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Rate coefficients measurement for the energy-pooling collisions Cs(5D) + Cs(5D) → Cs(6S) + Cs(nl = 9D, 11S, 7F)
Available online at www.sciencedirect.com
Optics Communications 281 (2008) 2112–2119 www.elsevier.com/locate/optcom
Rate coefficients measurement for the energy-pooling collisions Cs(5D) + Cs(5D) ? Cs(6S) + Cs(nl = 9D, 11S, 7F) Qian Wang a,b, Yifan Shen a,*, Kang Dai a a
Department of Physics, Xinjiang University, Urumqi 830046, China School of Science, Xi’an Jiaotong University, Xi’an 710049, China
b
Received 5 June 2007; received in revised form 8 December 2007; accepted 10 December 2007
Abstract We report experimental rate coefficients for the energy-pooling collisions Cs(5D) + Cs(5D) ? Cs(6S) + Cs(nl = 9D, 11S, 7F). In the experiment the Cs(5D) state was populated via photodissociation of Cs2 molecules using an argon-ion laser at wavelength 488.0 nm. We also consider the competing process 6P1/2 + 7S ? 6S + (nl = 9D, 11S, 7F) that might also populate 9D, 11S and 7F. An intermodulation technique was used to select the fluorescence contributions due only to the process 6P1/2 + 7S ? 6S + (nl = 9D, 11S, 7F). The excited atom (nlJ) density and spatial distribution were mapped by monitoring the absorption of a counterpropagating probe laser beam tuned to various nlJ ! n0 l0J0 transitions. The measured excited atom densities are combined with measured fluorescence ratios to yield rate coefficients for the energy-pooling collisions Cs(5D) + Cs(5D) ? Cs(6S) + Cs(nl = 9D, 11S, 7F). The rate coefficients for nl = 9D, 11S, 7F are (4.1 ± 2.0) 1010 cm3 s1, (1.6 ± 0.8) 1010 cm3 s1 and (3.6 ± 1.8) 1010 cm3 s1, respectively. The contributions to the rate coefficients from other energy transfer processes are also discussed. Ó 2007 Elsevier B.V. All rights reserved. PACS: 34.90.+q; 34.50.Rk Keywords: Energy-pooling collisions; Fluorescence; Rate coefficient; Cesium atom; Intermodulation technique
1. Introduction ‘‘Energy-pooling” (EP) collisions where two excited atoms collide and produce one highly excited atom and one ground-state atom [1] have been studied extensively in alkali homonuclear [2,3] and heteronuclear systems [4], as well as in other metal vapors [5]. Measurements of EP rate coefficients at thermal energies provide important tests of theoretical molecular potential curves at large internuclear distances. The latter are of current interest in the study of ultracold atom collisions which occur at large separations. While the majority of previous work for alkali
*
Corresponding author. E-mail address: [email protected] (Y. Shen).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.12.021
homonuclear systems concentrated on the EP collisions between two alkali atoms in the resonance state [3,6], there has been little work on the EP collisions involving alkali atoms in excited states which are higher than the resonance state. Here we present experimental results of rate coefficients for the cesium energy-pooling process k nl
Csð5DÞ þ Csð5DÞ ! Csð6SÞ þ Csðnl ¼ 9D;11S;7FÞ
ð1Þ
where knl indicates the EP rate coefficient. To the best of our knowledge, the only previous study of cesium EP process (1) was presented in Ref. [7]. However, in that study, only a rough estimate for the EP rate coefficient for nl = 7F state was obtained at T = 570 K by neglecting the collisional interaction with several neighboring states.
Q. Wang et al. / Optics Communications 281 (2008) 2112–2119
In this work, we use an argon-ion laser at wavelength 488.0 nm to photodissociate Cs2 molecules to produce excited atoms in the 5D, 7S and 6P states. ! þ X 1 þ hmpump ! Cs2 !Csðnl ¼ 5D;7S;6PÞ Cs2 X g
In addition, we have also established the contributions of the energy transfer processes k 12
Csð9DÞ þ Csð6SÞ ¢ Csð11SÞ þ Csð6SÞ k 21 k 13
Csð9DÞ þ Csð6SÞ ¢ Csð7FÞ þ Csð6SÞ þ Csð6SÞ
k 0nl
Csð6P1=2 Þ þ Csð7SÞ ! Csð6SÞ þ Csðnl ¼ 9D;11S;7FÞ
k 31
ð2Þ
Since the combination 6P1/2 + 7S can reach 9D, 11S and 7F states directly (see Fig. 1), the following EP process must also be considered ð3Þ
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k 23
Csð11SÞ þ Csð6SÞ ¢ Csð7FÞ þ Csð6SÞ k 32
ð4Þ ð5Þ ð6Þ
k tr
CsðnlÞ þ Csð6SÞ ! Cs states other than CsðnlÞðnl ¼ 9D;11S;7FÞ
ð7Þ
An intermodulation technique [8] is used to select the fluorescence contributions due only to EP process (3).The excited atom densities and spatial distributions are measured using a probe laser absorption. Rate coefficients for EP process (1) are reported. 2. Experiment 2.1. Setup and fluorescence measurements
Fig. 1. (a) Energy levels of cesium showing transitions studied in this work; (b) 9D, 11S and 7F levels. 7S1/2 + 6P1/2 and 5DJ þ 5DJ0 energies are represented by dashed lines.
The experimental setup is shown in Fig. 2, together with excitation schemes. Cesium metal was contained in a cylindrical glass cell, 2 cm inner diameter and 6 cm long, with no buffer gas. The cell was sealed after baking and evacuating. The cell was placed inside an oven with four quartz windows, which was heated with resistive heater tapes. The temperature of the cell was measured with thermocouples. The densities of cesium atoms and cesium molecules in the vapor phase were calculated from the vapor pressure formula [9]. An argon-ion laser (pump laser 1) at wavelength 488.0 nm (160 mW) was used to produce excited atoms in the 5D, 7S and 6P levels via photodissociation of Cs2 molecules. Perpendicular to the direction of the laser beam, a pair of lenses was used to image fluorescence onto the slits of 0.15 m monochromator (Acton SP2105i) with 1200 groove/mm grating blazed at 500 nm. The volume from which fluorescence was collected was a strip of width DY 0.5 mm and length DL 5 mm oriented along the laser propagation (z) axis. A photomultiplier tube (PMT1) was used to detect the fluorescence. The signal from the photomultiplier was passed to a lock-in amplifier (SRS830). Fluorescence intensities from 9D ? 6P (584.7 nm, 566.6 nm), 11S ? 6P (574.8 nm, 557.0 nm) and 7F ? 5D (687.2 nm, 682.7 nm) and the intensity of the line 6P1/2 ? 6S1/2 were recorded. The wavelength-dependent relative detection system efficiency e was measured using a calibrated tungsten–halogen lamp. Since the recorded fluorescence from the nl (nl = 9D, 11S, 7F) state contains a contribution from energy-pooling process (3), We were required to carry out a second experiment to select the fluorescence contribution due only to EP process (3), where the 5D, 6P and 7S atoms were still produced by the argon-ion laser photodissociation of Cs2
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Fig. 2. (a) Experimental setup. ND represents neutral density filter; (b) showing the cell geometry and the region from which fluorescence was detected.
molecules and the 6P level was populated further by direct 6S1/2 ? 6P1/2 pumping with a single-mode diode laser (DL100, pump laser 2). However, the vapor is optically thick at the 6S1/2 ? 6P1/2 transition frequency, so that pump laser 2 had to be detuned considerably to avoid ionizing the vapor and to maximize the number of 6P1/2 atoms. This was done by monitoring the fluorescence from both the 6P1/2 state and the state nl (=9D, 11S, 7F) when pump laser 2 is scanned over the 6S1/2 ? 6P1/2 transition and the argon-ion laser is blocked. (With the argon-ion laser blocked, any nl fluorescence observed is likely due to ionization and recombination). The optimum detuning was determined by the location where the absorption of pump laser 2 beam was optimized to produce a maximum number of 6P1/2 atoms without significantly increasing the nl fluorescence due to ionization and recombination. The argon-ion laser beam and pump laser 2 beam were chopped at two different angular frequencies x1 and x2 (both prime numbers). The fluorescence from the nl (nl = 9D, 11S, 7F) was detected by PMT-1 and the signal was passed to the lock-in amplifier (the reference frequency was x1 + x2). Only fluorescence from the nl (=9D, 11S, 7F) state populated by the energy-pooling collisions Csð7S1=2 Þx1 þ Csð6P1=2 Þx2 were recorded. The intensity of the line 6P1/2 ? 6S1/2 was also measured (the reference frequency of the lock-in amplifier was x2 for this measurement).
2.2. Measurement of the excited atom density and distribution A single-mode diode laser (probe laser) was used to probe the 5D, 6P1/2 and 7S1/2 densities and spatial distributions. The power of the probe laser was cut down to 1 lW with neutral density filters. We directly measured the nlJ density by scanning the probe laser over the nlJ ! n0 l0J0 transition and monitoring its transmission using another photomultiplier tube (PMT-2). The probe laser was stepped across the cell parallel to the pump beam using the translating mirror shown in Fig. 2a. In this case, the probe beam was chopped, but the pump beam (or pump beams) was not. The transmitted intensity of the probe laser beam through a length L of the vapor is given by I m ðLÞ ¼ I m ð0Þe
k n0 l0
J0
nlJ ðvÞt
ð8Þ
for light of frequency m. Here Im(0) is the incident intensity and k n0 l0 0 nlJ ðmÞ is the frequency-dependent absorption coefJ ficient. The excited atom density nnlJ is related to the integral of k n0 l0 0 nlJ ðmÞ by J
Z k n0 l0 0 J
nlJ ðmÞdm ¼
ðkn0 l0 0 J
8p
nlJ Þ
2
gn0 l0 0 J
gnlJ
nnlJ Cn0 l0 0 J
nlJ
ð9Þ
Q. Wang et al. / Optics Communications 281 (2008) 2112–2119
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where gn0 l0 0 and gnlJ are the degeneracies of the n0 l0J0 and nlJ J states, respectively. Thus we can extract nnlJ from the position dependent probe transmission scans using Eqs. (8) and (9). The density of 5D (or 6P1/2, 7S1/2) atoms in each of the above two pumping schemes was determined with the probe laser scanning over the 5D3/2 ? 10F5/2 and 5D5/2 ? 10F7/2 (or 6P1/2 ? 9S1/2, 7S1/2 ? 13P3/2) transition. * In the above two pumping schemes, the n7S1=2 ð r Þ was the same because pump laser 1 was the same. 2.3. Measurement of energy transfer rate coefficients
Fig. 3. Experimental setup for measurement of energy transfer rate coefficients.
We performed a separated experiment to measure the rate coefficients of energy transfer processes (4)–(7). The setup is shown in Fig. 3. The argon-ion laser at wavelength 488.0 nm (160 mW) populated the 5D and 6P states by photodissociating Cs2 molecules. The 5D or 6P state was then excited to the 9D or 7F state by a dye laser tuned to the 6P3/2 ? 9D3/2 (566.4 nm, R6G dye) or 5D3/2 ? 7F5/2 (682.5 nm, LDS698 dye) transition. The fluorescence was dispersed by a 0.15 m monochromator and detected by a photomultiplier tube (PMT-1). The argon-ion laser beam was not chopped and the dye laser beam was chopped, and the PMT-1 signals were processed by a lock-in amplifier. The cell temperature was varied between 200 °C and 260 °C, Fluorescence intensities from 11S ? 6P, 7F ? 5D and 9D ? 6P were recorded at different cell temperatures, in the same volume and solid angle. 2.4. Monitoring of ionization The high-lying levels (nl = 9D, 11S, 7F) are populated through EP collisions or by recombination following associative, Penning or runaway ionization, so the validity of our measured rate coefficients depends upon the absence of significant ionization. In order to observe any ionization that might occur, another cell containing Cs with the same geometry and two electrodes was used to monitor electrical current in two pumping schemes. On both sides of the optical axis of the cell flat nickel electrodes were mounted. Distances between them were 1 cm. A 5–10 V DC bias voltage was placed on one electrode, while the other was coupled to a current-to-voltage amplifier (Keithley instruments). No electrical signal was detected in the electrode cell in either of the two pumping schemes, indicating that at the laser
powers and temperatures used, no appreciable ionization occurred in the cell. The amplifier’s detection limit was 1012 A, and corresponding to this detection limit, only about a few thousandths of the observed high-lying level fluorescence in the EP experiment could be produced from ionization. 3. Theory 3.1. Rate equations Following the argon-ion laser pumping, the EP processes (1), and (3) and energy transfer processes (4)–(7) both contribute to the population of the highly excited state 9D, 11S, 7F. The steady-state rate equations for the population in state nl (nl = 9D, 11S, 7F) are ½1=s9D þ ðk 12 þ k 13 þ k tr1 ÞN n9D k 21 Nn11S k 31 Nn7F C7F!9D n7F 1 ¼ k 09D n7S n6P1=2 þ k 9D n25D 2 ½1=s11S þ ðk 21 þ k 23 þ k tr2 ÞNn11S k 12 Nn9D k 32 Nn7F 1 ¼ k 011S n7S n6P1=2 þ k 11S n25D 2 ½1=s7F þ ðk 31 þ k 32 þ k tr3 ÞNn7F k 13 Nn9D k 23 Nn11S 1 ¼ k 07F n7S n6P1=2 þ k 7F n25D 2
ð10Þ
ð11Þ
ð12Þ
Here we have neglected ionization and ionic recombination which are negligible for our experimental conditions. We can extract the EP rate coefficients from equations (take k9D for example)
RR RR RR RR n7S ðxÞn6P ðxÞdx n9D ðxÞdx n11S ðxÞdx n7F ðxÞdx 0 1=2 R R R ½1=s9D þ ðk 12 þ k 13 þ k tr1 ÞN R R k 21 N R R k 31 N R R k 9D RR R k 9D ¼
R
n6P1=2 ðxÞdx
n6P1=2 ðxÞdx
2
R R 2R n5D ðxÞdx R R R R
n6P
1=2
ðxÞdx
R
n6P1=2 ðxÞdx
R
n6P1=2 ðxÞdx
ð13Þ
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Here we have neglected the C7F?9D term compared to k31N. This is reasonable since C7F?9D k31N for our experimental conditions. The ratio of fluorescence from the state nl (=9D, 11S, 7F) to fluorescence from the 6P1/2 is I nl!n0 l0 0 J
I 6P1=2 !6S1=2
¼
R Cnl!n0 l0 0 k6P1=2 !6S1=2 enl!n0 l0J0 rÞd3 r nnl ð~ J eff R knl!n0 l0 0 e6P1=2 !6S1=2 C6P !6S rÞd3 r n6P1=2 ð~ 1=2 1=2 J ð14Þ
In the present experiment, the detection volume is a thin strip oriented along the z (laser propagation) axis (see Fig. 2b). If we assume that the excitation is uniform along the z axis and because Dy is small, we write the volume integrals in Cartesian coordinates, and immediately carry out the integrations over y and z, we obtain RR
nnl ðxÞdx R RR n ðxÞdx R 6P1=2
¼
I nl!n0 l0 0 =e0nl!n0 l0 J
J0
knl!n0 l0 0 J
I 6P1=2 !6S1=2 =e6P1=2 !6S1=2 k6P1=2 !6S1=2
Ceff 6P1=2 !6S1=2 Cnl!n0 l0 0 J
Here we have neglected the C7F?9D compared to k31N. We can obtain rate coefficients of EP process (3) by the measured fluorescence intensity ratios, the excited atom densities and rate coefficients of energy transfer processes (4)–(7). In the measurement of energy transfer rate coefficients, presumably EP process (1) and (3) are swamped by the much larger contribution from the direct excitation of the highly excited state. Hence in steady state, for direct excitation of the transition 6P3/2 ? 9D3/2, the rate equations are n007F ½1=s7F þ ðk 31 þ k 32 þ k tr3 ÞN ¼ n009D k 13 N
ð20Þ
n0011S ½1=s11S þ ðk 21 þ k 23 þ k tr2 ÞN ¼ n009D k 12 N
ð21Þ
For direct excitation of the transition 5D3/2 ? 7F5/2, the rate equations are 000 n000 11S ½1=s11S þ ðk 21 þ k 23 þ k tr2 ÞN ¼ n7F k 32 N
000 000 n000 9D ½1=s9D þ ðk 12 þ k 13 þ k tr1 ÞN ¼ n7F k 31 N þ C7F!9D n7F
ð15Þ
where R is the cell radius. Ceff 6P1=2 !6S1=2 is calculated using Holstein’s theory of radiation trapping [10] and natural radiative rates is taken from the literature [11–13]. We can obtain rate coefficients of EP process (1) by the measured fluorescence intensity ratios, the excited atom densities, rate coefficients of EP process (3), and energy transfer processes (4)–(7). Following the argon-ion laser plus pump laser 2 pumping, EP process (1) does not appear in rate equations because highly excited atoms created by EP process (1) are not detected due to the intermodulation technique. The population of the 9D, 11S and 7F levels is only affected by EP process (3) and energy transfer processes (4)–(7), rate equations have the forms ½1=s9D þ ðk 12 þ k 13 þ k tr1 ÞN n09D k 21 Nn011S k 31 Nn07F C7F!9D n07F ¼ k 09D n7S n06P1=2
ð16Þ
ð22Þ
ð23Þ where n00nl (or n000 nl ) and N are the nl and ground-state density, respectively. The rate coefficients kij (i, j = 1, 2, 3, i 6¼ j) and kji for energy transfer processes (4)–(6) are connected by the detailed balance principle. Eqs. (20)–(23) can be rearranged to obtain n009D 1 1 k 31 þ k 32 þ k tr3 ¼ þ n007F s7F k 13 N k 13
ð24Þ
n009D 1 1 k 21 þ k 23 þ k tr2 ¼ þ n0011S s11S k 12 N k 12
ð25Þ
n000 1 1 k 21 þ k 23 þ k tr2 7F ¼ þ 000 n11S s11S k 32 N k 32
ð26Þ
n000 1 ðk 12 þ k 13 þ k tr1 ÞN 7F þ ¼ s k 31 N þ C7F!9D n000 ðk N þ C Þ 9D 31 7F!9D 9D
ð27Þ
½1=s11S þ ðk 21 þ k 23 þ k tr2 ÞN n011S k 12 Nn09D k 32 Nn07F ¼ k 011S n7S n06P1=2
ð17Þ
½1=s7F þ ðk 31 þ k 32 þ k tr3 ÞN n07F k 13 Nn09D k 23 Nn011S ¼ k 07F n7S n06P1=2
ð18Þ
We can extract the EP rate coefficients from equations (take k 09D for example):
k 09D
The ratios of excited atom density in Eqs. (24)–(27) can be determined from the ratios of the measured fluorescence intensity. The ratios of excited atom densities versus N1 (N is the cesium ground-state atom density) can be fitted by straight lines and the rate coefficients of energy transfer processes (4)–(7) are obtained from the gradient and the intercept.
R 0 * 3 R 0 R 0 * 3 n9D ð r Þd r n11S ð~ rÞd3 r n ð r Þd r R R R ½1=s9D þ ðk 12 þ k 13 þ k tr1 ÞN 0 k 21 N 0 k 31 N 0 7F * 3 * * n6P ð r Þd3 r n6P ð r Þd3 r n6P ð r Þd r 1=2 1=2 R1=2 * 0 ¼ * n7S ð r Þn6P ð r Þd3 r R 0 1=2 * 3 n6P
1=2
ð r Þd r
ð19Þ
Q. Wang et al. / Optics Communications 281 (2008) 2112–2119
3.2. Calculation of the effective radiative rate Radiation trapping can be expressed in terms of the escape factor g which relates the effective radiative rate to the natural radiative rate Ceff ¼ gCnat
ð28Þ
The g factor can be calculated in the relatively low optical depth limit (k0L < 10) from the theory of Milne and in the high optical depth limit (k0L 1) from the theory of Holstein. In the present work, we model our cell as a cylinder of radius R = 1 cm and length L, L R, and we model trapping of resonance radiation (6P1/2 ? 6S1/2) using the Holstein theory of radiation trapping. Holstein used the limiting approximation of high optical depth (k0L 1), only a single broadening mechanism present (pure Doppler, pure impact broadening), and complete frequency redistribution. He obtained the following fundamental-mode escape factors for infinite cylinder geometries in the limits of pure impact broadening 1:115 g ¼ pffiffiffiffiffiffiffiffiffiffi pk p R
ð29Þ
where kp is given by kp ¼
g6P k2 C6P!6S N g6S 2pCbr
ð30Þ
k tr2 ¼ 2:1 1010 cm3 s1 RR n ðxÞn06P1=2 ðxÞ dx R 7S RR 0 n ðxÞdx R 6P1=2
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k tr3 ¼ 3:8 1011 cm3 s1
can be obtained from measurement of the excited atom density and distribution under the argon-ion laser plus pump laser 2 pumping, then combined with the measured fluorescence ratios (corrected for detection system efficiency) and the rate coefficients kij, ktr to yield values of rate coefficients for the EP collisions 7S + 6P1/2 ? nl + 6S from Eq. (19) (see Table 1). Similarly, we can calculate rate coefficients for the EP collisions 5D + 5D ? nl + 6S from Eq. (13) (see Table 2). An estimate of the uncertainty in our measured rate coefficients can be obtained. The cesium ground-state atom density is obtained from the Nesmeyanov vapor pressure formula and the measured temperature, we estimate the uncertainty in density at 10%. Fluores the ground-state I 9D I 7F cence ratios II9D ; ; are accurate to 15%. Transition 7F I 11S I 11S probabilities used in this work are taken from Warner, Fabry and Theodosiou, we estimate them accurate to 10%. The uncertainty in kij (i, j = 1, 2, 3, i 6¼ j) is estimated as 20% and the uncertainty in ktr is estimated as 25%, which is due both to the uncertainty in kij (20%) and in the extrapolation of the straight lines in Fig. 4 to zero N1. The value of Cnl!n0 l0 0 J
and Cbr is the collisional broadening rate. In the limit of pure self-broadening, Cbr = kbrN, kbr is the collisional broadening rate coefficient of the resonance transition. In our experiment (T = 475 K) the cesium density was sufficiently high that the Holstein impact broadening limit was always appropriate, and the theory yields 6P1/2 ? 4 1 6S1/2 radiative rate Ceff independent 6P1=2 !6S1=2 ¼ 8 10 s of density (kbr is taken from Ref. [14]). This rate corresponds to a ‘‘fundamental-mode” excitation scheme which is realized with an unfocused laser beam roughly filling the cell cross section.
Ceff 6P1=2 !6S1=2
4. Results and discussion
and RR
Fig. 4 shows the ratios of excited atom density versus N1 for measurement of the energy transfer rate coefficients. It shows that the experimental points can be fitted by straight lines very well, hence in Eq. (27), C7F?9D can be neglected, (i.e. C7F?9D k31N). The rate coefficients of the energy transfer processes (4)–(7) were obtained from the gradient and the intercept k 12 ¼ 1:4 1011 cm3 s1
k 23 ¼ 3:1 109 cm3 s1
k 13 ¼ 1:2 1010 cm3 s1 k 21 ¼ 1:6 1010 cm3 s1 k 31 ¼ 2:2 1010 cm3 s1 k tr1 ¼ 2:5 1010 cm3 s1
k 32 ¼ 4:7 1010 cm3 s1
has a statistical error of 30%. The fluorescence ratio I nl!n0 l0 0 =enl!n0 l0 0 J
J
I 6P1=2 !6S1=2 =e6P1=2 !6S1=2 is accurate to 10%. We estimate the uncertainties in the ratio RR n ðxÞn6P1=2 ðxÞdx R 7S RR n ðxÞdx R 6P1=2
n25D ðxÞdx R RR n ðxÞdx R 6P1=2 to be approximately 15%. Considering these various sources of statistical uncertainty, we estimate overall errors of 50% in our measured EP rate coefficients. If the kij describing energy transfer processes in rate equations (10) and (12) are neglected and the data are substituted into new expressions for k9D and k7F, values of k9Dand k7F are 4.5 1010 and 3.9 1010 cm3 s1, respectively. The relative error introduced in k9D, k7F is only of order 10%. If energy transfer processes 11S ? 9D and 9D ? 11S are neglected in Eq. (11), we can obtain k11S = 1.5 1010 cm3 s1. The relative error in the k11S is only of order 5%.
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Q. Wang et al. / Optics Communications 281 (2008) 2112–2119
Fig. 4. Plot of the population ratios determined from observed fluorescence intensities against N1 (a) for excitation transition 6P3/2 ? 9D3/2; (b) for excitation transition 5D3/2 ? 7F5/2.
Table 1 Values of the rate coefficients and cross sections for collisions 7S + 6P1/2 ? nl + 6S (T = 202 °C), Cnl!n0 l0 0 and Cnl are taken from Refs. [11–13], J 4 1 Ceff 6P1=2 !6S1=2 ¼ 8 10 s R R I 0nl!n0 l0 =enl!n0 l0 nl Monitored transition k 0nl (cm3 s1) r0nl (cm2) n0 ðxÞn06P ðxÞdx J0 R 7S J0 R R 0 1=2 I0 =e6P !6S 6P1=2 !6S1=2
9D 11S 7F
1=2
1=2
R
6
1=2
1=2
1=2
R
9D 11S 7F
9D ? 6P3/2 11S ? 6P1/2 7F ? 5D5/2
5
3.85 10 2.94 107 1.85 105
ðxÞdx
8.4 1011 2.4 1011 8.2 1011
2.3 10 2.3 109 2.3 109
Table 2 Values of the rate coefficients and cross sections for 5D + 5D ? nl + 6S (T = 202 °C) RR I nl!n0 l0 =enl!n0 l0 nl Monitored transition n7S ðxÞn6P1=2 ðxÞdx J0 J0 R RR I 6P !6S =e6P !6S 1=2
1=2
9
2.94 10 2.01 108 1.41 106
9D ? 6P3/2 11S ? 6P1/2 7F ? 5D5/2
n6P
n6P1=2 ðxÞdx 9
1.1 10 1.1 109 1.1 109
RR 2 n ðxÞdx R RR 5D R
2.2 1015 6.2 1016 2.1 1015
knl (cm3 s1)
rnl (cm2)
4.1 1010 1.6 1010 3.6 1010
1.1 1014 4.1 1015 9.3 1015
n6P1=2 ðxÞdx
1.2 1010 1.2 1010 1.2 1010
Q. Wang et al. / Optics Communications 281 (2008) 2112–2119
Acknowledgement This work was supported by the National Science Foundation of China under Grant No. 10264004. References [1] M. Allegrini, G. Alzetta, A. Kopystynska, L. Moi, G. Orriols, Opt. Commun. 19 (1976) 96. [2] M. Allegrini, C. Gabbanini, L. Moi, J. Phys. Colloq. 46 (1985) C1-61. [3] Y.F. Shen, K. Dai, B.X. Mu, S.Y. Wang, X.H. Cui, Chin. Phys. Lett. 22 (2005) 2805. [4] S. Gozzini, S.A. Abdullah, M. Allegrini, A. Cremoncini, L. Moi, Opt. Commun. 63 (1987) 97.
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[5] D. Azinovic’, I. Labazan, S. Milosˇevic’, G. Pichler, Opt. Commun. 183 (2000) 425. [6] R.K. Namiotka, J. Huennekens, M. Allegrini, Phys. Rev. A 56 (1997) 514. [7] C. Vadla, K. Niemax, J. Brust, Z. Phys. D 37 (1996) 241. [8] G.De. Filippo, S. Guldberg-Kjr, S. Milosˇevic´, J.O.P. Pedersen, M. Allegrini, Phys. Rev. A 57 (1998) 55. [9] A.N. Nesmeyanov, Vapour Pressure of the Elements, Academic, New York, 1963. [10] T. Holstein, Phys. Rev. 83 (1951) 1159. [11] B. Warner, Mon. Not. R. Astron. Soc. 139 (1968) 115. [12] M. Fabry, J. Quant, Spectrosc. Radiat. Transfer 16 (1976) 127. [13] C.E. Theodosiou, Phys. Rev. A 30 (1984) 2881. [14] Z.J. Jabbour, J. Sagle, R.K. Namiotka, J. Huennekens, J. Quant. Spectrosc. Radiat. Transfer 54 (1995) 767.
Optics Communications 281 (2008) 2112–2119 www.elsevier.com/locate/optcom
Rate coefficients measurement for the energy-pooling collisions Cs(5D) + Cs(5D) ? Cs(6S) + Cs(nl = 9D, 11S, 7F) Qian Wang a,b, Yifan Shen a,*, Kang Dai a a
Department of Physics, Xinjiang University, Urumqi 830046, China School of Science, Xi’an Jiaotong University, Xi’an 710049, China
b
Received 5 June 2007; received in revised form 8 December 2007; accepted 10 December 2007
Abstract We report experimental rate coefficients for the energy-pooling collisions Cs(5D) + Cs(5D) ? Cs(6S) + Cs(nl = 9D, 11S, 7F). In the experiment the Cs(5D) state was populated via photodissociation of Cs2 molecules using an argon-ion laser at wavelength 488.0 nm. We also consider the competing process 6P1/2 + 7S ? 6S + (nl = 9D, 11S, 7F) that might also populate 9D, 11S and 7F. An intermodulation technique was used to select the fluorescence contributions due only to the process 6P1/2 + 7S ? 6S + (nl = 9D, 11S, 7F). The excited atom (nlJ) density and spatial distribution were mapped by monitoring the absorption of a counterpropagating probe laser beam tuned to various nlJ ! n0 l0J0 transitions. The measured excited atom densities are combined with measured fluorescence ratios to yield rate coefficients for the energy-pooling collisions Cs(5D) + Cs(5D) ? Cs(6S) + Cs(nl = 9D, 11S, 7F). The rate coefficients for nl = 9D, 11S, 7F are (4.1 ± 2.0) 1010 cm3 s1, (1.6 ± 0.8) 1010 cm3 s1 and (3.6 ± 1.8) 1010 cm3 s1, respectively. The contributions to the rate coefficients from other energy transfer processes are also discussed. Ó 2007 Elsevier B.V. All rights reserved. PACS: 34.90.+q; 34.50.Rk Keywords: Energy-pooling collisions; Fluorescence; Rate coefficient; Cesium atom; Intermodulation technique
1. Introduction ‘‘Energy-pooling” (EP) collisions where two excited atoms collide and produce one highly excited atom and one ground-state atom [1] have been studied extensively in alkali homonuclear [2,3] and heteronuclear systems [4], as well as in other metal vapors [5]. Measurements of EP rate coefficients at thermal energies provide important tests of theoretical molecular potential curves at large internuclear distances. The latter are of current interest in the study of ultracold atom collisions which occur at large separations. While the majority of previous work for alkali
*
Corresponding author. E-mail address: [email protected] (Y. Shen).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.12.021
homonuclear systems concentrated on the EP collisions between two alkali atoms in the resonance state [3,6], there has been little work on the EP collisions involving alkali atoms in excited states which are higher than the resonance state. Here we present experimental results of rate coefficients for the cesium energy-pooling process k nl
Csð5DÞ þ Csð5DÞ ! Csð6SÞ þ Csðnl ¼ 9D;11S;7FÞ
ð1Þ
where knl indicates the EP rate coefficient. To the best of our knowledge, the only previous study of cesium EP process (1) was presented in Ref. [7]. However, in that study, only a rough estimate for the EP rate coefficient for nl = 7F state was obtained at T = 570 K by neglecting the collisional interaction with several neighboring states.
Q. Wang et al. / Optics Communications 281 (2008) 2112–2119
In this work, we use an argon-ion laser at wavelength 488.0 nm to photodissociate Cs2 molecules to produce excited atoms in the 5D, 7S and 6P states. ! þ X 1 þ hmpump ! Cs2 !Csðnl ¼ 5D;7S;6PÞ Cs2 X g
In addition, we have also established the contributions of the energy transfer processes k 12
Csð9DÞ þ Csð6SÞ ¢ Csð11SÞ þ Csð6SÞ k 21 k 13
Csð9DÞ þ Csð6SÞ ¢ Csð7FÞ þ Csð6SÞ þ Csð6SÞ
k 0nl
Csð6P1=2 Þ þ Csð7SÞ ! Csð6SÞ þ Csðnl ¼ 9D;11S;7FÞ
k 31
ð2Þ
Since the combination 6P1/2 + 7S can reach 9D, 11S and 7F states directly (see Fig. 1), the following EP process must also be considered ð3Þ
2113
k 23
Csð11SÞ þ Csð6SÞ ¢ Csð7FÞ þ Csð6SÞ k 32
ð4Þ ð5Þ ð6Þ
k tr
CsðnlÞ þ Csð6SÞ ! Cs states other than CsðnlÞðnl ¼ 9D;11S;7FÞ
ð7Þ
An intermodulation technique [8] is used to select the fluorescence contributions due only to EP process (3).The excited atom densities and spatial distributions are measured using a probe laser absorption. Rate coefficients for EP process (1) are reported. 2. Experiment 2.1. Setup and fluorescence measurements
Fig. 1. (a) Energy levels of cesium showing transitions studied in this work; (b) 9D, 11S and 7F levels. 7S1/2 + 6P1/2 and 5DJ þ 5DJ0 energies are represented by dashed lines.
The experimental setup is shown in Fig. 2, together with excitation schemes. Cesium metal was contained in a cylindrical glass cell, 2 cm inner diameter and 6 cm long, with no buffer gas. The cell was sealed after baking and evacuating. The cell was placed inside an oven with four quartz windows, which was heated with resistive heater tapes. The temperature of the cell was measured with thermocouples. The densities of cesium atoms and cesium molecules in the vapor phase were calculated from the vapor pressure formula [9]. An argon-ion laser (pump laser 1) at wavelength 488.0 nm (160 mW) was used to produce excited atoms in the 5D, 7S and 6P levels via photodissociation of Cs2 molecules. Perpendicular to the direction of the laser beam, a pair of lenses was used to image fluorescence onto the slits of 0.15 m monochromator (Acton SP2105i) with 1200 groove/mm grating blazed at 500 nm. The volume from which fluorescence was collected was a strip of width DY 0.5 mm and length DL 5 mm oriented along the laser propagation (z) axis. A photomultiplier tube (PMT1) was used to detect the fluorescence. The signal from the photomultiplier was passed to a lock-in amplifier (SRS830). Fluorescence intensities from 9D ? 6P (584.7 nm, 566.6 nm), 11S ? 6P (574.8 nm, 557.0 nm) and 7F ? 5D (687.2 nm, 682.7 nm) and the intensity of the line 6P1/2 ? 6S1/2 were recorded. The wavelength-dependent relative detection system efficiency e was measured using a calibrated tungsten–halogen lamp. Since the recorded fluorescence from the nl (nl = 9D, 11S, 7F) state contains a contribution from energy-pooling process (3), We were required to carry out a second experiment to select the fluorescence contribution due only to EP process (3), where the 5D, 6P and 7S atoms were still produced by the argon-ion laser photodissociation of Cs2
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Q. Wang et al. / Optics Communications 281 (2008) 2112–2119
Fig. 2. (a) Experimental setup. ND represents neutral density filter; (b) showing the cell geometry and the region from which fluorescence was detected.
molecules and the 6P level was populated further by direct 6S1/2 ? 6P1/2 pumping with a single-mode diode laser (DL100, pump laser 2). However, the vapor is optically thick at the 6S1/2 ? 6P1/2 transition frequency, so that pump laser 2 had to be detuned considerably to avoid ionizing the vapor and to maximize the number of 6P1/2 atoms. This was done by monitoring the fluorescence from both the 6P1/2 state and the state nl (=9D, 11S, 7F) when pump laser 2 is scanned over the 6S1/2 ? 6P1/2 transition and the argon-ion laser is blocked. (With the argon-ion laser blocked, any nl fluorescence observed is likely due to ionization and recombination). The optimum detuning was determined by the location where the absorption of pump laser 2 beam was optimized to produce a maximum number of 6P1/2 atoms without significantly increasing the nl fluorescence due to ionization and recombination. The argon-ion laser beam and pump laser 2 beam were chopped at two different angular frequencies x1 and x2 (both prime numbers). The fluorescence from the nl (nl = 9D, 11S, 7F) was detected by PMT-1 and the signal was passed to the lock-in amplifier (the reference frequency was x1 + x2). Only fluorescence from the nl (=9D, 11S, 7F) state populated by the energy-pooling collisions Csð7S1=2 Þx1 þ Csð6P1=2 Þx2 were recorded. The intensity of the line 6P1/2 ? 6S1/2 was also measured (the reference frequency of the lock-in amplifier was x2 for this measurement).
2.2. Measurement of the excited atom density and distribution A single-mode diode laser (probe laser) was used to probe the 5D, 6P1/2 and 7S1/2 densities and spatial distributions. The power of the probe laser was cut down to 1 lW with neutral density filters. We directly measured the nlJ density by scanning the probe laser over the nlJ ! n0 l0J0 transition and monitoring its transmission using another photomultiplier tube (PMT-2). The probe laser was stepped across the cell parallel to the pump beam using the translating mirror shown in Fig. 2a. In this case, the probe beam was chopped, but the pump beam (or pump beams) was not. The transmitted intensity of the probe laser beam through a length L of the vapor is given by I m ðLÞ ¼ I m ð0Þe
k n0 l0
J0
nlJ ðvÞt
ð8Þ
for light of frequency m. Here Im(0) is the incident intensity and k n0 l0 0 nlJ ðmÞ is the frequency-dependent absorption coefJ ficient. The excited atom density nnlJ is related to the integral of k n0 l0 0 nlJ ðmÞ by J
Z k n0 l0 0 J
nlJ ðmÞdm ¼
ðkn0 l0 0 J
8p
nlJ Þ
2
gn0 l0 0 J
gnlJ
nnlJ Cn0 l0 0 J
nlJ
ð9Þ
Q. Wang et al. / Optics Communications 281 (2008) 2112–2119
2115
where gn0 l0 0 and gnlJ are the degeneracies of the n0 l0J0 and nlJ J states, respectively. Thus we can extract nnlJ from the position dependent probe transmission scans using Eqs. (8) and (9). The density of 5D (or 6P1/2, 7S1/2) atoms in each of the above two pumping schemes was determined with the probe laser scanning over the 5D3/2 ? 10F5/2 and 5D5/2 ? 10F7/2 (or 6P1/2 ? 9S1/2, 7S1/2 ? 13P3/2) transition. * In the above two pumping schemes, the n7S1=2 ð r Þ was the same because pump laser 1 was the same. 2.3. Measurement of energy transfer rate coefficients
Fig. 3. Experimental setup for measurement of energy transfer rate coefficients.
We performed a separated experiment to measure the rate coefficients of energy transfer processes (4)–(7). The setup is shown in Fig. 3. The argon-ion laser at wavelength 488.0 nm (160 mW) populated the 5D and 6P states by photodissociating Cs2 molecules. The 5D or 6P state was then excited to the 9D or 7F state by a dye laser tuned to the 6P3/2 ? 9D3/2 (566.4 nm, R6G dye) or 5D3/2 ? 7F5/2 (682.5 nm, LDS698 dye) transition. The fluorescence was dispersed by a 0.15 m monochromator and detected by a photomultiplier tube (PMT-1). The argon-ion laser beam was not chopped and the dye laser beam was chopped, and the PMT-1 signals were processed by a lock-in amplifier. The cell temperature was varied between 200 °C and 260 °C, Fluorescence intensities from 11S ? 6P, 7F ? 5D and 9D ? 6P were recorded at different cell temperatures, in the same volume and solid angle. 2.4. Monitoring of ionization The high-lying levels (nl = 9D, 11S, 7F) are populated through EP collisions or by recombination following associative, Penning or runaway ionization, so the validity of our measured rate coefficients depends upon the absence of significant ionization. In order to observe any ionization that might occur, another cell containing Cs with the same geometry and two electrodes was used to monitor electrical current in two pumping schemes. On both sides of the optical axis of the cell flat nickel electrodes were mounted. Distances between them were 1 cm. A 5–10 V DC bias voltage was placed on one electrode, while the other was coupled to a current-to-voltage amplifier (Keithley instruments). No electrical signal was detected in the electrode cell in either of the two pumping schemes, indicating that at the laser
powers and temperatures used, no appreciable ionization occurred in the cell. The amplifier’s detection limit was 1012 A, and corresponding to this detection limit, only about a few thousandths of the observed high-lying level fluorescence in the EP experiment could be produced from ionization. 3. Theory 3.1. Rate equations Following the argon-ion laser pumping, the EP processes (1), and (3) and energy transfer processes (4)–(7) both contribute to the population of the highly excited state 9D, 11S, 7F. The steady-state rate equations for the population in state nl (nl = 9D, 11S, 7F) are ½1=s9D þ ðk 12 þ k 13 þ k tr1 ÞN n9D k 21 Nn11S k 31 Nn7F C7F!9D n7F 1 ¼ k 09D n7S n6P1=2 þ k 9D n25D 2 ½1=s11S þ ðk 21 þ k 23 þ k tr2 ÞNn11S k 12 Nn9D k 32 Nn7F 1 ¼ k 011S n7S n6P1=2 þ k 11S n25D 2 ½1=s7F þ ðk 31 þ k 32 þ k tr3 ÞNn7F k 13 Nn9D k 23 Nn11S 1 ¼ k 07F n7S n6P1=2 þ k 7F n25D 2
ð10Þ
ð11Þ
ð12Þ
Here we have neglected ionization and ionic recombination which are negligible for our experimental conditions. We can extract the EP rate coefficients from equations (take k9D for example)
RR RR RR RR n7S ðxÞn6P ðxÞdx n9D ðxÞdx n11S ðxÞdx n7F ðxÞdx 0 1=2 R R R ½1=s9D þ ðk 12 þ k 13 þ k tr1 ÞN R R k 21 N R R k 31 N R R k 9D RR R k 9D ¼
R
n6P1=2 ðxÞdx
n6P1=2 ðxÞdx
2
R R 2R n5D ðxÞdx R R R R
n6P
1=2
ðxÞdx
R
n6P1=2 ðxÞdx
R
n6P1=2 ðxÞdx
ð13Þ
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Q. Wang et al. / Optics Communications 281 (2008) 2112–2119
Here we have neglected the C7F?9D term compared to k31N. This is reasonable since C7F?9D k31N for our experimental conditions. The ratio of fluorescence from the state nl (=9D, 11S, 7F) to fluorescence from the 6P1/2 is I nl!n0 l0 0 J
I 6P1=2 !6S1=2
¼
R Cnl!n0 l0 0 k6P1=2 !6S1=2 enl!n0 l0J0 rÞd3 r nnl ð~ J eff R knl!n0 l0 0 e6P1=2 !6S1=2 C6P !6S rÞd3 r n6P1=2 ð~ 1=2 1=2 J ð14Þ
In the present experiment, the detection volume is a thin strip oriented along the z (laser propagation) axis (see Fig. 2b). If we assume that the excitation is uniform along the z axis and because Dy is small, we write the volume integrals in Cartesian coordinates, and immediately carry out the integrations over y and z, we obtain RR
nnl ðxÞdx R RR n ðxÞdx R 6P1=2
¼
I nl!n0 l0 0 =e0nl!n0 l0 J
J0
knl!n0 l0 0 J
I 6P1=2 !6S1=2 =e6P1=2 !6S1=2 k6P1=2 !6S1=2
Ceff 6P1=2 !6S1=2 Cnl!n0 l0 0 J
Here we have neglected the C7F?9D compared to k31N. We can obtain rate coefficients of EP process (3) by the measured fluorescence intensity ratios, the excited atom densities and rate coefficients of energy transfer processes (4)–(7). In the measurement of energy transfer rate coefficients, presumably EP process (1) and (3) are swamped by the much larger contribution from the direct excitation of the highly excited state. Hence in steady state, for direct excitation of the transition 6P3/2 ? 9D3/2, the rate equations are n007F ½1=s7F þ ðk 31 þ k 32 þ k tr3 ÞN ¼ n009D k 13 N
ð20Þ
n0011S ½1=s11S þ ðk 21 þ k 23 þ k tr2 ÞN ¼ n009D k 12 N
ð21Þ
For direct excitation of the transition 5D3/2 ? 7F5/2, the rate equations are 000 n000 11S ½1=s11S þ ðk 21 þ k 23 þ k tr2 ÞN ¼ n7F k 32 N
000 000 n000 9D ½1=s9D þ ðk 12 þ k 13 þ k tr1 ÞN ¼ n7F k 31 N þ C7F!9D n7F
ð15Þ
where R is the cell radius. Ceff 6P1=2 !6S1=2 is calculated using Holstein’s theory of radiation trapping [10] and natural radiative rates is taken from the literature [11–13]. We can obtain rate coefficients of EP process (1) by the measured fluorescence intensity ratios, the excited atom densities, rate coefficients of EP process (3), and energy transfer processes (4)–(7). Following the argon-ion laser plus pump laser 2 pumping, EP process (1) does not appear in rate equations because highly excited atoms created by EP process (1) are not detected due to the intermodulation technique. The population of the 9D, 11S and 7F levels is only affected by EP process (3) and energy transfer processes (4)–(7), rate equations have the forms ½1=s9D þ ðk 12 þ k 13 þ k tr1 ÞN n09D k 21 Nn011S k 31 Nn07F C7F!9D n07F ¼ k 09D n7S n06P1=2
ð16Þ
ð22Þ
ð23Þ where n00nl (or n000 nl ) and N are the nl and ground-state density, respectively. The rate coefficients kij (i, j = 1, 2, 3, i 6¼ j) and kji for energy transfer processes (4)–(6) are connected by the detailed balance principle. Eqs. (20)–(23) can be rearranged to obtain n009D 1 1 k 31 þ k 32 þ k tr3 ¼ þ n007F s7F k 13 N k 13
ð24Þ
n009D 1 1 k 21 þ k 23 þ k tr2 ¼ þ n0011S s11S k 12 N k 12
ð25Þ
n000 1 1 k 21 þ k 23 þ k tr2 7F ¼ þ 000 n11S s11S k 32 N k 32
ð26Þ
n000 1 ðk 12 þ k 13 þ k tr1 ÞN 7F þ ¼ s k 31 N þ C7F!9D n000 ðk N þ C Þ 9D 31 7F!9D 9D
ð27Þ
½1=s11S þ ðk 21 þ k 23 þ k tr2 ÞN n011S k 12 Nn09D k 32 Nn07F ¼ k 011S n7S n06P1=2
ð17Þ
½1=s7F þ ðk 31 þ k 32 þ k tr3 ÞN n07F k 13 Nn09D k 23 Nn011S ¼ k 07F n7S n06P1=2
ð18Þ
We can extract the EP rate coefficients from equations (take k 09D for example):
k 09D
The ratios of excited atom density in Eqs. (24)–(27) can be determined from the ratios of the measured fluorescence intensity. The ratios of excited atom densities versus N1 (N is the cesium ground-state atom density) can be fitted by straight lines and the rate coefficients of energy transfer processes (4)–(7) are obtained from the gradient and the intercept.
R 0 * 3 R 0 R 0 * 3 n9D ð r Þd r n11S ð~ rÞd3 r n ð r Þd r R R R ½1=s9D þ ðk 12 þ k 13 þ k tr1 ÞN 0 k 21 N 0 k 31 N 0 7F * 3 * * n6P ð r Þd3 r n6P ð r Þd3 r n6P ð r Þd r 1=2 1=2 R1=2 * 0 ¼ * n7S ð r Þn6P ð r Þd3 r R 0 1=2 * 3 n6P
1=2
ð r Þd r
ð19Þ
Q. Wang et al. / Optics Communications 281 (2008) 2112–2119
3.2. Calculation of the effective radiative rate Radiation trapping can be expressed in terms of the escape factor g which relates the effective radiative rate to the natural radiative rate Ceff ¼ gCnat
ð28Þ
The g factor can be calculated in the relatively low optical depth limit (k0L < 10) from the theory of Milne and in the high optical depth limit (k0L 1) from the theory of Holstein. In the present work, we model our cell as a cylinder of radius R = 1 cm and length L, L R, and we model trapping of resonance radiation (6P1/2 ? 6S1/2) using the Holstein theory of radiation trapping. Holstein used the limiting approximation of high optical depth (k0L 1), only a single broadening mechanism present (pure Doppler, pure impact broadening), and complete frequency redistribution. He obtained the following fundamental-mode escape factors for infinite cylinder geometries in the limits of pure impact broadening 1:115 g ¼ pffiffiffiffiffiffiffiffiffiffi pk p R
ð29Þ
where kp is given by kp ¼
g6P k2 C6P!6S N g6S 2pCbr
ð30Þ
k tr2 ¼ 2:1 1010 cm3 s1 RR n ðxÞn06P1=2 ðxÞ dx R 7S RR 0 n ðxÞdx R 6P1=2
2117
k tr3 ¼ 3:8 1011 cm3 s1
can be obtained from measurement of the excited atom density and distribution under the argon-ion laser plus pump laser 2 pumping, then combined with the measured fluorescence ratios (corrected for detection system efficiency) and the rate coefficients kij, ktr to yield values of rate coefficients for the EP collisions 7S + 6P1/2 ? nl + 6S from Eq. (19) (see Table 1). Similarly, we can calculate rate coefficients for the EP collisions 5D + 5D ? nl + 6S from Eq. (13) (see Table 2). An estimate of the uncertainty in our measured rate coefficients can be obtained. The cesium ground-state atom density is obtained from the Nesmeyanov vapor pressure formula and the measured temperature, we estimate the uncertainty in density at 10%. Fluores the ground-state I 9D I 7F cence ratios II9D ; ; are accurate to 15%. Transition 7F I 11S I 11S probabilities used in this work are taken from Warner, Fabry and Theodosiou, we estimate them accurate to 10%. The uncertainty in kij (i, j = 1, 2, 3, i 6¼ j) is estimated as 20% and the uncertainty in ktr is estimated as 25%, which is due both to the uncertainty in kij (20%) and in the extrapolation of the straight lines in Fig. 4 to zero N1. The value of Cnl!n0 l0 0 J
and Cbr is the collisional broadening rate. In the limit of pure self-broadening, Cbr = kbrN, kbr is the collisional broadening rate coefficient of the resonance transition. In our experiment (T = 475 K) the cesium density was sufficiently high that the Holstein impact broadening limit was always appropriate, and the theory yields 6P1/2 ? 4 1 6S1/2 radiative rate Ceff independent 6P1=2 !6S1=2 ¼ 8 10 s of density (kbr is taken from Ref. [14]). This rate corresponds to a ‘‘fundamental-mode” excitation scheme which is realized with an unfocused laser beam roughly filling the cell cross section.
Ceff 6P1=2 !6S1=2
4. Results and discussion
and RR
Fig. 4 shows the ratios of excited atom density versus N1 for measurement of the energy transfer rate coefficients. It shows that the experimental points can be fitted by straight lines very well, hence in Eq. (27), C7F?9D can be neglected, (i.e. C7F?9D k31N). The rate coefficients of the energy transfer processes (4)–(7) were obtained from the gradient and the intercept k 12 ¼ 1:4 1011 cm3 s1
k 23 ¼ 3:1 109 cm3 s1
k 13 ¼ 1:2 1010 cm3 s1 k 21 ¼ 1:6 1010 cm3 s1 k 31 ¼ 2:2 1010 cm3 s1 k tr1 ¼ 2:5 1010 cm3 s1
k 32 ¼ 4:7 1010 cm3 s1
has a statistical error of 30%. The fluorescence ratio I nl!n0 l0 0 =enl!n0 l0 0 J
J
I 6P1=2 !6S1=2 =e6P1=2 !6S1=2 is accurate to 10%. We estimate the uncertainties in the ratio RR n ðxÞn6P1=2 ðxÞdx R 7S RR n ðxÞdx R 6P1=2
n25D ðxÞdx R RR n ðxÞdx R 6P1=2 to be approximately 15%. Considering these various sources of statistical uncertainty, we estimate overall errors of 50% in our measured EP rate coefficients. If the kij describing energy transfer processes in rate equations (10) and (12) are neglected and the data are substituted into new expressions for k9D and k7F, values of k9Dand k7F are 4.5 1010 and 3.9 1010 cm3 s1, respectively. The relative error introduced in k9D, k7F is only of order 10%. If energy transfer processes 11S ? 9D and 9D ? 11S are neglected in Eq. (11), we can obtain k11S = 1.5 1010 cm3 s1. The relative error in the k11S is only of order 5%.
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Q. Wang et al. / Optics Communications 281 (2008) 2112–2119
Fig. 4. Plot of the population ratios determined from observed fluorescence intensities against N1 (a) for excitation transition 6P3/2 ? 9D3/2; (b) for excitation transition 5D3/2 ? 7F5/2.
Table 1 Values of the rate coefficients and cross sections for collisions 7S + 6P1/2 ? nl + 6S (T = 202 °C), Cnl!n0 l0 0 and Cnl are taken from Refs. [11–13], J 4 1 Ceff 6P1=2 !6S1=2 ¼ 8 10 s R R I 0nl!n0 l0 =enl!n0 l0 nl Monitored transition k 0nl (cm3 s1) r0nl (cm2) n0 ðxÞn06P ðxÞdx J0 R 7S J0 R R 0 1=2 I0 =e6P !6S 6P1=2 !6S1=2
9D 11S 7F
1=2
1=2
R
6
1=2
1=2
1=2
R
9D 11S 7F
9D ? 6P3/2 11S ? 6P1/2 7F ? 5D5/2
5
3.85 10 2.94 107 1.85 105
ðxÞdx
8.4 1011 2.4 1011 8.2 1011
2.3 10 2.3 109 2.3 109
Table 2 Values of the rate coefficients and cross sections for 5D + 5D ? nl + 6S (T = 202 °C) RR I nl!n0 l0 =enl!n0 l0 nl Monitored transition n7S ðxÞn6P1=2 ðxÞdx J0 J0 R RR I 6P !6S =e6P !6S 1=2
1=2
9
2.94 10 2.01 108 1.41 106
9D ? 6P3/2 11S ? 6P1/2 7F ? 5D5/2
n6P
n6P1=2 ðxÞdx 9
1.1 10 1.1 109 1.1 109
RR 2 n ðxÞdx R RR 5D R
2.2 1015 6.2 1016 2.1 1015
knl (cm3 s1)
rnl (cm2)
4.1 1010 1.6 1010 3.6 1010
1.1 1014 4.1 1015 9.3 1015
n6P1=2 ðxÞdx
1.2 1010 1.2 1010 1.2 1010
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