5 Results

5.1 Depolarization constants of the $^\mathsfsl{1}$P excited state of Mg 8pt10ptI, Ca 8pt10ptI and Sr 8pt10ptI

Alkali earth atoms in a ¹P state have an electronic angular momentum L=1 and an electronic spin S=0, so the angular momentum of the radiating atom is J=1. Combining J with the angular momentum j₂=1/2 of the H atom, we obtain the total angular momentum j=1/2,3/2. The relevant channels for a total angular momentum $J^{\rm T}$ are given in Appendix A. So, for each kinetic energy, two sets of three-channel scattering equations were solved for a given value of the total angular momentum $J^{\rm T}$ ranging from 1.5 to some suitable large value (up to 700.5 at the highest energies) in order to obtain convergence of the summation.
As inelastic cross sections to other states are negligible, we have only to consider the collisional relaxation rates of rank 0, 1 and 2. The corresponding cross sections are given by:

$$\sigma^0(1)$$

$$\sigma^1(1) \frac{1}{2}B(11;1)+\frac{5}{2}B(11;2)$$

$$\sigma^2(1) \frac{3}{2}B(11;1)+\frac{3}{2}B(11;2).$$ (9)

Coefficients $g^{\kappa}$ are presented in Figs. 4, 5 and 6. As expected, we find similarities in the three systems: all the rates increase smoothly with temperature and in all cases, the relaxation rate of orientation g¹ is larger than the relaxation rate of alignment g². As the reduced masses of the three systems are close, one may conclude that the relative differences between the results are due to the differences in the interaction potential energy curves, reflecting essentially the relative atomic sizes.

Figure 4: Relaxation rates $g^1/n_{\rm H}($cm3 s-1) and $g^2/n_{\rm H}($cm3 s-1) versus temperature T(K).

Figure 5: Relaxation rates $g^1/n_{\rm H}($cm3 s-1) and $g^2/n_{\rm H}($cm3 s-1) versus temperature T(K).

Figure 6: Relaxation rates $g^1/n_{\rm H}($cm3 s-1) and $g^2/n_{\rm H}($cm3 s-1) as function of temperature T(K).

5.2 Collisional relaxation and transfer rates in the Na $^\mathsfsl{2}$P $_\mathsfsl{1/2}$ and $^\mathsfsl{2}$P $_\mathsfsl{3/2}$ excited states

The excited ²P_J states of the Na  I atom correspond to an electronic angular momentum L=1 and an electronic spin S=1/2, so that the total electronic angular momentum is J=1/2 or J=3/2. For each energy, two sets of six coupled equations of the radiating atom (see Appendix B for the definition of the collisional channels) were solved for a total angular momentum $J^{\rm T}$ ranging from a minimum value equal to 2 (due to a condition of non negative $\ell$ value, see also Appendix B) to a large value (up to 700) insuring negligible contributions of larger $J^{\rm T}$ values.

We give here the explicit expressions of the $\Lambda^{\kappa}$ in terms of the Grawert coefficients B were given by Reid (1973) for J=1/2 and J=3/2.

Explicit expressions of the tensorial tranfer cross sections between J=1/2 and J=3/2 states are the following:

$$\sigma^0\left(\frac{3}{2}\rightarrow\frac{1}{2}\right) \frac{1}{2\sqrt{2}}\left[3B\left(\frac{1}{2}\frac{3}{2};1\right)+5B\left(\frac{1}{2}\frac{3}{2};2\right)\right]$$ =

$$\sigma^1\left(\frac{3}{2}\rightarrow\frac{1}{2}\right) \sigma^1\left(\frac{1}{2}\rightarrow\frac{3}{2}\right)=\frac{\sqr... ...t(\frac{1}{2}\frac{3}{2};1\right)-B\left(\frac{1}{2}\frac{3}{2};2\right)\right]$$ =

$$\sigma^2\left(\frac{3}{2}\rightarrow\frac{1}{2}\right) \sigma^2\left(\frac{1}{2}\rightarrow\frac{3}{2}\right)=0 .$$ = (10)

The usual fine structure transfer cross section is:

$$\sigma\left(\frac{1}{2}\rightarrow \frac{3}{2}\right)=\frac{5}{2}... ...rac{1}{2}\frac{3}{2};2\right)+\frac{3}{2}B\left(\frac{1}{2}\frac{3}{2};1\right)$$ (11)

and is thus proportional to $\sigma^0\left(\frac{3}{2} \rightarrow \frac{1}{2}\right)\cdot$

The calculated relaxation rates, polarization transfer rates and fine structure transfer rate are presented in Figs. 7, 8 and 9. All these rates increase slowly with the temperature. As mentioned previously, the relaxation rates are larger than the transfer rates, the polarization transfer rate g¹(1/2,3/2) being almost negligible in comparison with the $g^{\kappa}(1/2)$ and $g^{\kappa}(3/2)$ coefficients. We remark that the differences between g¹(1/2) and g³(3/2) are nowhere discernible to the resolution of the figure, this results from compensation of the two components of the relaxation cross sections (see Eq. (6)). The same trend was obtained by Wilson & Shimoni (1975) for Na(3p ²P $)-{\rm He}$ collisions. Table 2 shows a comparison of our calculated fine structure rate with previous results. We note a significant difference in these values due to the different potential energy curves used, nevertheless our values are very similar to those obtained by Monteiro et al. 1985 from potential energy curves that include correctly the ionic configuration in the ${^{1}{ \Sigma }^{+}}$ molecular states. This indicates the importance of an accurate description of the potential energy curves in the intermediate and large internuclear distances.

 

Table 2: Temperature-dependent fine structure rate $g(1/2\rightarrow 3/2)$ of Na. The rates are in units of $10^{-9}~\mbox{cm}^3 \mbox{ s}^{-1}$.

T(K)	4000	4500	5000	6000	7000	8000	9000	10 000
This work	7.45	8.87	8.26	8.93	9.49	9.95	10.33	10.65
(Monteiro et al. 1985)	-	-	8.06	8.62	9.16	9.7	10.04	-
(Roueff 1974)	-	5.12	-	5.61	5.95	-	6.45	6.75
(Lewis et al. 1971)	-	-	3.03	-	-	-	-	-

5.3 Relaxation among the hyperfine levels of the Na atom

The generalization of Eq. (2) with J=J' to include the effects of nuclear spin has been studied by (Omont 1977). We consider the case of a single hyperfine multiplet, with electronic angular momentum J and with nuclear spin I. The total angular momentum F takes the values $F= {\mid J-I \mid} , \mid J-I \mid +1, ..., J+I$ and we may consider possible off diagonal elements of the corresponding density matrix. The density matrix $\hat{\rho}$ can be expanded as (Omont 1977):

$$\hat{\rho}= \sum_{ FF'} \sum_{KQ} \rho ^{K}_{Q}( FF') ~ T^{K}_{Q} (FF')^{\dagger }.$$ (12)

Where

$$\sum_{M_F}(-1)^{F-M_{F}} (2K+1)^{1/2}$$ T^K_Q(FF') =

$$\times \left ( \begin{array}{ccc} F & F' & K \\ M_F & -M_F' & -Q \end{array} \right ) \mid FM_F\rangle \langle F'M_F'\mid$$ (13)

( ) is a 3j-coefficient (Fano & Racah 1959; Brink & Satchler 1962). The density matrix for the electronic states can be expanded in terms of the tensors $T^{\kappa_{J}}_{q_{J}}$ ( $0 \leq \kappa_{J} \leq 2J$) and the density matrix for the nuclear states have the tensors $T^{\kappa_{I}}_{q_{I}}$ ( $0 \leq \kappa_{I} \leq 2I$) as a basis.

The unitary transformation from the basis T^K_Q(FF') to the basis $T^{\kappa_{J}}_{q_{J}} \otimes T^{\kappa_{I}}_{q_{I}}$ is given by:

$$\sum_{\kappa_{J}q_{J}} \sum_{\kappa_{I}q_{I}} (-1)^{ \kappa_{J}-\kappa_{I}-Q} [(2F+1)(2F'+1)]^{1/2}$$ T^K_Q(FF') =

$$\times~ [(2\kappa_{J}+1)(2\kappa_{I}+1)(2K+1)]^{1/2}$$

$$\times~ \left ( \begin{array}{ccc} \kappa_{J} & \kappa_I & K \\ ... ...{J} & \kappa_{I} & K \end{array} \right \} T^{\kappa_J}_{q_J}T^{\kappa_I}_{q_I}$$ (14)

where $\{ \}$ denotes a Wigner 9j-coefficient (Fano & Racah 1959; Brink & Satchler 1962). As mentionned by Omont (1977), collisions are brief enough for the hyperfine coupling to be neglected. Then the collision S-matrix is just the direct product of the identity in the nuclear spin space and the collision matrix relative to the electronic coordinates computed in the absence of nuclear spin (see Paper I). As the relaxation matrix is known in the "uncoupled'' basis $T^{\kappa_J}_{q_J}(JJ) T^{\kappa_I}_{q_I}(II)$, in the case of an electronic multiplet $\{\lambda J\}$ we obtain the new matrix in the "coupled'' basis T^(K)_Q((JI)F,(JI)F') from the following unitary transformations:

$$\rho^{K}_{Q}(F,F') \sum_{\kappa_Jq_J }\sum_{\kappa_Iq_I }(-1)^{ \kappa_{J}-\kappa_{I}-Q}[(2\kappa_J+1) (2\kappa_I+1)]^{1/2}$$ =

$$\times ~ [(2K+1)(2F+1)(2F'+1)]^{1/2}$$

$$\times ~ \left ( \begin{array}{ccc} \kappa_{J} & \kappa_I & K \\ ... ...K \end{array} \right \} \rho^{ \kappa_{J}}_{ q_{J}}\rho^{ \kappa_{I}}_{q_{I}} .$$ (15)

The time variation due to collisional relaxation of the element $\rho ^{K}_{Q}(FF')$ is given by:

$${\frac{{\rm d} [\hfill^{(\lambda J)}\rho^{K}_{Q}(FF')]}{{\rm d}t} =-\sum _{ F''F'''}G_{K}(FF',F''F''')[\hfill^{(\lambda J)}\rho ^{K}_{Q}(F''F''')]}$$

$$+\sum_{ F''F'''}\sum_{J'\ne J}Q_{K}(FF',F''F''')[\hfill^{(\lambda'J')}\rho ^{K}_{Q} (F''F''')]$$ (16)

where the relaxation rates G_K are related to the electronic relaxation rate $g^{\kappa_{J}}(\lambda J)$ (Omont 1977):

$$\times ~[(2F''+1)(2F'''+1)]^{1/2} \sum_{\kappa_{J}\kappa_{I}} (2\kappa_{J}+1)(2\kappa_{I}+1)$$

$$\times ~\left \{ \begin{array}{ccc} J & I & F \\ J & I & F' \\ ... ... \kappa_{J} & \kappa_{I} & K \end{array} \right \} g^{\kappa_{J}}( \lambda J) .$$ (17)

In the same way, we obtain transfer hyperfine rates pertaining to the electronic transfer rate $g^{\kappa_{J}}(\lambda J, \lambda' J')$:

$$\times ~[(2F''+1)(2F'''+1)]^{1/2} \sum_{\kappa_I\kappa_{J} } (2\kappa_J+1)(2\kappa_I+1)$$

$$\times ~ \left \{ \begin{array}{ccc} J & I & F \\ J & I & F' \\ ... ..._{J} & \kappa_{I} & K \end{array} \right \} g^{\kappa_J}(\lambda J,\lambda'J').$$ (18)

These expressions are useful for any atom with nuclear spin I. The nuclear spin of the Na atom is I=3/2, which yields two hyperfine levels in the J=1/2 state, F=1 and F=2 and four hyperfine levels in the J=3/2 state, F=0, 1, 2, 3. We can apply Eqs. (2) and (5) to obtain the relaxation rates of the Zeeman multiplet and then deduce from Eqs. (17) and (18) all the relaxation rates between the hyperfine levels. Since there are many values and combinations of F and F', we give in Appendix C the values for (F=F' and K=0, 2, 4) related with the G_K and Q_K values.

As previously mentioned in the case of the multipole relaxation of the hyperfine levels in the ground state the algebra coefficients are a fraction of one and thus all the collisional rates among the excited hyperfine levels are of the same order of magnitude as the $g^{\kappa}$ coefficients of the fine structure ²P_1/2 and ²P_3/2 levels.

Figure 7: Relaxation rates $g^1/n_{\rm H}($cm3 s-1), $g^2/n_{\rm H}($cm3 s-1) and $g^3/n_{\rm H}($cm3 s-1) versus temperature T(K).

Figure 8: Polarization transfer rate $g^1(3/2,1/2)/n_{\rm H}($cm3 s-1) as function of temperature T(K).

Figure 9: fine structure rate versus temperature T(K).

5.4 Analytical representation

Cross sections were averaged over a Maxwell distribution of velocities. The variation of the relaxation rates with the temperature is very smooth and was found to increase as $T^\alpha$ and is given as follows for 200 K $\le$ $T \le 10000$ K.

Mg(¹P$)+{\rm H}$:

$$1.3836\times 10^{-8}n_{\rm H}\left(\frac{T}{5000}\right)^{0.329}(\mbox{s}^{-1})$$

$$1.079\times 10^{-8}n_{\rm H}\left(\frac{T}{5000}\right)^{0.367}(\mbox{s}^{-1})$$ (19)

Ca(¹P)+H:

$$1.3826\times10^{-8}n_{\rm H}\left(\frac{T}{5000}\right)^{0.321}(\mbox{s}^{-1})$$

$$1.1861\times 10^{-8}n_{\rm H}\left(\frac{T}{5000}\right)^{0.345}(\mbox{s}^{-1})$$ (20)

Sr(¹P$)+{\rm H}$:

$$8.9875\times 10^{-9}n_{\rm H}\left(\frac{T}{5000}\right)^{0.376}(\mbox{s}^{-1})$$

$$8.0015\times10^{-9}n_{\rm H}\left(\frac{T}{5000}\right)^{0.386}(\mbox{s}^{-1})$$ (21)

Na(²P$)+{\rm H}$:

$$g^{1}=1.0914\times10^{-8}n_{\rm H}\left(\frac{T}{5000}\right)^{0.388}(\mbox{s}^{-1})$$ J=1/2 (22)

$$g^{1}=9.5005 \times10^{-9}n_{\rm H}\left(\frac{T}{5000}\right)^{0.377}(\mbox{s}^{-1})$$ J=3/2

$$g^{2}=1.1183 \times10^{-8}n_{\rm H}\left(\frac{T}{5000}\right)^{0.384}(\mbox{s}^{-1})$$

$$g^{3}=1.1292\times10^{-8}n_{\rm H}\left(\frac{T}{5000}\right)^{0.442}(\mbox{s}^{-1})$$ (23)

$$g^{0}(1/2\rightarrow 3/2) g^{0}(3/2\rightarrow 1/2)$$ =

$$5.5685\times10^{-9}n_{\rm H}\left(\frac{T}{5000}\right)^{0.379}(\mbox{s}^{-1})$$ =

$$g(1/2~\rightarrow~ 3/2)~ 2\times g(3/2\rightarrow 1/2)$$ =

$$7.8751\times10^{-9}n_{\rm H}\left(\frac{T}{5000}\right)^{0.379}(\mbox{s}^{-1})$$ =

$$g^{1}(1/2\rightarrow 3/2) g^{1}(3/2\rightarrow 1/2)$$ =

$$-0.6898\times10^{-9}n_{\rm H}\left(\frac{T}{5000}\right)^{0.302}(\mbox{s}^{-1}).$$ = (24)