are able to significantly increase the temperature, especially in the presence of a strong magnetic field [1], and so break finely tuned configuration of a quantum system. As a result in many problems of current research information on the rate of thermal ionization of Rydberg (highly excited) states in atoms has become of primary importance. To make numerical data on these systems more easily accessible, simple analytic parametrizations are valuable. In this article we present results of calculations of the BBR-induced photoionization probabilities for Cs atoms in ns-, np- and nd-states for a wide range of principal quantum numbers n. On
* Corresponding author. Address: House 73, Box 19, Rostovskaya street, Voronezh 394090, Russian Federation. Tel.: +7 4732 445719; fax: +7 4732 208755. E-mail address: GlukhovOffi[email protected] (I.L. Glukhov). 0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.10.036
the basis of the calculated data, we propose an analytical formula for approximating the numerical values of the BBR-induced ionization rates. The least squares method was used to determine the coefficients of a second-order polynomial in powers of a level’s binding energy which provides interpolation of the ionization rate for a series of states with a given angular momentum and different principal quantum numbers at a fixed temperature. Given the values of coefficients for different temperatures, a similar interpolation procedure is applied for determining their values at arbitrary temperatures. The interpolation equation for each coefficient is presented in terms of a second-order polynomial in powers of T1/2, where T is the absolute temperature in Kelvin. So, the nine coefficients of the three temperature-dependent polynomials for a given series of an atom are sufficient to provide bulk of numerical data for the rate of ionization of Rydberg states by the ambient BBR. The deviation of results determined in this double interpolation procedure for wide ranges of the principal quantum number n and temperature T from those of exact calculations does not exceed 3–5% in overwhelming majority of cases. The atomic system of units (e = m = ⁄ = 1) is used throughout this article, if not otherwise stated explicitly. 2. Method of calculations Even for T = 10,000 K the energy e of greater part of BBR photons does not exceed 0.1 a.u., that is far below the second (two-electron) ionization potential in all series of Cs. Therefore, a single-electron approach is appropriate for calculating numerical characteristics of the process. In the dipole approximation the thermal photoionization rate of an nl-state is:
311
I.L. Glukhov, V.D. Ovsiannikov / Nuclear Instruments and Methods in Physics Research B 267 (2009) 310–312
Pnl ¼
4 3 a 3
Z
1
l lþ1 e3 M2nl!El1 þ M2nl!Elþ1 de; 2l þ 1 2l þ 1 expðe=kTÞ 1
jEnl j
ð1Þ where a = 1/137.036 — the fine-structure constant, T is the temperature of the ambient BBR, measured in Kelvin, k = 3.1668 106 a.u./K is the Boltzmann constant, jEnl j is the ionization potential of the nl-state with principal quantum number n and orbital momentum l, E ¼ e jEnl j is the energy of photoelectron. Mnl!El0 is dipole matrix element for a transition between the jnli bound state 0 and the state jEl i of continuum:
Mnl!El0 ¼
Z
Table 1 Rates of photoionization (in sec1) from Rydberg s-states in the Cs atom by BBR at T = 300 and 600 K (data from figure) according to Beterov et al. [4]. T
20
30
40
50
60
70
80
300 600
100 650
150 540
120 380
95 270
80 200
60 155
50 120
every nl state and go down to zero (together with the maximum of Pnl) when T ? 0.
1 0
REl0 ðrÞRnl ðrÞr3 dr;
3. Analytical approximations for Pnl
where r is the distance from the atomic center of mass. We have calculated these matrix elements with the wave functions generated using the Fues model potential [2]. This potential allows us to work with analytical formulas outside the atomic core space region where the Rydberg states exist. For a valence electron with nr nodes in its radial wave function the effective principle quantum number is
To present results of our calculations in a compact analytical form, we use a three-term approximation, recently implemented in the case of helium [5]. This generalizes the Spencer’s equation [6] for the approximate thermal photoionization rate Panl of the state jnli to
a0 þ a1 x þ a2 x2 ; m~4 ½expðxÞ 1
m
jEnl j 15:789 ; ¼ ~2 kT Tm
pffiffiffiffiffiffiffiffiffiffiffi m ¼ Z= 2jEnl j;
Panl ¼
where Z is the net charge of atomic ion (for example, Z = 1 for neutral atom). The corresponding effective orbital quantum number is
where a0, a1, a2 are fitted coefficients. a0 determines the general form of the photoionization curve and its asymptotic behaviour for high n, a1 fixes the correct position of the photoionization maximum in a given series of states, a2 adjusts a correct value of Panl at its maximum (Fig. 1). Eq. (2) is valid for states with parameter x below a certain critical value xc (depending on the series, xc = 12) and always covers the area of the photoionization maximum. At temperatures lower than 100 K the maximum of the photoionization curve for a series shifts to high n 40–45 where the smallness of intervals between energy levels smears the maximum position. This circumstance in combination with a shallow slope results in an uncertainty of the ai-coefficients and makes Eq. (2) inapplicable. All ai-coefficients depend slightly on temperature, and to extend the temperature ranges for applicability of Eq. (2), the following temperature approximation may be proposed for T > 100 K:
k ¼ m nr 1: The radial wave function then takes the next form:
2Z 3=2
Rnr l ðrÞ ¼
m2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2k þ 2Þnr Zr=m 2Zr k 2Zr ; e 1 F 1 nr ; 2k þ 2; nr !Cð2k þ 2Þ m m
where 1F1(a;b;x) is a confluent hypergeometric function. For a continuum electron with energy E and effective orbital 0 0 quantum number k , which is a limit of the respective l -series in 1 the atom , the wave function is:
REl0 ðrÞ ¼
rffiffiffiffiffiffi jCðk0 þ 1 þ iZ=,Þj pZ 2, 2, e p Cð2k0 þ 2Þ 0 Z ð2,rÞk ei,r 1 F 1 k0 þ 1 þ i ; 2k0 þ 2; 2i,r ;
,
where the wavenumber is
pffiffiffiffiffiffi
, ¼ 2E: This function is normalized in energy according to:
Z
1
REl ðrÞRE0 l ðrÞr 2 dr ¼ dðE E0 Þ:
0
An extra node has been added to radial wave functions of sstates. Such modification provides a correct model of a non-penetrating ground-state orbital for a valence electron in alkali atoms [3]. Matrix elements calculated with these functions gave photoionization rates (1) for Rydberg Cs s-, p-, d-states in a good agreement with the latest available theoretical and experimental data [4] and Table 1 in this paper. The calculated numerical values of Pnl for T = 300 and 600 K for states with l = 0,1,2 and n from 10 to 100 are presented in Table 2. The table shows the existence of a maximum in the n-dependence of Pnl. For T = 300 K this maximum is near n = 30 and for T = 600 K it shifts to the region of n = 20. For higher T it moves down to yet lower n. As T tends to absolute zero, n ? 1. The absolute values of Pnl increase with temperature for
10.
0
;
where
x¼
s¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi T=100:
ð2Þ
ð3Þ
This expansion reflects correctly the growth of magnitudes for ai-coefficients under growing temperature at 100–1000 K thanks to the opposite signs of bi0 and bi1. At high temperatures (>2000 K) the second and third terms in the approximate expansion (3) become insignificant, and a0-coefficient tends to b00, which is a constant. Such behaviour of the high-temperature asymptotic is a right one. We have evaluated (1) for states with x < xc, n up to 45, and temperatures from 100 K to 1000 K By fitting (2) and (3) to these results using the least squares method, we have determined the following values of for bik for s-, p-, d- states of Cs, respectively, (applicability conditions are presented on the right-hand side of the matrices):
0
2:1
B bik ¼ @ 5:3 0
0:075 0:59 10:0
3:4
8:0
8:7
8:1
B 3:8 bik ¼ @ 4:2 0:71 0:22 0 7:5 7:2 B bik ¼ @ 0:160 7:3 3:6
1
C 5:5 A x < xc ¼ 1:05; 4:9 1 2:9 C 0:90 A x < xc ¼ 1:6; 0:27 1 2:6 C 6:2 A x < xc ¼ 1:4: 2:9
0
An effective orbital quantum number k achieves its limit in the l -series for nr = 8–
100
ai ðTÞ ¼ bi0 þ bi1 s1 þ bi2 s2 ;
0:51 1
m~ ¼
Here the units are sec1.
312
I.L. Glukhov, V.D. Ovsiannikov / Nuclear Instruments and Methods in Physics Research B 267 (2009) 310–312
Table 2 Rates of photoionization (in sec1) from Rydberg states of the Cs atom by BBR at T = 300 and 600 K. Series
s s p p d d
Temperature
Rydberg state principal quantum number n
T, K
10
20
30
40
50
60
70
80
90
100
300 600 300 600 300 600
1.85 2.49 0.095 107 3.19 704
119 647 499 2974 499 2859
192 606 624 2170 517 1824
162 434 474 1383 385 1141
124 308 341 915 276 752
92.7 221 247 632 202 522
70.8 164 183 454 151 377
54.8 125 139 337 115 281
43.1 97.1 107 256 89.7 215
34.4 76.8 84.6 199 71.0 168
Fig. 1. The n-dependence of approximate (solid curve) and exact (crosses) ionization rates for p-states at 300 K. Fig. 2. The temperature dependence of approximate (n = 40-thick solid curve, n = 55-dashed curve, n = 70-thin solid curve) and exact (n = 40-crosses, n = 55circles, n = 70-diamonds) photoionization rates of p-states.
Table 3 Deviations at Tba. Series
d60l (%)
d70l (%)
d80l (%)
d90l (%)
d100l (%)
ns np nl
6.1 0.5 1.86
5.7 3.9 4.7
4.3 8 8.0
2.1 12.5 12
0.56 17.4 16
a
For s- and p-states Tb = 1000 K, and for d-states Tb = 700 K.
Table 4 Deviations at T = 10,000 K. Series
d20l (%)
d40l (%)
d60l (%)
d80l (%)
d100l (%)
ns np nd
60 9.4 20
0.24 3.5 11
6.7 3.8 1.2
5.6 14 9.5
1.1 26 21
4. Deviations of the parametrization from the exact calculated values Eqs. (2) and (3) and matrices of bik-coefficients give a simple parametrization of photoionization rates for s-, p- and d- states of Cs up to n = 100 at temperatures T = 100 10,000K (under the stated condition x < xc). The deviations dnl ¼ ðP nl Panl Þ=Pnl between the data of calculations (1) Pnl and the approximate values P anl as determined from (2) do not exceed 6% (see Fig. 1) for n < 55 at temperatures from T = 100 K to a certain ‘‘boundary” temperature Tb. For higher n-states (up to n = 100) the magnitudes of deviations change
smoothly (see Table 3.) and the approximation (2) accounts for the correct qualitative behaviour. At temperatures higher than Tb, where the contribution of the second and third terms in (2) vanish, the deviations remain low (about 6%) only for 10–20 states in each series. For the other states the deviations gradually increase with temperature (see Table 4. and Fig. 2), but an asymptotical behaviour of P anl remains correct. Acknowledgements This work was supported by the Russian Foundation for Basic Research (RFBR Grant No. 07-02-00279a) and by the joint program BRHE (Basic Research and Higher Education) of the US CRDF (Civilian Research & Development Foundation) and Russian Ministry of Education & Science (Grant No. RUXO-010-VZ-06, Annex BP2M10). References [1] [2] [3] [4]
S.X. Hu, J. Phys. B At. Opt. Mol. Phys. 41 (2008) 081009. N.L. Manakov, V.D. Ovsiannikov, L.P. Rapoport, Phys. Rep. 141 (1986) 319. A.A. Kamenski, V.D. Ovsiannikov, J. Phys. B At. Mol. Opt. Phys. 39 (2006) 2247. I.I. Beterov, D.B. Tretyakov, I.I. Ryabtsev, V.M. Entin, A. Ekers, N.N. Bezuglov, Available from :. [5] I.L. Glukhov, V.D. Ovsiannikov, in: M. Fedorov et al. (Eds.), International Conference on Coherent and Nonlinear Optics, 2007, 28 May–1 June, Minsk, Proceedings of the SPIE, Vol. 6726, 2007, p. 67261F-1 . [6] W.P. Spencer, A.G. Vaidyanathan, D. Kleppner, T.W. Ducas, Phys. Rev. A 24 (1982) 1490.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Thermal ionization of Cs Rydberg states I.L. Glukhov *, V.D. Ovsiannikov Department of Physics, Voronezh State University, University Square 1, Voronezh 394006, Russian Federation
a r t i c l e
i n f o
Available online 21 October 2008 PACS: 32.80.Fb 44.40.+a Keywords: Rydberg states Black-body radiation Thermal photoionization
a b s t r a c t Rates Pnl of photoionization from Rydberg ns-, np-, nd-states of a valence electron in Cs, induced by blackbody radiation, were calculated on the basis of the modified Fues model potential method. The numerical data were approximated with a three-term expression which reproduces in a simple analytical form the dependence of Pnl on the ambient temperature T and on the principal quantum number n. The comparison between approximate and exactly calculated values of the thermal ionization rate demonstrates the applicability of the proposed approximation for highly excited states with n from 20 to 100 in a wide temperature range of T from 100 to 10,000 K. We present coefficients of this approximation for the s-, p- and d-series of Rydberg states. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Black-body radiation (BBR) is a ubiquitous perturbation factor affecting every quantum system unless environmental temperature is absolute zero. BBR photons cause redistribution of state populations by means of induced upward and downward electron transitions, shifts of energy levels and sublevels, induce thermal photoionization of neutral atoms. In recent years access to fast computers and suitable software have made possible the study of photoionization of Rydberg states with n = 20–50. These states are of interest at normal temperatures where the maximum of the Planck distribution for BBR coincides with their ionization potentials. On the one hand, photoionization of Rydberg atoms induced by BBR is a very convenient method for ultracold plasma formation, while, on the other hand, the liberated photoelectrons are able to significantly increase the temperature, especially in the presence of a strong magnetic field [1], and so break finely tuned configuration of a quantum system. As a result in many problems of current research information on the rate of thermal ionization of Rydberg (highly excited) states in atoms has become of primary importance. To make numerical data on these systems more easily accessible, simple analytic parametrizations are valuable. In this article we present results of calculations of the BBR-induced photoionization probabilities for Cs atoms in ns-, np- and nd-states for a wide range of principal quantum numbers n. On
* Corresponding author. Address: House 73, Box 19, Rostovskaya street, Voronezh 394090, Russian Federation. Tel.: +7 4732 445719; fax: +7 4732 208755. E-mail address: GlukhovOffi[email protected] (I.L. Glukhov). 0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.10.036
the basis of the calculated data, we propose an analytical formula for approximating the numerical values of the BBR-induced ionization rates. The least squares method was used to determine the coefficients of a second-order polynomial in powers of a level’s binding energy which provides interpolation of the ionization rate for a series of states with a given angular momentum and different principal quantum numbers at a fixed temperature. Given the values of coefficients for different temperatures, a similar interpolation procedure is applied for determining their values at arbitrary temperatures. The interpolation equation for each coefficient is presented in terms of a second-order polynomial in powers of T1/2, where T is the absolute temperature in Kelvin. So, the nine coefficients of the three temperature-dependent polynomials for a given series of an atom are sufficient to provide bulk of numerical data for the rate of ionization of Rydberg states by the ambient BBR. The deviation of results determined in this double interpolation procedure for wide ranges of the principal quantum number n and temperature T from those of exact calculations does not exceed 3–5% in overwhelming majority of cases. The atomic system of units (e = m = ⁄ = 1) is used throughout this article, if not otherwise stated explicitly. 2. Method of calculations Even for T = 10,000 K the energy e of greater part of BBR photons does not exceed 0.1 a.u., that is far below the second (two-electron) ionization potential in all series of Cs. Therefore, a single-electron approach is appropriate for calculating numerical characteristics of the process. In the dipole approximation the thermal photoionization rate of an nl-state is:
311
I.L. Glukhov, V.D. Ovsiannikov / Nuclear Instruments and Methods in Physics Research B 267 (2009) 310–312
Pnl ¼
4 3 a 3
Z
1
l lþ1 e3 M2nl!El1 þ M2nl!Elþ1 de; 2l þ 1 2l þ 1 expðe=kTÞ 1
jEnl j
ð1Þ where a = 1/137.036 — the fine-structure constant, T is the temperature of the ambient BBR, measured in Kelvin, k = 3.1668 106 a.u./K is the Boltzmann constant, jEnl j is the ionization potential of the nl-state with principal quantum number n and orbital momentum l, E ¼ e jEnl j is the energy of photoelectron. Mnl!El0 is dipole matrix element for a transition between the jnli bound state 0 and the state jEl i of continuum:
Mnl!El0 ¼
Z
Table 1 Rates of photoionization (in sec1) from Rydberg s-states in the Cs atom by BBR at T = 300 and 600 K (data from figure) according to Beterov et al. [4]. T
20
30
40
50
60
70
80
300 600
100 650
150 540
120 380
95 270
80 200
60 155
50 120
every nl state and go down to zero (together with the maximum of Pnl) when T ? 0.
1 0
REl0 ðrÞRnl ðrÞr3 dr;
3. Analytical approximations for Pnl
where r is the distance from the atomic center of mass. We have calculated these matrix elements with the wave functions generated using the Fues model potential [2]. This potential allows us to work with analytical formulas outside the atomic core space region where the Rydberg states exist. For a valence electron with nr nodes in its radial wave function the effective principle quantum number is
To present results of our calculations in a compact analytical form, we use a three-term approximation, recently implemented in the case of helium [5]. This generalizes the Spencer’s equation [6] for the approximate thermal photoionization rate Panl of the state jnli to
a0 þ a1 x þ a2 x2 ; m~4 ½expðxÞ 1
m
jEnl j 15:789 ; ¼ ~2 kT Tm
pffiffiffiffiffiffiffiffiffiffiffi m ¼ Z= 2jEnl j;
Panl ¼
where Z is the net charge of atomic ion (for example, Z = 1 for neutral atom). The corresponding effective orbital quantum number is
where a0, a1, a2 are fitted coefficients. a0 determines the general form of the photoionization curve and its asymptotic behaviour for high n, a1 fixes the correct position of the photoionization maximum in a given series of states, a2 adjusts a correct value of Panl at its maximum (Fig. 1). Eq. (2) is valid for states with parameter x below a certain critical value xc (depending on the series, xc = 12) and always covers the area of the photoionization maximum. At temperatures lower than 100 K the maximum of the photoionization curve for a series shifts to high n 40–45 where the smallness of intervals between energy levels smears the maximum position. This circumstance in combination with a shallow slope results in an uncertainty of the ai-coefficients and makes Eq. (2) inapplicable. All ai-coefficients depend slightly on temperature, and to extend the temperature ranges for applicability of Eq. (2), the following temperature approximation may be proposed for T > 100 K:
k ¼ m nr 1: The radial wave function then takes the next form:
2Z 3=2
Rnr l ðrÞ ¼
m2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2k þ 2Þnr Zr=m 2Zr k 2Zr ; e 1 F 1 nr ; 2k þ 2; nr !Cð2k þ 2Þ m m
where 1F1(a;b;x) is a confluent hypergeometric function. For a continuum electron with energy E and effective orbital 0 0 quantum number k , which is a limit of the respective l -series in 1 the atom , the wave function is:
REl0 ðrÞ ¼
rffiffiffiffiffiffi jCðk0 þ 1 þ iZ=,Þj pZ 2, 2, e p Cð2k0 þ 2Þ 0 Z ð2,rÞk ei,r 1 F 1 k0 þ 1 þ i ; 2k0 þ 2; 2i,r ;
,
where the wavenumber is
pffiffiffiffiffiffi
, ¼ 2E: This function is normalized in energy according to:
Z
1
REl ðrÞRE0 l ðrÞr 2 dr ¼ dðE E0 Þ:
0
An extra node has been added to radial wave functions of sstates. Such modification provides a correct model of a non-penetrating ground-state orbital for a valence electron in alkali atoms [3]. Matrix elements calculated with these functions gave photoionization rates (1) for Rydberg Cs s-, p-, d-states in a good agreement with the latest available theoretical and experimental data [4] and Table 1 in this paper. The calculated numerical values of Pnl for T = 300 and 600 K for states with l = 0,1,2 and n from 10 to 100 are presented in Table 2. The table shows the existence of a maximum in the n-dependence of Pnl. For T = 300 K this maximum is near n = 30 and for T = 600 K it shifts to the region of n = 20. For higher T it moves down to yet lower n. As T tends to absolute zero, n ? 1. The absolute values of Pnl increase with temperature for
10.
0
;
where
x¼
s¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi T=100:
ð2Þ
ð3Þ
This expansion reflects correctly the growth of magnitudes for ai-coefficients under growing temperature at 100–1000 K thanks to the opposite signs of bi0 and bi1. At high temperatures (>2000 K) the second and third terms in the approximate expansion (3) become insignificant, and a0-coefficient tends to b00, which is a constant. Such behaviour of the high-temperature asymptotic is a right one. We have evaluated (1) for states with x < xc, n up to 45, and temperatures from 100 K to 1000 K By fitting (2) and (3) to these results using the least squares method, we have determined the following values of for bik for s-, p-, d- states of Cs, respectively, (applicability conditions are presented on the right-hand side of the matrices):
0
2:1
B bik ¼ @ 5:3 0
0:075 0:59 10:0
3:4
8:0
8:7
8:1
B 3:8 bik ¼ @ 4:2 0:71 0:22 0 7:5 7:2 B bik ¼ @ 0:160 7:3 3:6
1
C 5:5 A x < xc ¼ 1:05; 4:9 1 2:9 C 0:90 A x < xc ¼ 1:6; 0:27 1 2:6 C 6:2 A x < xc ¼ 1:4: 2:9
0
An effective orbital quantum number k achieves its limit in the l -series for nr = 8–
100
ai ðTÞ ¼ bi0 þ bi1 s1 þ bi2 s2 ;
0:51 1
m~ ¼
Here the units are sec1.
312
I.L. Glukhov, V.D. Ovsiannikov / Nuclear Instruments and Methods in Physics Research B 267 (2009) 310–312
Table 2 Rates of photoionization (in sec1) from Rydberg states of the Cs atom by BBR at T = 300 and 600 K. Series
s s p p d d
Temperature
Rydberg state principal quantum number n
T, K
10
20
30
40
50
60
70
80
90
100
300 600 300 600 300 600
1.85 2.49 0.095 107 3.19 704
119 647 499 2974 499 2859
192 606 624 2170 517 1824
162 434 474 1383 385 1141
124 308 341 915 276 752
92.7 221 247 632 202 522
70.8 164 183 454 151 377
54.8 125 139 337 115 281
43.1 97.1 107 256 89.7 215
34.4 76.8 84.6 199 71.0 168
Fig. 1. The n-dependence of approximate (solid curve) and exact (crosses) ionization rates for p-states at 300 K. Fig. 2. The temperature dependence of approximate (n = 40-thick solid curve, n = 55-dashed curve, n = 70-thin solid curve) and exact (n = 40-crosses, n = 55circles, n = 70-diamonds) photoionization rates of p-states.
Table 3 Deviations at Tba. Series
d60l (%)
d70l (%)
d80l (%)
d90l (%)
d100l (%)
ns np nl
6.1 0.5 1.86
5.7 3.9 4.7
4.3 8 8.0
2.1 12.5 12
0.56 17.4 16
a
For s- and p-states Tb = 1000 K, and for d-states Tb = 700 K.
Table 4 Deviations at T = 10,000 K. Series
d20l (%)
d40l (%)
d60l (%)
d80l (%)
d100l (%)
ns np nd
60 9.4 20
0.24 3.5 11
6.7 3.8 1.2
5.6 14 9.5
1.1 26 21
4. Deviations of the parametrization from the exact calculated values Eqs. (2) and (3) and matrices of bik-coefficients give a simple parametrization of photoionization rates for s-, p- and d- states of Cs up to n = 100 at temperatures T = 100 10,000K (under the stated condition x < xc). The deviations dnl ¼ ðP nl Panl Þ=Pnl between the data of calculations (1) Pnl and the approximate values P anl as determined from (2) do not exceed 6% (see Fig. 1) for n < 55 at temperatures from T = 100 K to a certain ‘‘boundary” temperature Tb. For higher n-states (up to n = 100) the magnitudes of deviations change
smoothly (see Table 3.) and the approximation (2) accounts for the correct qualitative behaviour. At temperatures higher than Tb, where the contribution of the second and third terms in (2) vanish, the deviations remain low (about 6%) only for 10–20 states in each series. For the other states the deviations gradually increase with temperature (see Table 4. and Fig. 2), but an asymptotical behaviour of P anl remains correct. Acknowledgements This work was supported by the Russian Foundation for Basic Research (RFBR Grant No. 07-02-00279a) and by the joint program BRHE (Basic Research and Higher Education) of the US CRDF (Civilian Research & Development Foundation) and Russian Ministry of Education & Science (Grant No. RUXO-010-VZ-06, Annex BP2M10). References [1] [2] [3] [4]
S.X. Hu, J. Phys. B At. Opt. Mol. Phys. 41 (2008) 081009. N.L. Manakov, V.D. Ovsiannikov, L.P. Rapoport, Phys. Rep. 141 (1986) 319. A.A. Kamenski, V.D. Ovsiannikov, J. Phys. B At. Mol. Opt. Phys. 39 (2006) 2247. I.I. Beterov, D.B. Tretyakov, I.I. Ryabtsev, V.M. Entin, A. Ekers, N.N. Bezuglov, Available from :