Relative-Velocity Distributions for Two Effusive Atomic Beams in Counterpropagating and Crossed-Beam Geometries

Abstract

Formulas are presented for calculating the relative velocity distributions in effusive, orthogonal crossed beams and in effusive, counterpropagating beams experiments, which are two important geometries for the study of collision processes between atoms. In addition formulas for the distributions of collision rates and collision energies are also given.

1. Introduction

Studies of atomic collisions—and in particular collisions between laser-excited atoms—have produced a large amount of information on atomic interactions. Numerous investigations on excited-atom collisions have been performed at thermal temperatures in vapor cells [1–6] and in atomic beams with different geometries [2, 7–13]. Typically, the cell experiments give results averaged over all collision directions and over the thermal velocity distribution, whereas the beam experiments allow more information to be extracted from the collision process, for example, the dependence of the collision dynamics on the atomic polarization.

The collision velocity is an important parameter in a collision process, and a knowledge of the relative collision velocities in the experimental setups is therefore essential for a careful analysis of a crossed beams experiment. This is often done using numerical simulations, but more insight can be obtained from analytical solutions using kinetic gas theory.

Two important cases are experiments where the two effusive beams are either crossed at right angles or counterpropagating, and for these two cases a number of useful formulas are presented for the distribution of relative velocities, collision velocities, and collision energies.

2. Velocity Distributions

In the following analysis of two crossed or counter-propagating beams, intrabeam collisions will be neglected, that is, the collisions between the atoms within a single beam (“head-tail” collisions). Intrabeam collisions in a single beam have been analyzed by Baylis [14], where formulas for the relative-velocity distribution and moments of the distribution are given. For crossed beams, numerical results and some related integrals have been given by Berkling et al. [15] and by Meijer [16]. A single molecular beam from an oven has been treated by Leiby and Besse [17], and supersonic nozzle beams have been described by Haberland et al. [18]. Recently Battaglia et al. [19] have described the velocity distribution of thermionic electrons.

2.1. Relative-Velocity Distributions

The determination of the distribution of relative velocities requires a knowledge of the velocity distribution of each beam. We will assume that the beams are produced from an effusive source with a thin-walled orifice and the velocity distribution can then be calculated from kinetic gas theory as a Maxwell-Boltzmann distribution, which is confined to one direction through an aperture. The distribution becomes [14] 𝑓(𝑣,𝑚,𝑇)=2𝑚³𝜋𝑘³𝑇^31/2𝑣²𝑒^{−𝑚𝑣2/2𝑘𝑇}, (2.1) where 𝑣≥0 because it is a beam in one direction. Measurements on atomic beams have shown that (2.1) is a good description of the velocity distribution [20].

For the general case of two beams, one with a distribution 𝑓(𝑣₁,𝑚₁,𝑇₁) and one with a distribution 𝑓(𝑣₂,𝑚₂,𝑇₂), the distribution function 𝐹 of the relative velocity is found by convoluting the two distributions. Using 𝑣_rel for the relative velocity, the convolution is written as follows: 𝐹𝑣_rel=^∞₀𝑑𝑣₁^∞₀𝑑𝑣₂𝑓𝑣₁,𝑚₁,𝑇₁𝑓𝑣₂,𝑚₂,𝑇₂𝛿𝑣_rel−^→𝑣₁−^→𝑣₂. (2.2)

In an experiment with well-collimated counter-propagating (cp) beams, the length of the velocity vector |^→𝑣₁−^→𝑣₂| is given by ^→𝑣₁−^→𝑣₂^cp=𝑣₁+𝑣₂, (2.3) whereas with orthogonal crossed beams (cb) the length is ^→𝑣₁−^→𝑣₂^cb=𝑣²₁+𝑣²₂. (2.4) In a single beam (sb), the length is ^→𝑣₁−^→𝑣₂^sb=𝑣₁−𝑣₂, (2.5) and as mentioned in the introduction an analytical expression of 𝐹^sb(𝑣_rel) has been given by Baylis [14] for the case when 𝑇₁ = 𝑇₂.

With counter-propagating beams, after inserting (2.3) in (2.2) and integrating over 𝑣₂, we get (note that the limit of integration has been changed as 𝑣₁≤𝑣_rel, because 0≤𝑣₂=𝑣_rel−𝑣₁) 𝐹^cp𝑣_rel=^𝑣rel₀𝑑𝑣₁𝑓𝑣₁,𝑚₁,𝑇₁𝑓𝑣_rel−𝑣₁,𝑚₂,𝑇₂. (2.6)

The integration (Several of the integrations were performed using Mathematica.) yields the result:𝐹^cp𝑣_rel=2𝜋𝑘³𝑚³₁𝑚³₂×𝑘𝑇₁𝑇₂𝑣_rel−4𝑘𝑚²₂𝑇²₁−3𝑘𝑚₁𝑚₂𝑇₁𝑇₂+𝑘𝑚²₁𝑇²₂+𝑚₁𝑚²₂𝑇₁𝑣²_rel𝑚₁𝑇₂+𝑚₂𝑇₁⁴exp𝑚₁𝑣²_rel/2𝑘𝑇₁+𝑘𝑇₁𝑇₂𝑣_rel−4𝑘𝑚²₁𝑇²₂−3𝑘𝑚₁𝑚₂𝑇₁𝑇₂+𝑘𝑚²₂𝑇²₁+𝑚²₁𝑚₂𝑇₂𝑣²_rel𝑚₁𝑇₂+𝑚₂𝑇₁⁴exp𝑚₂𝑣²_rel/2𝑘𝑇₂+𝜋𝑘/23𝑘²𝑚²₂𝑇³₁𝑇₂+6𝑘²𝑚₁𝑚₂𝑇²₁𝑇²₂+3𝑘²𝑚²₁𝑇₁𝑇³₂+𝑘𝑚³₂𝑇³₁𝑣²_rel𝑚₁𝑇₂+𝑚₂𝑇₁^9/2exp𝑚₁𝑚₂𝑣²_rel/2𝑘𝑚₁𝑇₂+𝑚₂𝑇₁+𝜋𝑘/2−3𝑘𝑚₁𝑚²₂𝑇²₁𝑇₂𝑣²_rel−3𝑘𝑚²₁𝑚₂𝑇₁𝑇²₂𝑣²_rel+𝑘𝑚³₁𝑇³₂𝑣²_rel+𝑚²₁𝑚²₂𝑇₁𝑇₂𝑣⁴_rel𝑚₁𝑇₂+𝑚₂𝑇₁^9/2exp𝑚₁𝑚₂𝑣²_rel/2𝑘𝑚₁𝑇₂+𝑚₂𝑇₁×𝑚erf²𝑇₁𝑣_rel2𝑘𝑇₂𝑚₁𝑇₂+𝑚₂𝑇₁𝑚+erf¹𝑇₂𝑣_rel2𝑘𝑇₁𝑚₁𝑇₂+𝑚₂𝑇₁. (2.7)

The unit of the distribution function is s⋅m⁻¹, and the function has been integrated analytically from 0 to ∞ to verify that the result is 1 as expected from the normalization. Unfortunately with typical values for 𝑚_1,2 and 𝑇_1,2, the argument of the error-function is not close to 0 or ∞, so the factor cannot be simplified further.

For orthogonal crossed beams, the result of inserting (2.4) in (2.2) and performing the integration gives 𝐹^cb𝑣_rel=𝑚₁𝑚₂4𝑘²𝑇₁𝑇₂^3/2𝑣⁵_rel−𝑚exp¹𝑇₂+𝑚₂𝑇₁𝑣²_rel4𝑘𝑇₁𝑇₂⁰𝐹^reg₁𝑚;2;¹𝑇₂−𝑚₂𝑇₁²𝑣⁴_rel64𝑘²𝑇²₁𝑇²₂, (2.8) where ₀𝐹^reg₁ is the regularized hypergeometric function given by ₀𝐹^reg₁()=1;𝑎;𝑧Γ(𝑎)^∞_𝑛=0𝑧^𝑛(𝑎)_𝑛.𝑛! (2.9)

The expressions (2.7) and (2.8) can be integrated to find the mean value of the relative velocity, which is given as 𝑣_rel=∫^∞₀𝑑𝑣_rel𝑣_rel𝐹𝑣_rel∫^∞₀𝑑𝑣_rel𝐹𝑣_rel. (2.10)

Note that because of the normalization of 𝐹, the denominator is just 1. The result of inserting (2.7) in (2.10) and performing the integration yields the surprisingly simple result: 𝑣_rel^cp=8𝑘𝑇₁𝜋𝑚₁+8𝑘𝑇₂𝜋𝑚₂, (2.11) which, however, can be understood as stating that the mean relative velocity is just the sum of the two mean velocities, ⟨𝑣_MB⟩_1,2, of the Maxwell-Boltzmann distribution from the two beam sources. This value is found from (2.1) and yields 𝑣_MB=8𝑘𝑇.𝜋𝑚 (2.12)

For orthogonal crossed beams, the expression for the mean relative velocity, using (2.8) and (2.10), becomes 𝑣_rel^cb=15√𝜋𝑚₁𝑚₂𝑇₁𝑇₂𝑚₁𝑇₂+𝑚₂𝑇₁^23/2𝑘𝑇₁𝑇₂𝑚₁𝑇₂+𝑚₂𝑇₁²𝐹₁74;94𝑚;2;²𝑇₁−𝑚₁𝑇₂²𝑚₁𝑇₂+𝑚₂𝑇₁², (2.13) where ₂𝐹₁ is the hypergeometric function given by ₂𝐹₁(𝑎;𝑏;𝑐;𝑧)=^∞_𝑛=0(𝑎)_𝑛(𝑏)_𝑛(𝑐)_𝑛𝑧^𝑛.𝑛! (2.14)

In the particular case of two identical crossed beams with 𝑚=𝑚₁=𝑚₂ and 𝑇=𝑇₁=𝑇₂, the expressions above can be simplified considerably. For counter-propagating beams, the relative-velocity distribution (2.7) becomes 𝐹^cp𝑣_rel=𝑚𝑣_rel−6𝑘𝑇+𝑚𝑣²_rel4𝜋𝑘²𝑇²exp𝑚𝑣²_rel+√/2𝑘𝑇𝑚12𝑘²𝑇²−4𝑘𝑚𝑇𝑣²_rel+𝑚²𝑣⁴_rel8√𝜋𝑘^5/2𝑇^5/2exp𝑚𝑣²_rel/4𝑘𝑇erf𝑚𝑣²_rel,4𝑘𝑇 (2.15) and the mean relative velocity is given as 𝑣_rel^cp=32𝑘𝑇𝑣𝜋𝑚=2_MB. (2.16)

For identical orthogonal crossed beams, (2.8) reduces to 𝐹^cb𝑣_rel=𝑚2𝑘𝑇³𝑣⁵_relexp−𝑚𝑣²_rel,2𝑘𝑇 (2.17) and (2.13) reduces to 𝑣_rel^cb=15𝜋𝑣32^MB. (2.18) Note that 15𝜋/32=1.47≃√2 as would have been expected for orthogonal crossed beams with the singular velocity ⟨𝑣_MB⟩. The expression (2.17) for identical crossed beams has also been obtained by Bezuglov et al. [21], Weiner et al. [22], and Huynh et al. [23]. Note, however, that equation (B6) in Huynh et al. [23] should be replaced by (2.2)–(2.5) of this paper. The expression (2.15) for identical counter-propagating beams has not been presented previously, but numerical integrations were performed in Meijer [16] and Huynh et al. [23]. Numerical results for ⟨𝑣_rel⟩ in identical beams can also be found in Meijer [16]. In Figure 1 examples of relative velocity distributions are shown for the case of identical beams (sodium with 𝑇=575 K) using (2.15) and (2.17) and different beams (sodium with 𝑇=575 K and potassium with 𝑇=475 K) using (2.7) and (2.8). The sodium temperature has been chosen to allow for a comparison with the numerical results of Meijer [16], and the potassium temperature has been selected to yield approximately the same density as a sodium vapor at 575 K.

(a)

(b)

Figure 1 

(a) shows relative velocity distributions for two identical beams (sodium with 𝑇=575 K), for crossed beams (cb) and for counter-propagating (cp) beams, respectively. The arrows indicate the mean relative velocity. (b) shows relative velocity distributions for the two configurations where a sodium beam with 𝑇=575 K collides with a potassium beam with 𝑇=475 K.

2.2. Collision-Rate Distributions

The distribution functions 𝐹(𝑣_rel) describe the distribution of relative velocities of the atom pairs. However, since atom pairs with low relative velocity will have a lower collision frequency compared to atom pairs with high relative velocity and since the atomic beams are used to study collision processes, a more important function is the collision-rate distribution 𝑛_col(𝑣_rel), that is, the collision rate as a function of relative velocities, which is given by 𝑛_col𝑣_rel=𝜎𝑣_rel𝑣_rel𝑛₀, (2.19) where 𝜎(𝑣_rel) is the collision cross-section and 𝑛₀ the number density.

If we assume that 𝜎(𝑣_rel) is velocity independent and that 𝑛₀ is a constant, we obtain (as also pointed out by Meijer [16]) that the distribution function for the collision rate simplifies to 𝐹_col𝑣_rel=𝑣_rel𝐹𝑣_rel. (2.20)

We note that under these assumptions, the distribution function for the collision rate is identical to the distribution function for the collision velocities and that 𝐹_col with this choice becomes dimensionless since the units of 𝐹 are s⋅m⁻¹.

Now the distribution of the collision rates (or collision velocities) for the two beam geometries can be found using (2.7) and (2.11) or (2.8) and (2.13). The expressions are not reproduced here, but in Figure 2 the distributions for the collision velocities for the same systems as in Figure 1 are shown. By comparing the set of figures it can be seen that the mean relative velocity is not identical to the mean collision velocity but that the distribution of collision rates as expected is shifted to higher velocities.

(a)

(b)

Figure 2 

(a) shows collision velocity distributions for identical crossed sodium beams (cb) and counter-propagating (cp) sodium beams with 𝑇=575 K. The arrows indicate the mean relative collision velocity. (b) shows relative collision velocity distributions for a sodium beam with 𝑇=575 K and a potassium beam with 𝑇=475 K.

A useful quantity is the mean value of the collision velocity, which is given by 𝑣_col=∫^∞₀𝑑𝑣_rel𝑣_rel𝐹_col𝑣_rel∫^∞₀𝑑𝑣_rel𝐹_col𝑣_rel=∫^∞₀𝑑𝑣_rel𝑣²_rel𝐹𝑣_rel∫^∞₀𝑑𝑣_rel𝑣_rel𝐹𝑣_rel=𝑣²_rel𝑣_rel. (2.21) To find ⟨𝑣_col⟩, the second moment of 𝑣_rel, has to be calculated, which for counter-propagating beams is given by (using that the normalization is 1) 𝑣²_rel^cp=^∞₀𝑑𝑣_rel𝑣²_rel𝐹^cp𝑣_rel=3𝑘𝑇₁𝑚₁+3𝑘𝑇₂𝑚₂+28𝑘𝑇₁𝜋𝑚₁8𝑘𝑇₂𝜋𝑚₂, (2.22) and the mean collision velocity is then found from (2.11), (2.21), and (2.22) as 𝑣_col^cp=8+(3/2)𝜋𝑚₁𝑇₂/𝑚₂𝑇₁+𝑚₂𝑇₁/𝑚₁𝑇₂4𝜋𝑚₁/8𝑘𝑇₁+𝜋𝑚₂/8𝑘𝑇₂. (2.23)

It is also possible to find the width of the collision velocity distribution from its definition: Δ𝑣_col=𝑣²_{rel𝐹𝑣col}−𝑣_rel²_𝐹𝑣col=𝑣³_rel𝑣_rel−𝑣²_rel𝑣_rel². (2.24) The third moment of the relative velocity is found just as the second: 𝑣³_rel=^∞₀𝑑𝑣_rel𝑣³_rel𝐹𝑣_rel. (2.25) However, the expression for the third moment is quite large, so the substitutions 𝑥=𝑚₁𝑇₂ and 𝑦=𝑚₂𝑇₁ are used. With this notation one obtains 𝑣³_rel^cp=𝑘𝑚₁𝑚₂^3/22𝑥𝑦√𝜋𝑥+𝑦^9/2×3√𝑥+𝑦𝑥^9/2−3𝑥^7/2𝑦+𝑥^5/2𝑦²+𝑥²𝑦^5/2−3𝑥𝑦^7/2+𝑦^9/2+𝑥1+𝑦8𝑥⁵+50𝑥⁴𝑦+120𝑥³𝑦²+137𝑥²𝑦³+89𝑥𝑦⁴+15𝑦⁵+𝑦1+𝑥15𝑥⁵+89𝑥⁴𝑦+137𝑥³𝑦²+120𝑥²𝑦³+50𝑥𝑦⁴+8𝑦⁵. (2.26) The width of the collision-velocity distribution then becomes Δ𝑣^cp_col=𝑘2𝑚₁𝑚₂√𝑥+√𝑦×√𝑥𝑦√𝑥+√𝑦𝑥+𝑦^9/2×3√𝑥+𝑦𝑥^9/2−3𝑥^7/2𝑦+𝑥^5/2𝑦²+𝑥²𝑦^5/2−3𝑥𝑦^7/2+𝑦^9/2+𝑥1+𝑦8𝑥⁵+50𝑥⁴𝑦+120𝑥³𝑦²+137𝑥²𝑦³+89𝑥𝑦⁴+15𝑦⁵+𝑦1+𝑥15𝑥⁵+89𝑥⁴𝑦+137𝑥³𝑦²+120𝑥²𝑦³+50𝑥𝑦⁴+8𝑦⁵−3𝜋𝑥+𝑦+16√𝑥𝑦4𝜋^1/2. (2.27)

For orthogonal crossed beams, the second moment of 𝐹^cb(𝑣_rel) is found as 𝑣²_rel^cb=3𝑘𝑚₁𝑇₂+𝑚₂𝑇₁𝑚₁𝑚₂, (2.28) and the mean collision velocity is found from (2.13), (2.21), and (2.28) as 𝑣_col^cb=√𝑘𝑚₁𝑇₂+𝑚₂𝑇₁^9/25√𝜋𝑚₁𝑚₂^5/2𝑇₁𝑇₂²₂1𝐹₁7/4;9/4;2;𝑚₁𝑇₂−𝑚₂𝑇₁²/𝑚₁𝑇₂+𝑚₂𝑇₁². (2.29) The third moment of the relative velocity is given as 𝑣³_rel^cb=210√𝜋𝑘𝑚₁𝑚₂𝑇²₁𝑇²₂^3/2𝑚₁𝑇₂+𝑚₂𝑇₁^29/2𝐹₁9/4;11/4;2;𝑚₁𝑇₂−𝑚₂𝑇₁²/𝑚₁𝑇₂+𝑚₂𝑇₁², (2.30) and the width of the collision-velocity distribution becomes Δ𝑣^cb_col=14𝑘𝑇₁𝑇₂𝑚₁𝑇₂+𝑚₂𝑇₁²𝐹₁9/4;11/4;2;𝑚₁𝑇₂−𝑚₂𝑇₁²/𝑚₁𝑇₂+𝑚₂𝑇₁²₂𝐹₁7/4;9/4;2;𝑚₁𝑇₂−𝑚₂𝑇₁²/𝑚₁𝑇₂+𝑚₂𝑇₁²−𝑘𝑚₁𝑇₂−𝑚₂𝑇₁⁹25𝜋𝑚₁𝑚₂⁵𝑇₁𝑇₂⁴₂1𝐹₁7/4;9/4;2;𝑚₁𝑇₂−𝑚₂𝑇₁²/𝑚₁𝑇₂+𝑚₂𝑇₁^221/2. (2.31)

The results are simplified by setting 𝑚=𝑚₁=𝑚₂ and 𝑇=𝑇₁=𝑇₂. The second moments of the relative velocity now become 𝑣²_rel^cp=8+3𝜋4𝑣_MB²,𝑣²_rel^cb=3𝜋4𝑣_MB². (2.32) The mean collision velocities reduce to 𝑣_col^cp=8+3𝜋8𝑣_MB,𝑣_col^cb=85𝑣_MB. (2.33)(2.34) By comparison with (2.16) and (2.18), it can be noted that ⟨𝑣_col⟩^cp is 8.9% higher than ⟨𝑣_rel⟩^cp and that ⟨𝑣_col⟩^cb is 8.6% higher than ⟨𝑣_rel⟩^cb.

The third moments are 𝑣³_rel^cp=13𝜋4𝑣_MB³,𝑣³_rel^cb=105𝜋²𝑣256^MB3, (2.35) and the widths of the distributions reduce to Δ𝑣^cp_col=56𝜋−64−9𝜋²𝑣64^MB,Δ𝑣^cb_col=7𝜋8−64𝑣25^MB. (2.36)(2.37)

It is interesting to observe that the ratio Δ𝑣_col/⟨𝑣_col⟩ is independent on 𝑇 and 𝑚 and has a value of about 27% for both configurations. Similar relations for Δ𝑣_rel can easily be found using the formulas in this section. Again one finds that the ratio Δ𝑣_rel/⟨𝑣_rel⟩ is independent on 𝑇 and 𝑚 and has a value of about 30% for the two configurations considered here.

2.3. Collision-Energy Distributions

A relevant quantity for the physics in the collision processes is collision energy, which is given by 𝐸_rel=(1/2)𝜇𝑣²_rel, where 𝜇=𝑚₁𝑚₂/(𝑚₁+𝑚₂) is the reduced mass. To find the distribution function for the collision energy, we note that the collision rate 𝑛_col can be written as a function of the relative energy, and using the chain rule and (2.20), one obtains 𝑛_col𝐸_rel=𝑛_col𝑣_rel𝑑𝑣_rel𝑑𝐸_rel=𝑛_col𝑣_rel1𝜇𝑣_rel=𝐹𝑣_rel𝜇. (2.38) Since 𝜇 has dimensions of a mass, 𝑛_col(𝐸_rel) has the dimensions s·m⁻¹·kg⁻¹.

To find the normalization for the distribution of the collision energy we use (2.38) to calculate ^∞₀𝑑𝐸_rel𝑛_col𝐸_rel=^∞₀𝑑𝑣_rel𝑣_rel𝐹𝑣_rel=𝑣_rel, (2.39) and the distribution function 𝐹_𝐸col for the collision energy is therefore 𝐹_𝐸col𝑣_rel=𝑛_col𝐸_rel𝑣_rel=𝐹𝑣_rel𝜇𝑣_rel. (2.40)

The graphs for 𝐹_𝐸col(𝑣_rel) can then be obtained by a simple transformation of the ordinate in Figure 2. Another option is to show the distribution using 𝐸_rel as the argument. The result of transforming the abscissa in Figure 2 from 𝑣_rel to 𝐸_rel and using (2.40) for the two beam geometries is shown in Figure 3, again for the two systems, Na-Na and Na-K.

(a)

(b)

Figure 3 

(a) shows distributions of collision energies for identical crossed sodium beams (cb) and counter-propagating (cp) sodium beams with 𝑇=575 K. The arrows indicate the mean collision energy. (b) shows collision energy distributions for a sodium beam with 𝑇=575 K and a potassium beam with 𝑇=475 K.

The mean collision energy is then 𝐸_col=𝐸_{rel𝐹𝐸col}=^∞₀𝑑𝐸_rel𝐸_rel𝐹_𝐸col𝑣_rel=1𝜇𝑣_rel^∞₀𝑑𝑣_rel𝜇𝑣_rel12𝜇𝑣²_rel𝐹𝑣_rel=𝜇2𝑣³_rel𝑣_rel. (2.41) By comparison with (2.21), it can be noted that this is different from (1/2)𝜇⟨𝑣_col⟩². Finally, after deriving ⟨𝐸²_col⟩ analogous to (2.41), the width of the collision energy distribution can be found as Δ𝐸_col=𝐸²_{rel𝐹𝐸col}−𝐸_rel²_𝐹𝐸col=𝜇²4⋅𝑣⁵_rel𝑣_rel−𝜇2⋅𝑣³_rel𝑣_rel². (2.42)

Using (2.11) and (2.26), the mean collision energy for counter-propagating beams is found from (2.41) as follows: 𝐸_col^cp=𝑘√𝑥𝑦4𝑚₁+𝑚₂√𝑥+√𝑦𝑥+𝑦^9/2×3√𝑥+𝑦𝑥^9/2−3𝑥^7/2𝑦+𝑥^5/2𝑦²+𝑥²𝑦^5/2−3𝑥𝑦^7/2+𝑦^9/2+𝑥1+𝑦8𝑥⁵+50𝑥⁴𝑦+120𝑥³𝑦²+137𝑥²𝑦³+89𝑥𝑦⁴+15𝑦⁵+𝑦1+𝑥15𝑥⁵+89𝑥⁴𝑦+137𝑥³𝑦²+120𝑥²𝑦³+50𝑥𝑦⁴+8𝑦⁵. (2.43)

The fifth moment is given by 𝑣⁵_rel^cp=32𝜋𝑘𝑚₁𝑚₂^5/21𝑥+𝑦^9/2×𝑥^1/2𝑦^5/2𝑥1+𝑦466𝑥⁴+744𝑥³𝑦+631𝑥²𝑦²+295𝑥𝑦³+35𝑦⁴+𝑥^5/2𝑦^1/2𝑦1+𝑥35𝑥⁴+295𝑥³𝑦+631𝑥²𝑦²+744𝑥𝑦³+466𝑦⁴+√𝑥+𝑦16𝑥^13/2+15𝑥⁶𝑦^1/2+144𝑥^11/2𝑦−15𝑥⁵𝑦^3/2+5𝑥⁴𝑦^5/2+√𝑥+𝑦5𝑥^5/2𝑦⁴−15𝑥^3/2𝑦⁵+144𝑥𝑦^11/2+15𝑥^1/2𝑦⁶+16𝑦^13/2, (2.44) and using (2.42), the width of the collision-energy distribution becomes Δ𝐸^cp_col=𝑘4𝑚₁+𝑚₂√𝑥+√𝑦×6√𝑥+√𝑦𝑥+𝑦^9/2×𝑥^1/2𝑦^5/2𝑥1+𝑦466𝑥⁴+744𝑥³𝑦+631𝑥²𝑦²+295𝑥𝑦³+35𝑦⁴+𝑥^5/2𝑦^1/2𝑦1+𝑥35𝑥⁴+295𝑥³𝑦+631𝑥²𝑦²+744𝑥𝑦³+466𝑦⁴+√𝑥+𝑦16𝑥^13/2+15𝑥⁶𝑦^1/2+144𝑥^11/2𝑦−15𝑥⁵𝑦^3/2+5𝑥⁴𝑦^5/2+√𝑥+𝑦5𝑥^5/2𝑦⁴−15𝑥^3/2𝑦⁵+144𝑥𝑦^11/2+15𝑥^1/2𝑦⁶+16𝑦^13/2−𝑥²𝑦²𝑥+𝑦⁹3√𝑥+𝑦𝑥^9/2−3𝑥^7/2𝑦+𝑥^5/2𝑦²+𝑥²𝑦^5/2−3𝑥𝑦^7/2+𝑦^9/2+𝑥1+𝑦8𝑥⁵+50𝑥⁴𝑦+120𝑥³𝑦²+137𝑥²𝑦³+89𝑥𝑦⁴+15𝑦⁵+𝑦1+𝑥15𝑥⁵+89𝑥⁴𝑦+137𝑥³𝑦²+120𝑥²𝑦³+50𝑥𝑦⁴+8𝑦^521/2. (2.45)

For orthogonal crossed beams, the corresponding relations are 𝐸_col^cb=7𝑘𝑚₁𝑚₂𝑇₁𝑇₂𝑚₁+𝑚₂𝑚₁𝑇₂+𝑚₂𝑇₁²𝐹₁9/4;11/4;2;𝑚₁𝑇₂−𝑚₂𝑇₁²/𝑚₁𝑇₂+𝑚₂𝑇₁²₂𝐹₁7/4;9/4;2;𝑚₁𝑇₂−𝑚₂𝑇₁²/𝑚₁𝑇₂+𝑚₂𝑇₁², (2.46) the fifth moment is 𝑣⁵_rel^cb=3780√𝜋𝑘^5/2𝑚₁𝑚₂^3/2𝑇₁𝑇₂⁴𝑚₁𝑇₂+𝑚₂𝑇₁^211/2𝐹₁𝑚11/4;13/4;2;¹𝑇₂−𝑚₂𝑇₁²𝑚₁𝑇₂+𝑚₂𝑇₁², (2.47) and the width of the collision-energy distribution is Δ𝐸^cp_col=√7𝑘𝑚₁𝑚₂𝑇₁𝑇₂𝑚₁+𝑚₂𝑚₁𝑇₂+𝑚₂𝑇₁₂1𝐹₁7/4;9/4;2;𝑚₁𝑇₂−𝑚₂𝑇₁²/𝑚₁𝑇₂+𝑚₂𝑇₁²×9₂𝐹₁𝑚7/4;9/4;2;¹𝑇₂−𝑚₂𝑇₁²𝑚₁𝑇₂+𝑚₂𝑇₁²²𝐹₁𝑚11/4;13/4;2;¹𝑇₂−𝑚₂𝑇₁²𝑚₁𝑇₂+𝑚₂𝑇₁²−7₂𝐹₁𝑚9/4;11/4;2;¹𝑇₂−𝑚₂𝑇₁²𝑚₁𝑇₂+𝑚₂𝑇₁^221/2. (2.48)

Finally, the reduced formulas obtained by setting 𝑚=𝑚₁=𝑚₂ and 𝑇=𝑇₁=𝑇₂ are found as 𝐸_col^cp=134𝐸𝑘𝑇,_colcb=74𝑣𝑘𝑇,_5relcp=219𝜋²𝑣32^MB5,𝑣⁵_rel^cb=945𝜋³𝑣2048^MB5,Δ𝐸^cp_col=5√8𝑘𝑇,Δ𝐸^cb_col=78𝑘𝑇. (2.49)(2.50)(2.51)(2.52)(2.53)(2.54)

Note that Δ𝐸_col/⟨𝐸_col⟩ is a constant and approximately equal to 0.54 for the two geometries. As also discussed by Meijer [16], the collision energy may be related to a velocity (different from the mean collision velocity) given by 𝑣𝐸_col=2𝐸_col𝜇, (2.55) and for the two beam configurations, one then obtains 𝑣𝐸_col^cp=13𝜋8𝑣_MB,𝑣𝐸_col^cb=𝑣_MB. (2.56)(2.57)

By comparison with the results above, these velocities are found to be a factor of 1.04 larger than the mean collision velocities and a factor of 1.13 larger than the mean relative velocities.

2.4. Numerical Values

In Table 1 some numerical results for two identical counter-propagating sodium beams at three different temperatures are given (using (2.16), (2.33), (2.36), (2.49), (2.53), and (2.56)). The lowest temperature of 575 K is chosen to allow a comparison with the numerical results obtained by Meijer [16], and our formulas reproduce his results exactly. The higher temperature of 823 K is a typical working temperature in the laboratory [24] and has been included together with results for 1,000 K to indicate the temperature dependence of the expressions.

Table 1 

Values (see text) for counter-propagating identical sodium beams with temperatures 𝑇=575, 823 and 1,000 K.

3. Conclusion

Atomic (and molecular) beam studies provide valuable information on collision studies, and the collision parameters such as collision velocities and collision energies are important quantities. In this work a number of useful and relatively simple analytic formulas are presented for the cases where two effusive beams are either crossed at right angles or counterpropagating. In particular the difference between the mean relative velocity and mean collision velocity has been underlined.

Often an important number in an experiment is the fraction of collision energies that will be above the threshold for an endothermic reaction. This number can be obtained using the formula given in the paper and by integrating the distribution function in (2.40).