Energy transfer reaction K(4s) + K(7s) → K(4s) + K(5f), theory compared with experiment

Journal of Quantitative Spectroscopy & Radiative Transfer 227 (2019) 152–169

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Energy transfer reaction K(4s) + K(7s) → K(4s) + K(5f), theory compared with experiment M. Głódz´ a,∗, A. Huzandrov a, S. Magnier b, L. Petrov c,a, I. Sydoryk a, J. Szonert a, J. Klavins d, K. Kowalski a a

Institute of Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland Univ. Lille, CNRS, UMR 8523- PhLAM- Laboratoire de Physique des Lasers, Atomes et Molécules, F-59000 Lille, France Institute of Electronics, Bulgarian Academy of Sciences, 1784 Sofia; Boul. Tsarigradsko Shosse 72, Bulgaria d Institute of Atomic Physics and Spectroscopy, University of Latvia, 1586 Riga, Latvia b c

a r t i c l e

i n f o

Article history: Received 29 December 2017 Revised 19 December 2018 Accepted 9 January 2019 Available online 11 January 2019 Keywords: Potassium Atomic inelastic collisions Thermal energies Quasi-molecular treatment Theoretical vs experimental cross sections

a b s t r a c t A comparison between theory and experiment, concerning the K(4 s) + K(7s)→K(4 s) + K(5f) reaction of excitation energy transfer in thermal collisions, is presented. The cross sections for this process are calculated for the potassium vapour temperatures in the range of 310–10 0 0 K. The calculations are based on the theoretical adiabatic K2 potential energy curves and on the use of the multicrossing LandauZener model. The experiment was carried out using the method of spectroscopy with resolution in time. The signals of the direct-fluorescence decay from pulsed-laser-excited 7s state, and of the sensitizedfluorescence from 5f state, were registered and analysed. In the temperature range of the experiment, of 428–451 K, the calculated cross sections vary little, from 1.994 × 10−14 cm2 to 1.992 × 10−14 cm2 , and agree well with the value 1.8(7)×10−14 cm2 , which is the average of the corresponding experimental cross section results. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The efficiency of the processes of excitation energy transfer (ET) in atomic thermal collisions is of interest in various fields of science. The most obvious fields of this kind are: spectroscopy of atoms in a gas phase including determination of atomic states natural-lifetimes, the study of kinetics related to chemical processes, the laboratory-plasma physics. This also applies to astrophysics of cosmic plasmas [1–3]. Among other elements, also alkali metal atoms (including potassium atoms) play a role in cosmic collisional encounters. Characteristics of atomic collisions are also crucial in some branches of technology, where there is a need to asses either a positive or negative impact of collision initiated processes. For example, the operation principles of the collision lasers operating on atomic transitions, are directly founded on ET collisions [4], while the development of the high-power diode-pumped alkali vapour lasers (DPALs) may suffer from a kind of ET process, which is energy pooling (EP)1 . Principles of DPALs rely also on ET in collisions ∗

Corresponding author. ´ E-mail address: [email protected] (M. Głódz). 1 The EP reaction starts with two colliding atoms, both initially in excited states, and ends up with one atom in a highly excited state, and the other in the ground state [5]. https://doi.org/10.1016/j.jqsrt.2019.01.009 0022-4073/© 2019 Elsevier Ltd. All rights reserved.

with buffer gas, in the scheme involving the ground- and firstexcited-states of alkali metal atoms in the following transitions: laser-diode photo-pumping (D2 ) LASING (D1 ) ET s1/2 −−−−−−−−−−−−−−−−−−− −−→ p3/2 → p1/2 −−−−−−→ s1/2 . EP is one of the processes which may excite, for example, higher nd and their neighbouring n’s states in collisions among p1/2 and p3/2 atoms. Then the D1 laser- and D2 pump- photons can photoionize alkali atoms from these higher nd and n’s states, which leads to loss of neutral alkali atoms in the DPAL gain medium, thus the DPAL performance can be significantly degraded [6,7]. Alkali metal atoms, characterized by single valence electrons, are most often used in energy transfer studies. In this article, an investigation is presented of the ET process in thermal vapour-cell collisions of the excited potassium atoms K(7s) with the parent ground state atoms K(4 s) for the case of reaction

K(4s ) + K(7s ) → K(4s ) + K(5f ) − 332cm−1

(1)

Over the last few decades, a lot of work on alkali atom collisions has been done on energy pooling (and on the reverse energy pooling) also with participation of potassium atoms (see, e.g., [5,8–11]. Less often reactions of ET in collisions with ground state atoms were studied. This particularly refers to the involvement of atoms in an f-state (n, l = 3) with the exception of widely investigated specific kind of ET, namely, the quasi-elastic l-mixing

M. Głód´z, A. Huzandrov and S. Magnier et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 227 (2019) 152–169

between nearly degenerate alkali states of the same hydrogen-like manifold, see e.g., Chapter 5.5.2 in [12], and [13]. To the best of our knowledge, reaction (1) has not been studied before, neither theoretically nor experimentally. In fact, for potassium K-K∗ collisions it is even hard to find in the literature reports on other analogous reactions. An experiment concerning the reaction

K(4s ) + K(6s ) → K(4s ) + K(4d ) + 53cm−1

(2)

was reported in Ref. [14] and the determined cross section (relatively small) was discussed in the review of s→d transfer reactions [15]. One of the aims of the present work is to contribute at filling gaps in characteristics of K-K∗ collisions and, in particular, to address the case of a considerable difference in the angular momentum quantum numbers l = 3 between the final f state and the initial s-state. From the theoretical perspective, it is rare to reach so high alkali excited states (as are f states) in calculations, because often only the two first asymptotes, s + s and s + p, are needed in calculations concerning cold molecules or transfer reactions. Usually, people look at molecular states located below the first p + p limit and, in particular, at those that dissociate into ns + np. In view of this, the calculated cross sections for a collisional process involving f states, as confronted with measured ones, could, in principle, serve as a test to check the accuracy of the relevant molecular theoretical data; ever more so since the spectroscopy of molecular states correlated with f states is extremely rare. The possible role, in this respect, of the experimental results, presented in this article, is discussed in its final section (Section 4). A theoretical interpretation of the process of collisional reaction (1) is presented in Section 2. The experimental procedures, data processing and the results are described in Section 3. Finally, the theoretical results are compared with the experimental ones in Section 4. 2. Theoretical Potential energy curves (PECs) involved in the interpretation of the excitation energy transfer reaction (1) have been determined in the framework of a pseudopotential method, in a similar way as in Ref. [16]. One s and one p Gaussian-type orbitals have been added to the basis set used in Ref. [16] to reach the K(4s) + K(7s) and K(4s) + K(5f) asymptotes. An averaged value of the difference between computed and experimental energies is found to be equal to 12 cm−1 for the 7 dissociation limits located above the K(4p) + K(4p) asymptote (i.e. up to K(4 s) + K(5f)), while in the previous calculations this value was equal to 13 cm−1 for the 5 asymptotes above K(4p) + K(4p) (i.e. up to K(4 s) + K(5d)). The dissociation energy of the X1  g + ground state is now estimated to be 4373 cm−1 at Re = 7.37a0 and has been slightly improved (4332 cm−1 at 7.35a0 for previous calculations and 4440 cm−1 (4451 cm−1 ) at 7.42a0 for experiments given in Ref. [16]). Equilibrium position, transition and dissociation energies are reported in Table 1 for 50 1,3  g,u + electronic states and most of these results have never been published before. A satisfying agreement with previous experimental and theoretical data is obtained in the particular in case of the electronic states located below the K(4p) + K(4p) asymptote. Equilibrium positions and depths of potential energy wells are well reproduced at short and intermediate distances even if molecular states are experimentally unknown. Large differences between these two calculations appear mainly in the case of higher excited states as, for example, the 121  g + or the 123  u + states correlated to K(4 s) + K(5d) where the structures are not the same in the previous and present calculations. These changes are mainly due to the basis sets that

153

have been used and are illustrated by more or less potential wells and/or energy barriers. For all these computed states, correlated from K(4 s) + K(4 s) up to K(4 s) + K(5f), the averaged errors were previously equal to Re = 0.08a0 , Te = 76cm−1 and De = 81cm−1 (in Ref. [16]) and are now found to be Re = 0.08a0 , Te = 68cm−1 and De = 74cm−1 . In the case of the most excited dissociation limits (i.e. from K(4p) + K(4p) up to K(4 s) + K(5f)), we have obtained Re = 0.13a0 , Te = 26cm−1 and De = 73cm−1 to be compared with Re = 0.13a0 , Te = 34cm−1 and De = 85cm−1 (in Ref. [16]). This analysis shows that the present potential energy curves can be used to interpret collisional processes and experiments on the spectroscopy of highly excited states of the K2 molecule. These potential energy curves should be very useful to validate the presence of double inner wells. + PECs for 1,3 g,u molecular states correlated to the asymptotes from K(4s) + K(6p) and up to K(4s) + K(5f) have been considered and are displayed in Figs. 1 and 2 for 1  g,u + and 3  g,u + , respectively. Numerous avoided crossings are present, in particular at intermediate internuclear distances. They are partly due to the ioniccovalent interaction, the ionic state being in the ground state or in an excited state. Landau-Zener parameters (position Rn , diabatic potential energy Un , splitting between the two PECs Vn , difference of the slopes Fn at Rn ) have been extracted from the present potential energy curves for each avoided crossing. They are listed in Table 2, and the Landau-Zener probability value Pn , corresponding to a transition between the two curves involved in the avoided crossing n, is given for one value of the impact parameter (b = 0) and one collisional energy (E = 400 K). The probabilities are estimated to be ≈ 0.990 for several impact parameters and collisional energies, which leads to a nearly full population transfer from one curve to the other at the corresponding avoided crossing. All avoided crossings are then considered as diabatic and contribute to the process. Total cross sections σ (υ ) for each molecular symmetry have been determined by numerically integrating the final population of each possible exit channel correlated with 4s + 5f over different impact parameters. These values are still dependent on the centreof-mass collision energy and, consequently, on the relative velocity υ of the colliding partners. Having in mind the vapour-cell experiment in which thermally averaged collisional transfer rates are measured (see Section 3), the relevant thermally averaged theoretical cross sections have been calculated. They are defined as

σ¯ (T ) = σ (υ )υ/υ¯

(3)

where σ (υ )υ = k is the thermally averaged rate constant corresponding to a certain vapour temperature T. The conventional Maxwell distribution over relative collision velocities has been used. υ¯ is the mean relative velocity of K atoms at temperature T

υ¯ =



8kB T /π μ

(4)

with μ - their reduced mass, and kB - the Boltzmann constant. From now on we introduce the abbreviation σ¯ (T ) ≡ σ¯ . Sum and contributions of each molecular symmetry are displayed in Fig. 3 for the thermally averaged cross sections. The major contribution comes from the avoided crossings of 3 g+ and 1  + molecular states. g Comparison of the theoretical results with the experimental ones, obtained for a range of temperatures centred at about 440 K, is very satisfying, as presented in Section 4.

154

M. Głód´z, A. Huzandrov and S. Magnier et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 227 (2019) 152–169 Table 1 Main spectroscopic constants (Re (a0 ), De and Te in cm-1 for all 1,3  g,u + molecular states, correlated from K(4s)+K(4s) up to K(4s)+K(5f) asymptotes. Present results are compared with available theoretical and experimental data as listed in Ref. [16]. Te (cm−1 )

De (cm−1 )

Molecular state and asymptote

Determination

Re (a0 )

11  g + (4s+4s)

Expt. Previous calculations Present work

7.42 7.35 7.37

21  g + (4s+4p)

Previous calculations Present work

9.63 9.63

14377 14391

2979 2994

31  g + (4s+5s)

Previous calculations Present work

8.58 8.60

20348 20355

5002 5030

41  g + (4s+3d)

Previous calculations Present work

8.98 8.99

21410 21438

4458 4443

51  g + (4s+5p)

Expt. Previous calculations Present work Previous calculations Present work Previous calculations Present work

8.68 8.56 8.56 14.80 14.82 22.29 22.29

25376 25308 25339 27014 27030 26427 26445

3793 3742 3732

Expt. Previous calculations Present work Previous calculations Present work Previous calculations Present work

8.42 8.36 8.38 19.90 19.88 32.80 32.85

25882 25819 25844 29652 29673 29169 29191

Previous calculations Present work Previous calculations Present work Previous calculations Present work

8.61 8.54 9.80 9.60 10.79 10.79

27964 27985 28102 28120 27963 27981

2417 2416

81  g + (4s+4d)

Expt. Previous calculations Present work

8.84 8.52 8.60

28065 28052 28047

3783 3678 3699

91  g + (4s+6s)

Expt. Previous calculations Present work Previous calculations Present work Previous calculations Present work

8.46 8.46 8.35 13.30 13.53 14.39 14.38

28233 28206 28210 30791 30807 30780 30798

3668 3579 3603

1004 1015

101  g + (4s+4f)

Previous calculations Present work

9.80 9.83

29473 29436

3033 3093

111  g + (4s+6p)

Previous calculations Present work Present work

9.17 8.79 20.46

30031 29759 32978

3307 3600

Previous calculations Present work Previous calculations Present work Previous calculations Present work

8.72 9.13 10.20 21.29 11.37 Not found

31075 30131 31323 33982 31236

3450 4408

131  g + (4s+7s) Hump Outer well

Present work Present work Present work

8.86 9.90 10.81

31478 31718 31566

3159 3072

141  g + (4s+5f)

Present work

9.87

31877

3902

11  u + (4s+4p)

Expt. Previous calculations Present work

8.59 8.56 8.56

11108 11041 11073

6328 6314 6312

21  u + (4s+5s)

Expt. Previous calculations Present work Expt. Previous calculations Present work Expt. Previous calculations Present work

9.28 9.20 9.25 10.51 10.40 10.45 14.04 14.05 14.06

22019.9 21956 21983 22081.8 21994 22018 21726.5 21634 21657

3772 3717 3728

Expt. Previous calculations Present work

8.91 8.83 8.84

23863 23573 23604

2123 2295 2277

Hump Outer well 61  g + (4p+4p)

Hump Outer well 71  g + (4p+4p) Hump Outer well

Hump Outer well

Hump 121  g + (4s+5d) Hump Outer well

Hump

Outer well

31  u + (4s+3d)

4440 (4451) 4332 4373

2639 2626 4621 4560 4552

1208 1205 2416 2412

3290

5002 3402

(continued on next page)

M. Głód´z, A. Huzandrov and S. Magnier et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 227 (2019) 152–169 Table 1 (continued) Molecular state and asymptote

Determination

Re (a0 )

Te (cm−1 )

4 u

Previous calculations Present work Previous calculations Present work Previous calculations Present work Previous calculations Present work Previous calculations Present work

8.73 8.74 10.20 10.03 11.65 11.66 15.20 15.20 22.35 22.38

26502 26535 26623 26654 26498 26521 26900 26920 26425 26443

Previous calculations Present work Previous calculations Present work Previous calculations Present work

8.55 8.55 15.20 15.24 32.75 32.80

27294 27322 30257 30281 29164 29186

2570 2560

61  u + (4s+6s)

Previous calculations Present work

9.90 9.91

28367 28393

3417 3414

71  u + (4s+4f)

Previous calculations Present work

8.56 8.54

29233 29193

3274 3336

81  u + (4s+6p)

Previous calculations Present work Previous calculations Present work Previous calculations Present work Present work

9.25 9.27 12.80 12.87 14.34 14.31 21.32

29669 29514 31865 31900 31729 31753 32954

3669 3336

8.52 9.52 10.34 10.67 21.47

30410 30442 31220 31197 34132

4115 4099

Hump Outer well Hump

Previous calculations Present work Present work Present work Present work

101  u + (4s+7s) Hump Outer well

Present work Present work Present work

8.84 11.97 12.49

30690 32412 32322

2315

111  u + (4s+5f)

Present work

10.08

31367

4412

13  g + (4s+4p)

Previous calculations Present work

8.98 8.99

13570 13597

3786 3788

23  g + (4s+5s)

Previous calculations Present work

8.05 8.06

19376 19412

5874 5974

33  g + (4s+3d)

Previous calculations Present work

8.48 8.49

23564 23596

2304 2285

43  g + (4s+5p)

Expt. Previous calculations Present work

8.27 8.31 8.32

25547 25577 25610

3618 3472 3461

53  g + (4s+4d)

Previous calculations Present work

8.50 8.51

27102 27129

4628 4617

63  g + (4s+6s)

Previous calculations Present work

8.59 8.62

28036 28047

3705 3760

73  g + (4s+4f)

Previous calculations Present work

8.39 8.41

28239 28266

4268 4263

83  g + (4s+6p)

Previous calculations Present work

8.65 8.48

29342 29034

3996 4326

93  g + (4s+5d)

Previous calculations Present work

8.40 8.67

30609 29996

3916 4545

1

+

(4s+5p)

Hump Outer well (1) Hump Outer well (2) 51  u + (4s+4d) Hump Outer well

Hump Outer well Hump 91  u + (4s+5d)

De (cm−1 ) 2564 2536

2569 2550

2642 2628 4436 4424

1609 1607

3344 3947

103  g + (4s+7s)

Present work

8.73

30947

3610

113  g + (4s+5f) Hump Outer well

Present work Present work Present work

8.45 9.45 9.48

32140 32336 32029

2910 3021

13  u + (4s+4s)

Expt. Previous calculations Present work

10.91 10.82 10.73

4196 4091 4082

254 241 278

23  u + (4s+4p)

Dissociative state

33  u + (4s+5s)

Previous calculations Present work

9.98 9.98

21777 21790

3574 3595

43  u + (4s+3d)

Previous calculations Present work

9.32 9.33

24192 24217

1676 1664

53  u + (4s+5p)

Previous calculations Present work

8.52 8.52

26329 26357

2720 2714

(continued on next page)

155

156

M. Głód´z, A. Huzandrov and S. Magnier et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 227 (2019) 152–169 Table 1 (continued) Molecular state and asymptote

Determination

Re (a0 )

Te (cm−1 )

63  u + (4p+4p)

Previous calculations Present work

8.59 8.60

27384 27408

2996 2987

73  u + (4p+4p)

Previous calculations Present work

8.47 8.48

28646 28633

1734 1763

83  u + (4s+4d)

Previous calculations Present work

8.61 8.59

29243 29206

2486 2539

93  u + (4s+6s)

Previous calculations Present work Present work Present work Present work Present work

8.50 8.55 11.56 12.17 12.60 13.49

30407 30171 31275 31164 31230 31101

1377 1642

Previous calculations Present work Present work Present work Present work

9.42 8.52 10.95 11.42 13.91

30745 30444 31540 31449 32422

1762 2085

Previous calculations Present work Present work Present work

8.83 9.49 11.54 12.11

31386 30918 32112 31926

1952 2441

Hump Outer well Hump Outer well

Previous calculations Present work Previous calculations Present work Previous calculations Present work Present work Present work Present work Present work

10.11 8.42 11.00 9.26 11.57 10.78 13.04 13.72 15.21 15.73

32249 32412 32660 32388 32330 31787 33050 32917 33438 33384

133  u + (4s+7s) Hump Outer well Hump Outer well

Present Present Present Present Present

work work work work work

9.36 10.29 11.52 14.23 15.01

32893 33427 32487 33561 33480

143  u + (4s+5f) Hump Outer well Hump Outer well

Present Present Present Present Present

work work work work work

9.08 9.98 10.34 11.82 13.00

33572 33479 33628 34283 33695

Hump Outer well Hump Outer well 103  u + (4s+4f) Hump Outer well Hump 113  u + (4s+6p) Hump Outer well 123  u + (4s+5d) Hump Outer well

De (cm−1 )

648 712

1080

1433 2275 2129

2194 2754 1624 1157 1745 2150 1157 1479 1422 1355

Fig. 1. Potential energy curves of 1  g + (solid line) and 1  u + (dashed line) electronic states dissociating into K(4s)+K(6p) and up to K(4s) + K(5f). Energy values and internuclear distances are given in atomic units.

M. Głód´z, A. Huzandrov and S. Magnier et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 227 (2019) 152–169

157

Fig. 2. Potential energy curves of 3  g + (solid line) and 3  u + (dashed line) electronic states dissociating into K(4s) + K(6p) and up to K(4s) + K(5f). Energy values and internuclear distances are given in atomic units.

Table 2 Landau–Zener parameters (Rn , Un , Vn , Fn ) of each avoided crossing n present in the 1,3 g+,u potential energy curves involved in the energy transfer reaction K(4s)  + K(7s)→K(4s) +K(5f).

2E n| Probability value (Pn (b, E ) = exp(−2π υ (|Vb,E μ (1 − )Fn ) with υRn (b, E ) = Rn dicated for b = 0 and E = 400K. Parameter values are given in atomic units. 2

n 1



+ g

1 2 3 4 5 6 1

u+

1 2 3 4 5 6 7 8 9 3

g+

1 2 3 4 5 6 7 3

u+

1 2 3 4 5 6 7 8 9 10 11 12

b2 Rn 2

−

Un E

)) is in-

States

Rn

Un

Vn

Fn

Pn (0,400K)

4s6p − 4s5d 4s5d − 4s7s 4s5d − 4s7s 4s7s − 4s5f 4s7s − 4s5f 4s7s − 4s5f

13.55 07.34 20.64 09.88 18.54 29.33

−0.192929 −0.193067 −0.183821 −0.194020 −0.183670 −0.181303

3.72 × 10−4 1.32 × 10−3 3.07 × 10−4 3.62 × 10−4 1.47 × 10−4 6.94 × 10−5

3.97 × 10−2 1.22 × 10−1 3.08 × 10−2 3.64 × 10−2 1.52 × 10−2 7.02 × 10−3

0.9933 0.9727 0.9944 0.9931 0.9971 0.9986

4s6p − 4s5d 4s6p − 4s5d 4s6p − 4s5d 4s5d − 4s7s 4s5d − 4s7s 4s5d − 4s7s 4s5d − 4s7s 4s7s − 4s5f 4s7s − 4s5f

07.11 12.73 18.71 09.64 10.20 20.89 21.45 10.24 16.15

−0.195479 −0.193568 −0.187946 −0.198114 −0.196582 −0.183528 −0.183382 −0.196058 −0.184252

6.27 × 10−4 9.16 × 10−5 1.77 × 10−3 2.83 × 10−4 2.04 × 10−4 4.23 × 10−8 5.62 × 10−7 2.39 × 10−4 4.38 × 10−5

5.36 × 10−2 1.08 × 10−2 1.78 × 10−1 3.11 × 10−2 2.24 × 10−2 3.66 × 10−4 2.24 × 10−4 2.61 × 10−2 5.53 × 10−3

0.9861 0.9985 0.9664 0.9951 0.9965 0.9999 0.9999 0.9958 0.9993

4s6p − 4s5d 4s6p − 4s5d 4s6p − 4s5d 4s5d − 4s7s 4s5d − 4s7s 4s7s − 4s5f 4s7s − 4s5f

05.09 13.87 19.12 11.72 15.69 05.32 35.52

−0.139405 −0.191652 −0.186644 −0.195442 −0.187088 −0.144991 −0.181107

1.07 × 10−3 9.85 × 10−4 3.71 × 10−4 5.13 × 10−4 7.32 × 10−5 1.58 × 10−3 2.29 × 10−5

1.89 × 10−2 1.01 × 10−1 3.74 × 10−2 5.38 × 10−2 8.34 × 10−3 1.08 × 10−1 2.32 × 10−3

0.8726 0.9817 0.9928 0.9907 0.9987 0.9500 0.9995

4s6p − 4s5d 4s6p − 4s5d 4s6p − 4s5d 4s6p − 4s5d 4s5d − 4s7s 4s5d − 4s7s 4s5d − 4s7s 4s7s − 4s5f 4s7s − 4s5f 4s7s − 4s5f 4s7s − 4s5f 4s7s − 4s5f

05.65 11.12 13.83 15.76 09.32 11.97 15.09 08.45 10.31 13.18 18.69 35.76

−0.157717 −0.193503 −0.189092 −0.186836 −0.189729 −0.190228 −0.186456 −0.185452 −0.186137 −0.185999 −0.181810 −0.181101

1.75 × 10−4 6.39 × 10−5 2.28 × 10−4 4.73 × 10−5 7.12 × 10−4 5.13 × 10−4 1.29 × 10−4 2.56 × 10−4 4.65 × 10−4 7.38 × 10−4 1.05 × 10−4 2.35 × 10−5

−2.19 × 10−2 9.58 × 10−3 2.38 × 10−2 4.92 × 10−2 7.05 × 10−2 3.47 × 10−2 1.36 × 10−2 2.17 × 10−2 4.60 × 10−2 7.44 × 10−2 1.09 × 10−2 2.41 × 10−3

1.0 0 0 0 0.9992 0.9957 0.9991 0.9862 0.9945 0.9976 0.9941 0.9909 0.9858 0.9980 0.9995

158

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Fig. 3. Variation of the thermally averaged cross sections for the K(4s) + K(7s) → K(4s) + K(5f) reaction as a function of the centre of mass collision energy expressed in Kelvins.

3. Experiment 3.1. Experimental system and data acquisition The experiment regarding ET reaction (1) was carried out in a spectral cell with potassium vapour. The K atoms were excited to the 7s state by laser light pulses at the wavelength λ = 660.4 nm in the two-photon 4s→7s transition. The development in time of the direct-fluorescence I7s ≡ I7s (t) and of the collisionally sensitized fluorescence I5f ≡ I5f (t) was registered on the respective transitions with unresolved fine-structure components (wavelengths in brackets): 7s→4p (579.5 nm) and 5f→3d (1102.1 nm). For K levels and transitions see Fig. 4. Fig. 5 shows the main components of the experimental system. The spectral cell was essentially a glass cylinder (length of about 85 mm, and internal diameter of 25 mm) with both ends blinded. The wall of the cylinder was equipped with two flat windows, one at the input of the laser beam to the cell, and the other at its exit on the opposite side of the cylinder. The alkali resistant 1720 Corning glass was used to produce the cell. This type of glass is also characterized by high level of impermeability to atmospheric helium. After the standard preliminary cleaning procedure applied to the cell-inside, the cell was connected to the vacuum system and baked, surrounded by a furnace. Outgassing lasted for over one month, initially at slightly lower temperature, and then, during the last 2.5 weeks, at 900 K until the base pressure dropped to the level of < 1 × 10−8 Torr. High purity potassium was then distilled into the cell under the vacuum. Afterwards, the cell, sealed off from the vacuum system, was vertically positioned in the oven of the experimental system. In order to reduce the Earth- and stray- magnetic fields in the zone of the experiment, the cell was surrounded, inside the oven, by two layers of effective magnetic shielding foils with appropriate holes. The shielding resulted in a field reduction to less than 10 mGs (as measured). The oven consisted of two chambers, one above the other. Temperature of each chamber was independently adjusted and stabilised. The measurements were carried out for different values of potassium vapour temperature. Temperature Tu of the upper part

of the cell was changed in the range of 427.8 to 451.2 K. The lower part of the cell, which was stuck in the lower chamber of the oven, was heated to the temperature Tl (index “l” from “lower”), about 4 to 6 K less than the upper part (exact range of Tl was 423.5 to 445.0 K). Such temperature difference between Tu and Tl was sufficient to prevent potassium condensation on the internal walls of the upper part of the cell. The number density Nu of potassium vapor in the upper part was derived from the saturated vapor pressure pl above the melt pool in the bottom of the cell. The standard Nesmeyanov formula [17] for liquid potassium was used to obtain pl . It is worthwhile to notice that there exists a more recent formula (now considered to be standard) given by Alcock et al. in 1984 [18] and reprinted in the Handbook of CRC [19]. For the range of Tl values of the present experiment this formula provides pl values bigger by 19% to 17% than those obtained from the Nesmeyanov formula (Fig. 6 dotted curve). The authors of Ref. [18] state that their equation “reproduces the observed vapour pressures to an accuracy of ± 5% or better”. At about the same time (in 1983), Shirinzadeh et al. [20] gave an equation based on their absorption measurements, which provides, “no worse than 3%” uncertainty in potassium vapour values, according to these authors estimate. However, these values (dash-dotted curve in Fig. 6) for the range of Tl of our experiment, are only, by less than 3%, higher than those from Nesmeyanov formula. Similar compliance with the Nesmeyanov formula, i.e., up to a few percent difference (in Fig. 6 these limits are symbolized by two full red lines), was reported in 1993 by Horvatic et al. for their potassium vapor pressure values obtained in absorption measurements whose accuracy was estimated at ± 3% of statistical errors [21]. The latter measurements were done for temperatures exceeding the range of our Tl values, but it seems from Fig. 6 that the results of [21] could be extrapolated to lower temperatures as well, with a similar conclusion. It seems that the problem of the discrepancy between the potassium pressure values which can be obtained from [18] and those in articles [20] and [21] (the latter two in agreement with Nesmeyanov) could be solved only by new experiments. If necessary, the cross sections obtained in the present work could accordingly be rescaled.

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Fig. 4. Diagram of potassium levels, with transitions of the main interest in the experiment.

For the uncertainty  pl in determination of potassium pressure in our experiment, the value of 6% was conservatively adopted, by accounting also for existence of measurement results from two earlier sources [22,23]. Those results are rather scarce and scattered, but indicate ca 10% difference from Nesmeyanov formula for temperatures in our experiment. For  pl the mean of such relative differences is taken, given [20–23]. In order to determine the number density in the upper part of the cell, the ideal gas law N = p/kB T and the following relationships were considered according to Huennekens [24]:

Nu T = l Nl Tu Nu = Nl



for L  D Tl Tu

for L D

(5a)

(5b)

where L is the mean free path of atoms in the vapour, and D the diameter of the tube connecting the cell part at the lower temperature with that at the higher temperature, i.e., the internal diameter of the cell cylinder. L was calculated from the formula √ L = ( 2π d2 Nl )−1 , where d = 2r is the effective diameter of the Katom. The empirical K- radius, amounting to 220 pm as derived

by Slater [25], was assumed for the value of r. The ratio of L/D was estimated to be ca from 4 to 1, within the range of Nl of our experiment 9.47 × 1012 − 3.50 × 1013 cm−3 ; therefore, neither of the above two conditions was satisfied, and perhaps the right formula is “something in-between” (5a) and (5b). In calculations of Nu , the formula (5b) was used. However, given the temperature values of this experiment, and that Tu and Tl , differed only by about 5 K, this rather arbitrary choice is of minor importance, since the respective values for Nu practically do not depend, at the level of 0.5% tolerance, on which of these two above formulae was used. The determined Nu (from now on denoted by N) values are in the range 9.33 × 1012 −3.48 × 1013 cm−3 . A dye laser (with 7 ns pulse duration and ca 4 GHz of spectral bandwidth), pumped by XeCl excimer laser with repetition rate of 90 Hz, was used for pulsed excitation. By employing a lens (L1) with a long focal length the laser beam was partially focused inside the cell to a diameter estimated at about 0.3 mm (± 0.1 mm). The beam waist was sufficiently far behind the cell, not to create plasma on the cell windows. In order to determine whether the results do not depend on the laser pulse intensity, the measurements were carried out not only by changing the temperature of the measurement (and consequently N υ¯ values), but also the laser

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Fig. 5. Schematic of the experimental system. K cell – spectral cell, made of alkali resistant Corning 1720 glass, contains potassium vapour over pure metallic potassium; Oven – two chambers, one above the other, with separately regulated temperatures (in the figure, the cell and the oven are presented as a sketch of their cross-section views at the horizontal excitation-detection plane); PM – fast photomultiplier; PD – fast photodiode; L1-L3 – lenses; F – interchangeable colored glass filter, A – aperture; BS – beam splitter.

Fig. 6. The saturated vapor pressure px over the potassium melt pool from [18,20,21], presented as the relative (temperature-dependent) difference (px –pNesm )/pNesm . Values for pNesm , are derived from the formula in Nesmeyanov book [17]; values px for the dotted curve, are taken from [18]; px for blue dash-dotted curve, from [20]; two red lines represent limits for deviations of px from pNesm for px values determined in [21]. The range of the temperatures of our experiment is marked by two vertical lines and arrows; Tmp denotes the melting point of potassium. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

intensity. In this experiment the applicable range of the laser intensity Ilas inside the cell (i. e. the range in which such dependence was not observed) was found to be 3.3–6.6 MW/cm2 . We should notice that while the relative values of the laser intensity are quite reliable, the values estimated for the beam diameter 0.3 mm, which are given here and throughout the article, suffer quite a big uncertainty. Its main reason is the uncertainty in the

size of the beam (given above as ± 0.1 mm), which creates + 125% and −44% uncertainty in each Ilas value. For laser intensity valmax = 6.6 MW/cm2 , the use of the adopted model fails. ues above Ilas We attribute this to possible nonlinear processes which might occur or intensify with rising laser power. One of the symptoms is the change in the character of the fluorescence decay from the 7s max (e.g., I 2 state. When Ilas < Ilas las = 5.5 MW/cm , as in Fig. 7b) an

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Fig. 7. Examples of direct fluorescence I7s at the laser intensity (a) above the acceptable range (decay is not single-exponential) and (b) within the acceptable range (single exponential decay is expected). See main text for details.

essentially single-exponential decay is observed, whereas if max (e.g., I 2 Ilas > Ilas las = 13.2 MW/cm , as in the case of the decay in Fig. 7a) then the decay is no more a single–exponential decay. This will be discussed in Section 3.2.2, see Fig. 12. In accordance with the time scale of the output histograms from MCS, in this work time is often expressed in channel numbers n, whereas the channel width is 5 ns. The fluorescence (direct or sensitized) was registered at right angle to the exciting beam. It was imaged (reduced in size by ¾) with the help of the optical system of L2 and L3 lenses along the centre of the horizontally oriented slit of the monochromator. The system was also equipped with two interchangeable colored glass edge filters. The filters were selected for the highest possible attenuation of the laser light scattered at passing the cell windows, and at the same time for the highest transmittivity at the wavelength of the fluorescence, either from the 7s or 5f states. The slits were kept fully (i.e., to 1.5 mm) open what resulted in the 6 nm spectral resolution. Such widely open slits were used to minimise the escape of irradiating atoms out of the viewingrange. In this experiment, the mean velocity of potassium atoms, 8kB T /π m (m is the potassium atom mass), amounted to about 0.5 mm/μs. The applied slit-width allowed fluorescence from the states of the primary interest (7s and 5f) to decay for several lifetimes, before the atoms escaped out of the viewing range. The effective lifetimes for the 7s state determined in this work amount to about 149 ns, while the effective lifetimes for the 5f state measured in the course of the other experiment of this group [26] are about 112 ns (see Section 3.2.1). Fluorescence photons after passing through the edge filter and the monochromator, fell on the photocathode (of S1 type) of a fast

161

photomultiplier mounted behind the exit slit. The photocathode of the photomultiplier was cooled to 200 K by using N2 vapours over LN2 . Time-dependence of the I7s and I5f fluorescence was reconstructed with the help of a multichannel scaler (MCS) enabling photon counting in real time in a preset number of channels (each channel width was 5 ns). The averaging was accomplished by accumulation of the channel contents during each run of the MCS which was initiated by the pulse from the fast photodiode PD irradiated by a fraction of the laser pulse reflected on the beam splitter BS (Fig. 5). As the result, a histogram of the photon arrival times was recorded. The contents of the channels were checked for the possible occurrence of the pile-up. In addition to the main measurements (described below) the following supplementary measurements were also carried out: (i) The spectral sensitivity of the detection system was calibrated, with respect to the characteristics of the monochromator and of the two different color (edge) filters, which were alternately inserted into the fluorescence beam depending on which of the two signals I7 s or I5 f was currently recorded. A standard light source, which was a calibrated tungsten ribbon lamp, was used. (ii) The above mentioned, low resolution emission spectra were recorded by tuning the monochromator below and above the exciting laser wavelength (the spectra will be referred to as VIS- and IR-spectra, respectively) within the spectral sensitivity range of the S1 photocathode of the photomultiplier. For this purpose, a two-channel photon-counter was used. The channels differed in the time-gates setting. The channel-1 gate was open for the first 50 ns after the beginning of the fluorescence signals, in order to detect the “early spectrum”, and the channel-2 gate was open for the next at least 450 to 10,0 0 0 ns, in order to detect the “delayed spectrum”. (iii) The time dependence of the laser pulse intensity Ilas ≡ Ilas (t), scattered on the cell windows, was also registered (with monochromator tuned at 579.5 nm) after this intensity reaching the photomultiplier cathode was sufficiently reduced. By a (main) measurement two subsequent registrations of timeresolved fluorescence signals I7s and I5f , are understood, performed at the same potassium number density N and at the same laser intensity in the cell. For an example of such a pair see the experimental signals, presented in Figs. 8 and 9, which are registered in the course of the same measurement at N = 2.60 × 1013 cm−3 and Ilas = 5.5 MW cm−2 . It was observed during the preliminary measurements that the considerable electronic noise was registered in MCS at about the channel corresponding to the onset of the I7s and I5f signals. It was identified that this noise originates from the discharge sources in the excimer laser. The level of the noise varied in the time scale of one measurement. Despite various efforts, it was not possible to completely eliminate this noise by screening and grounding the elements of the set-up. Therefore, as a remedy, a delay line was installed between the photomultiplier and the MCS, in order to delay the registered signals of interest. As a result, the beginning of the registered fluorescence signals was moved to about channel 20 and the signals became well separated from this electronic noise. As expected, this noise can still be observed on the figures, registered in few first channels of the MCS. All registered I7s signals begin with a high, narrow peak as does the example-signal in Fig. 8. If this peak were (entirely) (i) a residual noise from the excimer laser (assuming that part of this noise still enters the MCS via signal cables from the photomultiplier) then a high peak would also be observed about channel 20 with the laser frequency detuned from the two-photon resonance. Similarly, if the peak were (entirely) (ii) from the residual laser light photons (which, scattered on the cell windows, somehow managed to reach the photocathode of the photomultiplier), then again such high peak should be seen not only by registering the I7s signal, but also with detuned laser frequency from the two-photon

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Fig. 8. Main plot: The experimental decay I7s (the same example as in Fig. 7(b)) with two red lines - results of fitting the single-exponential function (9a) with added background α . One of the plots (straight line) is for α =0 and the other for α , set as a free parameter of the fit (in this example the two fitted effective lifetimes τ 7s were obtained practically the same). Presented lines are for the case of fittings on truncated data with initial channel nini set to channel 35. Inset: The dependence of fitting results for τ 7s as a function of nini . The red line shows the mean in the considered range of nini . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. Experimental I5f . signal, registered in the same measurement as I7s . in Fig. 8, with two fitting results. The blue curve is the result of fitting function 9b) of the initial simple model (6) in which only the 7s state is assumed to be the source of population of the 5f state. The red curve is the result of fitting function (13) of the extended final model, in which the involvement of two additional sources is also assumed, the short- and long-living ones. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

resonance. The registrations were performed with the laser frequency detuned by 30 GHz, and while some negligibly small peaks could be seen at about channel 20, no high peaks were observed. Therefore, one could believe that peaks in I7s are neither of kind (i), nor of kind (ii). This conclusion is also confirmed by inspection of the early VIS-spectra registered under conditions of this experiment, where practically no background can be seen. The possible origin of the peak (peaks) as a result of collective or other non-linear processes will be discussed in Section 3.2.2 Unlike in I7s , the relative contribution of some long living or delayed fluorescence in weak I5f signals could be expected as significant (see the example of the experimental signal I5f in Figs. 9 and

10 and the fitting results). However, in the remote channels of I5f . signals its level is rather negligible. The mean contents of channels in the ranges 401–700 and 701–10 0 0 are < 0.3 and < 0.2, respectively. 3.2. Data processing 3.2.1. Preliminary simple model The first approach was done under the assumption that the direct collisional transfer 7s→5f, (see reaction (1)) can be isolated by disregarding some other processes which may follow the 7s state excitation and contribute to the 5f state population. Consequently,

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Fig. 10. Experimental I5f signal and the red curve (d), which is function (13) of the final approach fitted to this signal (both are the same as in Fig. 9), together with three contributions to (d) from three terms in (13). The navy blue curve (a), of the main interest, is related to the collisional population transfer 7s→5f from the singleexponentially (with ca natural decay rate), decaying part of 7s state population; the green (b) and violet (c) curves are related to populating the 5f state from a short-living and a long-living sources, respectively, whose origin is discussed in the main text. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

two simple equations can model the evolution of these state populations (in number densities) N7 s and N5f (see, for instance, equations (6) in Ref. [27])

dN7s = − 7s N7s , dt

(6a)

dN5f = R7s→5f N7s − 5f N5f . dt

(6b)

Here R7s → 5f is the collisional transfer rate related to the cross section σ 7s → 5f , the determination of which is the aim of our experiment:

R7s→5f = σ7s→5f N υ¯ ,

(7)

where N is the number density of the perturbing potassium atoms in the ground state and υ¯ is the temperature dependent mean relative velocity of colliding partners (see formula (4)). The rates

7s = 1/τ 7 s and 5f = 1/τ 5f are effective rates (inverse of respective effective lifetimes) of the total decay of the 7s or 5f state populations. They are expressed by the formula BBR

7s(5f) = 1/τ7snat(5f) + σ7stot(5f) Nυ¯ + 7s (5f ) ,

(8)

nat is the state natural lifetime and 1/τ nat nat where τ7s = 7s (5f ) 7s(5f ) (5f ) the

tot natural radiative rate, σ7s (5f ) is the cross section for the total collisional decay, which refers to all collisional decay channels, and BBR

7s (5f ) is the depopulation rate via transitions which are induced by thermal (black body) radiation present in the cell. Note that in BBR = 0, formula (8) becomes a Stern-Volmer relathe limit of 7s (5f )

tot tion, linear in N υ¯ in the range of T in which σ7s (5f ) can be considered as (nearly) independent of T. In the framework of such a model, the role of levels other than 7s and 5f was only to receive population via radiative and collisional transitions from these two states. In this model, (as well as in the one which was finally used), no back-stream of the population transfer was assumed. This seems well-grounded, given the conditions of our experiment.

Solutions of equations (6), under the initial conditions: 0 N7s (t = t0 ) = N7s and N5 f (t = t0 ) = 0, give single- and doubleexponential functions for the time dependence of N7 s and N5f , respectively; t0 is the moment of excitation of 7s state, in this experiment considered to be in the vicinity of the maximum of the exciting laser pulse intensity. By applying the proportionality relations between the intensity of detected fluorescence, and the number density of atoms in the upper states, namely I7s = A7s-4p N7s , and I5f = A5 f-3d N5f , where Ai-k denote the spontaneous emission rates of the observed transitions, these solutions can be expressed in terms of the time-dependence of I7s and I5f as 0 − 7s (t−t0 ) I7s = I7s e

I5 f = ξ

 A5f−3d 0 R7s→5f  − 7s (t−t0 ) I e − e− 5f (t−t0 ) . A7s−4p 7s 5f − 7s

(9a)

(9b)

The factor ξ in (9b) was additionally introduced to compensate for different spectral sensitivities of the detection system at the λ7s − 4p and λ5f − 3d wavelengths (which were known due to the performed spectral calibration, as mentioned in Section 3.1), as well as for the different number of the MCS runs used to register I7s and I5f . Function (9a) was numerically fitted to each experimental I7s signal. In the fitting procedure, n = 315 was taken to be the last considered channel of the dataset I7s . The effective lifetime 0 = I (t = t ) were the essential τ 7s = 1/ 7 s and the amplitude I7s 7s 0 free parameters of the fit. To take account of a possible background, averaged to a constant in the range of fitting, a parameter α was added to (9a), and it was set either to 0 or taken as an additional free parameter. Similarly as in Ref. [26], fitting to each registered I7s decay was repeated with the initial fragment of the decay-dataset successively truncated by 5 channels each time, starting the fitting procedure with dataset beginning from nini = 25 (thus ca 5 channels after t0 ). From nini , amounting 35 or 40, the results for τ 7 s stabilized, which indicates a single exponential dependence. The results of the series of fittings (with α = 0), performed

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with the experimental I7s signal (the same signal as in Fig. 7b), which is presented in the main part of Fig 8, are plotted as a function of nini in the inset of Fig. 8. The straight red line on the plot in the inset is the mean of the stable results of the series. The analogous fitting procedure was carried out with the parameter α freed, and the mean of these results was obtained quite similar to that on the plot (by considering all measurements, the difference of such two in pairs obtained values was on average ca 1.5%). For the result of the effective lifetime τ 7 s of a given measurement, the average was taken of these two mean values obtained in the above described way. In the main part of Fig. 8, along with the experimental decay I7s , two plots (in red) are presented of the fitted functions from nini = 35. One of the plots (straight line) is for α = 0 and the other for α free. The values obtained for the fitted α amounted a fraction of one count. This may be treated as a measure of a (very low) level of average contributions of long-lasting fluorescence which could possibly accompany the actual I7 s decay in the range of ca 300 channels of the fit. 0 = I (t = t ) of the singleThe result for the fitted amplitude I7s 0 7s exponential decay for a given measurement was obtained in a similar way as the corresponding results for τ 7 s . Some scatter of the obtained 7s = 1/τ 7 s values was smoothened by fitting a straight line to 7s (Nυ¯ ) dependence. The same was done with 5f = 1/τ 5f and data from the other experiment [26]. Such linearization is jusBBR tified in view of formula (8), and of small values estimated for 7s BBR

and also for 5f (according to Ref. [28]), which in the temperature range of the experiment amount to a fraction of 1% of respective

nat . Obtained values for effective τ 7 s span from 148 to 150 ns (with uncertainty of about ± 4 ns) in the range of N υ¯ of the experiment. Since, according to (8), the relation τ 7s ≤ τ7nat should hold, the τ 7s s nat = 155(6) ns as meavalues turn out to be in compliance with τ7s sured by Hart and Atkinson [29] and with the unpublished renat , determined by our group, and also with sult 153(3) ns for τ7s nat = 149 ns as calculated by Safronova and Safronova [30] (when τ7s the uncertainty limits in the experiment are taken into account). Corresponding values for τ 5f , amounting from 109 to 115 ns (± 4 ns), were obtained in a similar way from data recorded in another experiment of this group [26]. These values are in accornat = 117(3) ns and theoretical dance with both the experimental τ5f τ5fnat = 117(4) ns values presented in Ref. [26]. In an attempt to determine values of cross-sections σ 7s → 5f , under the approximation of equations (6), function (9b) was fitted to the experimental I5f signals. For the effective decay rates 7 s and

5f , determining the time dependence of I5f , the experimental values were taken as reported above. The amplitude of function (9b) 0 I5f =ξ

A5f−3d 0 R7s→5f I A7s−4p 7s 5f − 7s

(10)

0 , with R was taken for the free parameter of the fit. From I5f 7s→5f from (7), the value for the cross section σ 7s → 5f can be derived as:

σ7s→5f =

0 1 A7s−4p I5f

5f − 7s . 0 ξ A5f−3d I7s Nυ¯

(11)

All factors on the right hand side of (11) are known: the calibration factor ξ was determined; for A5f − 3d and A7s − 4p the val0 was determined toues recommended by NIST [31] were taken, I7s gether with 7s . Although the fitting procedure gives the chance to compare the fitted function with time dependence of I5f, taken from the experiment, the equivalent way to determine the cross section σ 7s → 5f (under the approximation of model (6)) is to use the following formula which is based on the ratio of time integrals of the sensitized

and direct fluorescences:

∞

σ

int 7s→5f

1 A7s−4p 0 = ξ A5f−3d ∞

I5f dt I7s dt

5f

N υ¯

.

(12)

0

This formula will be referred to below in Section 3.2.2. (see Fig. 12 and its description). The blue curve in Fig. 9 is the result of fitting the function (9b) to the I5f experimental signal registered in the same measurement as I7s in Fig. 8 (and Fig. 7b). As observed in the case of this example and in the case of other registered I5f signals, simple model (6), resulting in function (9b), turned out insufficient to well reproduce the time dependence of the experimental I5 f signal. It seemed necessary to consider the involvement of additional sources of feeding the 5f state population: a very quickly decaying one named here “state X” and a slowly decaying one named “state Y”. 3.2.2. The final extended model for the time dependence of fluorescence from the 5f state It has been assumed that the fluorescence from the 5f state is the result of populating this state from three additively contributing sources, each decaying single-exponentially. Consequently, in the extended model function (formula (13)), to the above considered formula (9b) with the amplitude I50f (see formula (10)), two 0{X }

analogous double-exponential functions with amplitudes I5f

and

0{Y } I5f

have been added in order to compensate for the incompleteness of model (6)





0 {X }

ext 0 I5f = I5f e− 7s (t−t0 ) − e− 5f (t−t0 ) + I5f 0{Y }

+ I5f





e− Y (t−t0 ) − e− 5f (t−t0 ) ,



e− 5f (t−t0 ) − e− X (t−t0 )



(13)

here X and Y designate decay rates of X and Y states, respecext denotes the extended model. tively; “ext” in the superscript of I5f The red line in Figs. 9 and 10 is the result of fitting function (13) to the experimental dependence. More details about the concept of this model, the fitting procedure and the results are given in what follows. The “state X” was associated with the existence of the narrow peak at about t = t0 which was observed in the I7s decay but neglected in the initial approach. As it was discussed in Section 3.1, this peak is neither the electronic noise from the excimer laser, nor it is due to the scattered laser photons. One may then expect that the peak is, or at least it contains, the fluorescence from the 7s state which, at about t = t0 , gets depopulated much faster than due to the natural decay. It is supposedly the result of a nonlinear mechanism or mechanisms as it will be discussed below. When formulating function (13), it was assumed that the 7s state, the fluorescence from which decays at two very different speeds, can be modeled by two different states: the “normal 7s state” and the fast decaying “state X”. Fluorescence from the normal 7s state decays single-exponentially with the rate 7 s from the 0 = I (t = t ). Because of the limited time-resolution amplitude I7s 7s 0 of our experiment, it was impossible to trace the actual timedependence of the fast decay. Having this in mind, as well as the simplicity of the model, a single exponential function was assumed for the “state X” population decay, therefore also for the fluorescence decay. The decay rate is denoted by X and X = 1/τ X . Just as the normal 7s state, the state X also transfers population to the 5f state in collisions, therefore the time-dependence of the second term in (13) should be analogous to that in the first term. The short-lasting peak about the moment of a Rydberg state excitation, observed in the off-axis registered fluorescence–decay

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from this state, may be the signature of various nonlinear processes, which generate light emissions (also the ones that may develop into cascades). These emissions are collinear or nearly collinear with the axis defined by the laser beam, at the frequencies of atomic transitions or near to them. The very presence of light pulses at various frequencies, which may propagate in the cell in addition to the exciting pulsed laser-beam and the fluorescence from the 7s and 5f states, could possibly influence the experiment; this will be discussed in the context of the origin of the long living state Y. Our experiment was neither aimed at studying these processes, nor our set-up was equipped with necessary tools for such an investigation (especially not with detectors reaching deep enough into infrared (IR) and ultraviolet (UV) to observe the pulses of emissions in these spectral regions. However, in our “early” spectra (registered within the time-gate open for 50 ns following excitation, (see Section 3.1, where the additional measurements were listed), besides the line of the direct 7s-4p fluorescence (unresolvable from the line 5d-4p) and the very weak line of the sensitized 5f-3d fluorescence, a few relatively strong potassium lines are present. They correspond to dipole-allowed transitions (from the states lying in energy below the 7s state), the wavelengths of which can be detected within the limited spectral range of our system (ca 400–1190 nm). Some faint lines from states above 7s were also searched for. These strong lines were identified as corresponding to the following transitions: (i) 5p-4s, (ii) two unresolvable transitions: 6s-4p with 4d-4p, (iii) 3d-4p, and (iv) 4p-4s. As registered at about t = t0 , they may correspond to short pulses of cascade emissions (scattered on the cell windows) related to a nonlinear process, or to the fluorescence from states populated by the last pulse of such a population transferring cascade. Let us first consider the possible processes involved. One of the mechanisms, which were taken into consideration, was the phenomenon of collective emission, referred to by the authors cited below as superradiance (SR).2 Under certain conditions, in the laser excited sample of Rydberg atoms, a spontaneous phase-locking of the atomic dipoles develops, and this results in a pulse of the collective emission along the axis of the laser beam [33]. The superradiant pulse is generated at the IR wavelength of the transition to a lower nearby state, and while this lower state gets abruptly and efficiently populated, the upper one gets abruptly depopulated, see e.g., [33–36], also [37] and references within. As a result of the abrupt depopulation, the generation of an SR pulse in IR is evidenced by a short lasting peak which develops, at about t = t0, in the off-axis observed fluorescence decay; for instance on VIS transition from the same upper state as SR emission. Since the upper state is only partially depopulated by SR, after the peak the fluorescence decays with the rate of spontaneous emission [35,38]. Often, with such first-step emission, further superradiant emissions occur as a superradiant cascade of pulses at the wavelengths of successive atomic transitions. SR is usually generated in IR, since IR (and microwave) transitions have small Doppler dephasing and consequently the low threshold for SR [33]. One can see the similarity between the above and our observations. It especially concerns the peak at the beginning of the sideway observed fluorescence decay, and the cascading pulses (observed also in other experiments on SR), which could fast deliver population from the directly excited state to the states related to the lines observed in our early spectra. However, authors usually observed superradiance in alkalies at the atomic number density N, lower or much lower than in our experiment; although e.g. in [36] the range of N partially overlaps with our lowest N-values. The

2 In more recent works, the term “superfluorescence” is often used for this process after [32].

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comparison with other conditions is in general difficult. In particular, we are not aware of any experiment on superradiance related to two photon excitation in alkalis, not to mention potassium. Other widely studied nonlinear mechanisms are also known, which generate coherent pulses of radiation in vapours of alkali metal atoms under a single-photon excitation as well as under two-photon excitation, and some of these mechanisms should be considered as perhaps more in line (than SR) with our case. To these phenomena belong: (i) amplified spontaneous emission (ASE), which is also referred to as mirorrless lasing (ASE develops due to the population inversion, often as cascade ASE emission which is conditioned by the presence of the population inversion in each successive step of atoms cascading down through their energy levels; the cascade channels may split); (ii) stimulated hyperRaman scattering (SHRS); (iii) four- and six-wave mixing (FWM and SWM, respectively); and (iv) parametric four- and six-wave mixing (PFWM and PSWM), e.g., [39,40]. The light beams participating in the latter two processes must satisfy either axially phase–matching or angle phase-matching conditions [41]. For the case of two-photon (4s-7s) excitation in potassium, the nonlinear processes were observed and analyzed [42], but only in the context of UV emissions, which are generated through (usually parametric) wave mixing under K(4s-ns) excitations at much higher potassium number densities N and much higher laser intensities Ilas than in our experiment (see complementary publications of Zhang et al. [42,43]). However, basing on publications [43] and of Armyras et al. [44], one can expect that, given the resonant two-photon excitation, relatively small N, and moderate Ilas in our experiment, ASE (or SHRS) and FWM, and not the parametric processes, are apt to be taken into account in discussing our experiment, unlike in the works where just the parametric processes in K were of interest [4043,45]. The potential role of the two processes ASE and FWM in our experiment will be considered below. Several works are known which deal with the K(4s-6 s) twophoton excitation which is analogous to K(4s-7s), see e.g., the above cited articles [40,41,44,45] and also many articles of the Greek group of Armyras et al., which are cited in [44] (as well as many more recent articles of this group which are not much in line with our work). Among these works the theoretical calculations in [44] seem to be most relevant for the comparison with our experiment, at least in the fragment with the smallest N value assumed in calculations which is close to the ones used in our experiment. However, in [44], besides data related to the 6s- and not to 7sstate, also some other assumed parameters and conditions differ from the ones of our experiment, not to mention the simplifications in the very model used in [44]. Therefore, qualitative rather than quantitative hints for our case can be expected from these simulations. In particular, for small N and Ilas values, the model in [44] predicts a narrow peak in the decay of the 6 s state population (N6s ) at about t = t0 . After some ringing, the time-dependence of N6s stabilizes at, presumably, natural decay with the rate of the 6 s state. This result of simulation was identified in [44] as related to FWM in the path 4s-6s-5p-4 s (the first path of transitions of only two, most efficient, paths considered in these simulations) in the following scheme: one photon generated at the fequency ω6s-5p in a nonlinear process is mixed with two laser photons what, due to FWM, leads to generation of the fourth photon at the fequency ω5p- 4 s . The second assumed path 4s-6s-4p-4s (in its case the process of ASE was expected in the 6s-4p step, and perhaps the process of lasing without inversion (LWI) in the 4p-4s step) was found to be activated only in the range of higher N and Ilas , where amplitudes of the first-path emissions evolve with elevating Ilas from a power-law dependence on Ilas , into saturation [44]. The latter, if applied to our case, may suggest that, providing our experiment is carried out in an analogous combination of small N and Ilas values,

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ASE at ω7s-4p, related to the path 4s-7s-4p-4s, would be weak, if present at all. Then the peak in 7s population at t = t0 , which is observed via the off-axis fluorescence 7s-4p (see our Figs. 7b or 8), should mainly be shaped due to emission generated in the path of 4s-7s-6p-4 s transitions, and perhaps also in the path of 4s-7s5p-4 s transitions. One should note that such paths may require involvement of FWM, because at moderate N and Ilas population inversion between levels np and 4 s (and thus ASE) is not expected. If in our case, ASE at ω7s-4p would not be quite negligible (unlike in the above mentioned predictions for second-path transition in [44]), then even weak pulses of ASE emission might bring some direct contribution to the shape of time evolution of the peak on I7s . This is because at least a part of the registered peak might be build up just from photons of the axial ASE pulses, which happen to be scattered on the cell windows. They might reach the photomultiplier, since they are generated at the same wavelengths ω7s-4p as fluorescence being detected. Of course, this admixture to the observed peak would have no influence on excitation transfer processes, because only this part of the peak, which is proportional to the (fast) decaying population is responsible for the collisional energy transfer X→5f, which corresponds to the second term in (13). For small N and Ilas values, simulations in [44] predict small population of the 4p state. This might be possibly in line with the fact that no considerable long living component, which could be due to radiation trapping, was observed in our I7s decays. In our experiment, where the population transfer is actually not limited to two (or three) paths only, the cascading ASE may be considered as developing in various branches down in energy of potassium states, similarly as in the case of the SR cascading. Several atomic states lower than 7s may become populated shortly after the exciting laser pulse, and pulses of photons at various wavelengths, released in ASE cascades propagating along the laser beam, are available for absorption in the cell, providing a transition at coinciding frequency exists. In this context, only one fragment of supposedly cascading branches of ASE emissions turned out to be important in the discussion concerning the origin of the Y state, namely the fragment 6p-4d-5p. As argued above, the short living state X may be related to the peak in the 7s state population decay, as observed off axis via the 7s-4p fluorescence, or at least to a part of the peak. The long living state Y has been searched among nd-states. If populated, the d-states lying in energy above the 5f state could transfer population to the 5f state by spontaneous emission, while the 5d state, which lies 421 cm−1 below the 5f state, by, for instance, collisional transfer. This latter channel of excitation energy transfer to 5f could be only effective with the 5d state considerably excited, which does not seem to be the case. This cannot be checked by searching for possible fluorescence from the 5d state in the early- and delayed-spectra, because, given the spectral resolution of our system, 5d-4p line spectrally overlaps with 7s-4p, while 5d-6p, 5d-4f and 5d-5p transitions are outside the spectral sensitivity range of the system. The fact, that practically no long living component could be seen as an admixture to the fluorescence decay from 7s, is a certain indication that the 5d cannot be much excited. Among d-states above the 5f state, the 10d state, if populated, could affect the measured σ 7s-5f also in a different way than only by populating the 5f state through spontaneous emission. Namely, the spectral positions of near-IR transitions 10d-5p (unresolvable with 12s-5p) lie pretty close to the position of the 5f-3d line and, if strong enough, they could partially overlap with the 5f-3d fluorescence. However, neither this near-IR fluorescence from 10d (12 s) was found in our early and delayed spectra, nor was the VIS fluorescence at transitions 10d-4p (with 12s-4p). The lines from 10d (12 s) were not found even in the spectra registered at the highest

laser intensity Ilas = 13.2 MW cm−2 applied; this Ilas value is quite above the applicable range of Ilas in our experiment. VIS fluorescence lines from other nd (with (n + 2)s) states to 4 s were also carefully searched for in the spectra. At Ilas = 13.2 MWcm−2 , lines corresponding to nd-4p (with (n + 2)s-4p), for n from 6 to 9, can be clearly discerned. At Ilas , within the range of the experiment, only one (quite faint) line 6d-4p (with 8s-4p) is present (especially in the early spectra), which indicates that the 6d state has to be taken into account as a possible candidate for state Y. From here on it is assumed that the 6d-4p fluorescence is present in this unresolved line, therefore the 6d state is populated and, consequently, transfers population to 5f by spontaneous emission. The 8 s state itself, if excited, is not expected to be involved in populating the 5f state. From here on the 8s-4p transition will be not referred to. According to unpublished materials of our group, the mean values of the effective lifetimes of the 5d and 6d states, in the range of N values of our experiment, are 592 ns (118 channels) and 755 ns (151 channels), respectively. The theoretical natural lifetime of the 10d (J = 2/3) state amounts ca 2586 ns (517 channels) [30]; therefore, the value for the effective lifetime τ 10d must be somewhat smaller. The population of the 6d state at about t = t0 may be related to the fact that the difference in the energies of 6d and 6p states is only by 11 cm−1 bigger than the difference between the 4d and 5p states. One may then expect that an (inefficient) excitation of the 6d state may occur from the 6p state (for instance, on the far wing of the absorption profile 6p-6d) by photons generated in the 4d5p step of the ASE cascade. With such expected pulsed excitation of the 6d state in mind, a single exponential function was assumed in (13) for the Y- state population decay. In the procedure of numerical fitting function (13) to the registered I5f signals, a parameter β , which corresponds to the constant noise background, was added to this three-component formula. For each measurement, for the value of β , an average content of remote channels (with numbers in several thousands) of the experimental I5f signal was calculated. The values β < 1 × 10−2 counts per channel were obtained. The fragment of the registered I5f signals used for fitting was limited to channels in the range of 21 to 10 0 0. The initial channel number (t0 in (13)) was fixed at about the position of the maximum of the laser pulse, i.e., usually at t0 = 20.5 channel. The 0 , also I 0{X } and I 0{Y } were taken as free parameamplitudes I5f 5f 5f ters of the fit. The effective rates, the inverse of the effective lifetimes: 7s = 1/τ 7s in the first term, and 5f = 1/τ 5f in all three terms, were fixed on inverses of the same experimental effective lifetimes, which were used in the preliminary approach (Section 3.2.1). The effective lifetimes τX = 1/ X and τY = 1/ Y were either both set as free parameters, or one of them fixed at some expected value. Fitting with 1/ X and 1/ Y as free parameters resulted in 3.4(1.0) and 217(24) channels for the mean of the obtained values for τ X and τ Y , respectively. A value of a few channels well corresponds to the duration of the peaks in I7s . The value of 217 channels for τ Y exceeds the value of 151 channels for the mean effective τ 6d , thus even more so the value of 118 channels for the mean effective τ 5d . The latter may indicate that the 5d state does not significantly contribute to the 5f population. The longer lifetime than expected for the 6d state can be explained as due to cumulative contributions also from nd-5f fluorescence which originates from longer living higher nd states (n > 6) whose transitionlines could lay below the noise level in the spectra. For example, the ionisation (followed by recombination) of the 7s state, or of other considerably excited states, could be one of the sources of their population (perhaps of the delayed character).

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Fig. 11. The final results for the experimental cross-sections plotted as a function of potassium number density.

0 are not very sensitive on the actual value of τ ; Values for I5f Y obtained in fittings with τ Y fixed at 150 channels (ca τ 6d ), differ from those with τ Y free by, on average, ca 10%. In Fig. 10, together with the example of the experimental I5f ext (13) (the red curve), fluorescence and the fitted to it function I5f both the same as in Fig. 9, the three separate contribution-terms of (13) are also shown as reconstructed by introducing the respective fitted parameters to each term. As in the initial simplified model (6), for each measurement carried out at a certain temperature, the cross-section σ 7s → 5f was 0 obtained in derived by using formula (11) but with the value for I5f 0 , this fitting approach. The other values of parameters, including I7s

7s and 5f , were taken the same as in the preliminary approach (Section 3.2.1). 0 (10) of the doubleOn average, the values of the amplitude I5f exponential function (9b) fitted in the preliminary approach (for an example of fitted (9b) see blue curve in Fig. 9) are obtained twice as big as values for the relevant amplitudes from the final approach; therefore, the values for the mean cross-sections σ 7s → 5f are in the same proportion. Note, that the function, contribution to (13) from 7s → 5f ET process, is plotted in navy blue in Fig. 10 (plot a). In Figs. 11 and 12 (inset), one can see that the final results (obtained within the range of acceptable parameters) show neither systematic dependence on potassium number density N, nor systematic dependence on the exciting laser intensity Ilas . In order to enable some comparison of measurements carried out within the acceptable range of Ilas < ∼6.6 MWcm−2 , with three measurements at Ilas > ∼6.6 MWcm−2 , it was decided to apply the simplified initial model (6) again; in consequence, the presence of all processes different than the collisional population transfer 7s→5f is not taken into account. This time, formula (12), based on the ratio of time integrals of the sensitizedand direct-fluorescence signals, is used to process signals in both ranges. int The results of such a procedure σ7s (“int” from integrals) →5f are plotted in the the main part of Fig. 12. For measurements at Ilas <∼6.6 MWcm-2 values are plotted as empty diamonds, while for measurements at Ilas >∼6.6 MWcm-2 , as full diamonds. One int should notice that σ7s values represented by empty diamonds →5f

are, on average, 1.5 times bigger than the final results in the inset, and, as the final results, they also do not depend of Ilas , unlike full diamonds, which exhibit strong dependence on Ilas . The full diamond at 8.8 MWcm−2 and the upper one of the two at 13.2 MWcm−2 come from the measurements carried out at the number density N = 2.6 × 1013 cm−3 , while the lower one at 13.2 MW at int N = 9.6 × 1012 cm−3 . The difference in σ7s value in the pair of →5f −2 full diamonds at 13.2 MW cm may indicate that above the acceptable range of Ilas , some processes, which depend both on the laser intensity and the number density of potassium, do not allow for simple modeling. The other indications of the presence of new or enhanced processes, mentioned above, that are acting when Ilas goes beyond the acceptable range, are: not single-exponential I7 s signals (see an example in Fig 7a) and the increasing density of lines observed in the registered spectra (e.g., the presence of unresolved lines from several nd with (n + 2)s states with n ≥ 6 in the spectra taken at I7s = 13.2 MWcm−2 . In the next Section, the values for σ 7s → 5f , obtained in the final approach related to fitting of the three-term function (13), are presented as the experimental results of this work. Their mean value is taken as the representative experimental cross section σ 7s → 5f for the range of the applied temperatures. In order to estimate the uncertainty limits of this mean cross section, the following contributions were taken into account, combined as square root of the sum-of-the-squares: (i) uncertainties due to a change of the cross section values, as the result of slightly changing fitting procedure, (e.g., changing t0 by + 0.3 or by −0.3, X or Y fitted or free, noise background estimated from the content of remote channels or taken as free parameter of the fit) (total contribution estimated for 26%); (ii) errors in estimations of the relative spectral sensitivity of the setup (19%); (iii) error in the difference 5f − 7s (13%): (iv) standard error- of- the- mean for the averaged cross sections (12%); (v) error in N υ¯ values, both due to uncertainty with respect to the choice of the source of the values of potassium saturated vapour pressure, i.e., also in N (as discussed in Section 3.1) and due to an error in the measured temperature conservatively assessed at 0.5 K (total error in N υ¯ 9%), (vi) error in the amplitude 0 (2%). I7s The total uncertainty in the mean experimental cross section from these contributions is 38%, which is 0.7 × 10−14 cm2 .

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Fig. 12. Dependence on laser intensity. Inset: The results for the final experimental cross-sections plotted as a function of exciting laser intensity Ilas . Main plot: Approximate results (empty red diamonds) are based on the ratio of time integrals of I5f and I7s , according to formula (12). The same formula was used for I5f and I7s from measurements at Ilas above the acceptable range for the experiment (Ilas > ∼6.6 MWcm−2 ); see full red diamonds. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 13. Main plot: Calculated thermally averaged cross sections, sum of contributions of each molecular symmetry ("Sum of contributions" in Fig. 3) as a function of temperature T. The range of T of the experiment is marked in red over the plot. Inset: The theoretical predictions in this range (full line) are compared with the experimental values (points). The dashed line marks the average of the experimental values. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

4. Results and conclusions Let us remind definitions required to show the correspondence between theoretical and experimental cross sections. The theoretical cross section σ¯ (T ) is given by σ¯ (T ) = σ (υ )υ/υ¯ (Formula (3)), where the rate constant is k = σ (υ )υ. The experimental cross section σ 7s → 5f is defined as (see formula (7))

σ7s→5f = R7s→5f /Nυ ,

(14)

and the rate constant k7s → 5f is related to the rate of the ET process R7s → 5f in the following way k7s → 5f = R7s → 5f /N. Since the nearly Maxwellian distribution of relative collision velocities of potassium atoms is expected in our vapour cell, for a given temperature T, and the Maxwellian sistribution is assumed in calculations, then there is a correspondence between the theoretical rate constant (see explanation below Formula (3)) and the experimental rate constant k7s →5f , thus also between the theoretical cross section σ¯ (T ) ≡ σ¯ and the experimental cross section σ 7s → 5f , defined by (14), with υ¯ taken from formula (4).

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In Fig. 13, the theoretical σ¯ results and experimental σ 7s → 5 f results are jointly displayed as σ values dependent on temperature T. The main figure presents the plot, denoted by “Sum of contributions” in Fig. 3. The range of T of the experiment is marked in red over the plot. In the inset, the theoretical results in this range are compared with the corresponding experimental ones. In the relatively small range of temperatures of the experiment, the cross section values are predicted by the present calculations to be nearly constant. Therefore for the representative experimental result in this range we have taken the average, amounting to 1.8(7)x10−14 cm2 . This value is in a satisfying agreement with the theoretical ones spanned from 1.994 × 10−14 cm2 to 1.992 × 10−14 cm2 . In conclusion, the thermally averaged cross sections for the excitation energy transfer K(4s) + K(7s) → K(4s) + K(5f) reaction have been calculated for the range of potassium vapour temperatures of 310–10 0 0 K. For the range of temperatures of our experiment 428– 451 K, the theoretical cross section values well reproduce the experimental ones. This agreement can be treated as a contribution towards checking and validating the accuracy of potential energy curves, determined in this work, of these high excited molecular states for which experimental molecular spectroscopy remains unknown. The fact that the experiment was performed with resolution in time and supplemented with spectroscopic measurements, made it possible to track the competing processes that accompany the process of interest in this work, and to compensate for them in the course of data processing. References [1] Klyucharev AN, Bezuglov NN, Matveev AA, Mihajlov AA, Ignjatovic´ LjM, Dimitrijevic´ MS. Rate coefficients for the chemi-ionization processes in sodiumand other alkali-metal geocosmical plasmas. New Astron Rev 2007;51:547–62. [2] Allard NF, Kielkopf JF, Allard F. Impact broadening of alkali lines in brown dwarfs. Eur Phys J D 2007;44:507–14. ´ [3] Mihajlov AA, Sreckovi c´ VA, Ignjatovic´ LjM, Dimitrijevic´ MS. Atom– Rydberg-atom chemi-ionization processes in solar and DB white-dwarf atmospheres in the presence of (n – n )-mixing channels. MNRAS 2016;458:2215–20. [4] Petrash GG. Collision lasers on atomic transitions. Quantum Electron 2009;39:111–24. [5] Bicchi P. Energy-pooling reactions. Riv Nuovo Cimento 1997;20:1–74. [6] Gao F, Chen F, Xie JJ, Li DJ, Zhang LM, Yang GL, Guo J, Guo LH. Review on diode-pumped alkali vapor laser. Optik 2013;124:4353–8. [7] An G, Wang Y, Han J, Cai H, Zhou J, Zhang W, Xue L, Wang H, Gao M, Jiang Z. Influence of energy pooling and ionization on physical features of a diode-pumped alkali laser. Opt Express 2015;23:26414–25. ´ [8] Allegrini M, Alzetta G, Kopystynska A, Moi L, Orriols G. Electronic energy transfer induced by collision between two excited sodium atoms. Opt Commun 1976;19:96–9. [9] Yurova IYu, Dulieu O, Magnier S, Masnou-Seeuws F, Ostrovskii VN. Structures in the long-range potential curves of Na2 : II. Application to the semiclassical study of the energy pooling process between two excited sodium atoms. J Phys B 1994;27:3659–75. [10] Guldberg-Kjær S, De Filipo G, Miloševic´ S, Magnier S, Pedersen JOP, Allegrini M. Reverse energy-pooling collisions: K(5D)+Na(3S) → K(4P)+Na(3P). Phys Rev A 1997;55:R2515–18. [11] Wang Q, Shen Y, Dai K. Rate coefficients measurement for the energypooling collisions Cs(5D)+Cs(5D) → Cs(6S)+Cs(nl=9D,11S,7F). Opt Commun 2008;281:2112–19. [12] Beigman IL, Lebedev VS. Collision theory of Rydberg atoms with neutral and charged particles. Phys Rep 1995;250:95–328. [13] Głódz´ M, Huzandrov A, Sydoryk I, Szonert J, Klavins J. All-experimental values of self-quenching cross-sections for nf states (n = 5-8) of potassium atoms. Acta Phys Pol A 2007;112:1185–94. [14] Ekers A, Alnis J. Non-dipole excitation transfer between the 62 S and 42 D states in potassium. Latvian Journal of Physics and Technical Sciences 1999;1:64–9.

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