# Elementary Statistical Models for Vector Collision-Sequence Interference Effects with Poisson-Distributed Collision Times

Authors: John Courtenay Lewis
Journal: International Journal of Spectroscopy
 (2010)

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## Abstract

In a recent paper (Lewis,  2008)  a class of models suitable for application to collision-sequence interference was introduced. In these models velocities are assumed to be completely
randomized in each collision. The distribution of velocities was assumed to be Gaussian. The
integrated induced dipole momentμk, for vector interference, or the scalar modulation μk, for scalar
interference, was assumed to be a function of the impulse (integrated force) fk, or its magnitude
fk, experienced by the molecule in a collision. For most of (Lewis,  2008) it was assumed that μk∝fk and μk∝fk, but it proved to be possible to extend the models, so that the magnitude of the
induced dipole moment is equal to an arbitrary power or sum of powers of the intermolecular force.
This allows estimates of the infilling of the interference dip by the disproportionality of the induced
dipole moment and force. One particular such model, using data from (Herman and Lewis,  2006), leads to the most realistic
estimate for the infilling of the vector interference dip yet obtained.
In (Lewis,  2008) the drastic assumption was made that collision times occurred at equal intervals. In the
present paper that assumption is removed: the collision times are taken to form a Poisson process.
This is much more realistic than the equal-intervals assumption.
The interference dip is found to be a Lorentzian in this model.


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## Body

## 1. Introduction

Spectra  resulting from dipole moments induced in molecular collisions typically have the form of broad bands with widths determined by the durations of those collisions [1]. However, these broad bands often exhibit narrow features, which result from the coherence or correlation of induced dipole moments extending over successive or sometimes many collisions. The most conspicuous such features are the vector intercollisional interference dips found in the fundamental bands of H2-X spectra and in the pure translational bands of mixtures of rare gas atoms [2]. Also well known are scalar collisional interference features found in the R and P transitions in the fundamental bands of HD-X spectra [3–5] and the corresponding R transitions in the pure rotational spectra [6, 7]. The terms “vector intercollisional interference” or “vector collision-sequence interference” refer to the fact that an internal H2 scalar transition operator is modulated by a vector function of intermolecular displacement. Thus only a Q branch is observed, with the intercollisional dynamics being those which describe the vector property (intermolecular force, to a good approximation) of the external coordinates.The present work is based on that of paper I, which will henceforth be referred to as paper I. In paper it was assumed that the collisions suffered by a molecule occur at equally spaced times. This drastic Ansatz allowed the use of the apparatus of discrete Fourier transforms. In this present work it is assumed that the collision times of a given molecule are distributed exactly as a Poisson process, which is in fact an excellent approximation to reality (see [9, 10]). A summary of this work has appeared in [11].


## 2. Poisson-Distributed Collision Times

At sufficiently low densities and for the study of interference phenomena collisions can be assumed to be instantaneous; the dipole moment induced in one atom or molecule by interaction with a bath of dissimilar atoms or molecules can be represented as(1)μ(t)=∑k‍μkδ(t-tk)e-ιω0t,
where binary collision k occurs at time tk and the dipole moment induced in collision k is μk. For the vector interference dips in Q branches the quantity μj is parallel to and approximately proportional in magnitude to the impulse (integrated force) fk experienced by a molecule in the collision. Our models will be expressed in terms of these impulses fk. A second assumption is that the velocities before and after a collision are uncorrelated and are Gaussian:(2)P(…,vk,vk+1,vk+2,…)=…P(vk)P(vk+1)P(vk+2)…,
where(3)P(v)={12πe-(1/2)v2in two dimensions,1(2π)3/2e-(1/2)v2in three dimensions.
Then the persistence of velocity is zero. This is a fair approximation in some useful cases, such as H2-H2, H2-He, and HD-He, and a good approximation for systems such as H2 in Ar. It is inappropriate for high-mass atoms or molecules dilute in a fluid of low-mass molecules, such as Ar in H2. Equations (2) and (3) are exact for hard spheres of equal mass, as was first shown by Clerk Maxwell  [12].In general, as stated above, the collision timestk approximate to a Poisson process, and in the present work it will be assumed that they are drawn from a true Poisson process, with frequency ν̅, whereas in paper I the collisions were assumed to occur at equal intervals.Equation (1) describes the transition moment for a transition with frequency ω0 in the absence of shifting and broadening mechanisms.It will be assumed initially thatN collisions lie in the finite-time time interval [0,T) such that 0≤t1≤⋯≤tN-1≤tN≤T. The time T is of course a random variable if N is fixed, but whenN is large the record length T can be taken equal to its expectation value, which will be N/ν̅. The Fourier transform of μ(t) is(4)a(ω)=∫-∞∞‍eιωtμ(t)dt=∑k‍μkeιω̃tk,
where ω̃≡ω-ω0. The unaveraged periodogram is given by(5)1T|a(ω)|2=1T∑k‍∑k′‍μk·μk′*eιω̃(tk-tk′),
whence the spectrum itself is given by
(6)S(ω)=limT→∞1T〈|a(ω)|2〉=ν̅[〈μk·μk*〉+2ℜ𝔢〈μk·μk+1*eιω̃(tk-tk+1)〉+2ℜ𝔢〈μk·μk+2*eιω̃(tk-tk+2)〉+⋯].
In the present class of models, as stated above, μk is expressed in terms of the impulse fk. Hence, in full generality,(7)μk=μk(vk,vk+1)
whence, by (2) and (3),(8)〈μk·μk+p〉=〈μk〉2=0forp≥2.
Then (6) for the spectrum becomes(9)S(ω)ν̅=〈μk·μk*〉+2ℜ𝔢〈μk·μk+1*eιω̃(tk-tk+1)〉.A principal assumption of the present model, and the feature in which it differs from the class of models discussed in paper I, is that the intervalsΔk≡tk+1-tk,k=1,2,…,N-1 between collisions are independent of the velocities of the particle, that is, the collision times  tk,…,tk′,… are random variables which constitute a Poisson process. Poisson-distributed collision times are a good approximation for real gases [9], even at high densities [10], though not exact.If the random variables…tk,…,tk′,… form a Poisson process, then the intervals Δi are exponentially distributed [13]:(10)P(Δi)=ν̅e-ν̅Δi.
The intervals [0,t1) and [tN,T) of durations Δ0 and ΔN, respectively, also follow the distribution law (11), this constitutes a well-known “paradox” in the theory of Poisson processes.From (10) it follows immediately that(11)〈eιω̃(tk-tk+1)〉=〈e-ιω̃Δi〉=ν̅∫0∞‍e-(ν̅+ιω̃)ΔdΔ=ν̅ν̅+ιω̃.
Then the spectrum is, from (9), given by(12)S(ω)ν̅=〈μk·μk*〉+2ℜ𝔢{〈μk·μk+1*〉ν̅ν̅+ιω̃}.The dipole moment or transition moment induced in a collision is roughly but not exactly proportional to the intermolecular force; the overlap parts differ in range by about 25%. For purposes of calculating the intercollisional interference the integrated induced dipole momentμk can be taken parallel to the intermolecular force fk, but with magnitude proportional to some nonlinear function of the magnitude of the intermolecular force; specifically, we consider(13)μk=fk(1+αfkβ)=(1+α|vk+1-vk|β)(vk+1-vk),
where α and β are constants, which will not in general be integerial. With this model for μk, the calculation of
(14)maxS=limω→∞S(ω)=〈μk·μk〉,minS=S(0)=〈μk·μk〉+2〈μk·μk+1〉
is exactly the same as in paper I.


### 2.1. Two Dimensions

For two dimensions it was found in paper I that(15)〈μk·μk〉=4+α2β+3Γ(β2+2)+α222β+2Γ(β+2),〈μk·μk+1〉=-2-2α𝒥(β+1,1)-α2𝒥(β+1,β+1),
where(16)𝒥(ν,ν′)≡〈(vk-vk-1)·(vk+1-vk)|vk-vk-1|ν-1×|vk+1-vk|ν′-1〉=3(ν+ν′)/2+1Γ(ν/2+3/2)Γ(ν′/2+3/2)8×2F1(ν2+32,ν′2+32;2;14).
Note that ν and ν′ are exponents, typically fractional, and are not connected in any way with the mean collision frequency ν̅.Because(17)𝒥(β+1,1)=2β+1Γ(β2+2),
it follows that(18)minS=2α2(22β+1Γ(β+2)-𝒥(β+1,β+1)).For the power-law model, which is a limiting case forα≫1 of (13) such that(19)μk=fkfkβ
with β≈-0.25 (for which value see [14]), it can be shown that(20)minS=2(22β+1Γ(β+2)-𝒥(β+1,β+1))=𝒪(β2)
while(21)maxS=22β+2Γ(β+2).
For the case β=-0.25, we obtain(22)minSmaxS|β=-0.25=0.0088399.


### 2.2. Three Dimensions

The necessary integrals to evaluateminS and maxS in three dimensions are evaluated in Appendix A. It is found that(23)〈μk·μk〉=6+2α2β+3πΓ(β2+52)+α222β+3πΓ(β+52),〈μk·μk+1〉=-𝒥3D(1,1)-2α𝒥3D(β+1,1)-α2𝒥3D(β+1,β+1)=-3-2α2β+2πΓ(β2+52)-α23β+3/22πΓ(β2+52)2×2F1(β2+52,β2+52;52;14).
Then from (14), it follows that(24a)minS=α2[22β+3πΓ(β+52)-3β+3/2πΓ(β2+52)2×2F1(β2+52,β2+52;52;14)],(24b)maxS=6+𝒪(α).
The fact that(25)minSmaxS∝α2
is in accord with discussions in [2, 15].For the power-law model given in (19),  we have(26a)minS=0.58β2+𝒪(β3),(26b)maxS=6+𝒪(β)
and, for β=-0.25, we obtain(27)minSmaxS|β=-0.25=0.0064980
which may be compared with the two-dimensional value given in (22).


## 2.1. Two Dimensions

For two dimensions it was found in paper I that(15)〈μk·μk〉=4+α2β+3Γ(β2+2)+α222β+2Γ(β+2),〈μk·μk+1〉=-2-2α𝒥(β+1,1)-α2𝒥(β+1,β+1),
where(16)𝒥(ν,ν′)≡〈(vk-vk-1)·(vk+1-vk)|vk-vk-1|ν-1×|vk+1-vk|ν′-1〉=3(ν+ν′)/2+1Γ(ν/2+3/2)Γ(ν′/2+3/2)8×2F1(ν2+32,ν′2+32;2;14).
Note that ν and ν′ are exponents, typically fractional, and are not connected in any way with the mean collision frequency ν̅.Because(17)𝒥(β+1,1)=2β+1Γ(β2+2),
it follows that(18)minS=2α2(22β+1Γ(β+2)-𝒥(β+1,β+1)).For the power-law model, which is a limiting case forα≫1 of (13) such that(19)μk=fkfkβ
with β≈-0.25 (for which value see [14]), it can be shown that(20)minS=2(22β+1Γ(β+2)-𝒥(β+1,β+1))=𝒪(β2)
while(21)maxS=22β+2Γ(β+2).
For the case β=-0.25, we obtain(22)minSmaxS|β=-0.25=0.0088399.


## 2.2. Three Dimensions

The necessary integrals to evaluateminS and maxS in three dimensions are evaluated in Appendix A. It is found that(23)〈μk·μk〉=6+2α2β+3πΓ(β2+52)+α222β+3πΓ(β+52),〈μk·μk+1〉=-𝒥3D(1,1)-2α𝒥3D(β+1,1)-α2𝒥3D(β+1,β+1)=-3-2α2β+2πΓ(β2+52)-α23β+3/22πΓ(β2+52)2×2F1(β2+52,β2+52;52;14).
Then from (14), it follows that(24a)minS=α2[22β+3πΓ(β+52)-3β+3/2πΓ(β2+52)2×2F1(β2+52,β2+52;52;14)],(24b)maxS=6+𝒪(α).
The fact that(25)minSmaxS∝α2
is in accord with discussions in [2, 15].For the power-law model given in (19),  we have(26a)minS=0.58β2+𝒪(β3),(26b)maxS=6+𝒪(β)
and, for β=-0.25, we obtain(27)minSmaxS|β=-0.25=0.0064980
which may be compared with the two-dimensional value given in (22).


## 3. Conclusions

In this paper, we have extended a class of model developed in paper I for the study of collision-sequence interference effects in collision-induced absorption, to include realistic distributions of collision times. In these models, a single particle is followed. Its collisions are supposed to be instantaneous. In paper I the collisions were assumed to occur at equally spaced times, whereas in the present work the collision times are distributed according to a Poisson process. Velocities are supposed to be completely randomized at each collision. It is supposed that the dipole momentμk or μk induced in a collision is proportional to the integrated intermolecular force fk or fk, respectively, or a power or sum of powers of the integrated intermolecular force.It is important that the model spectra can be determined analytically, or at worst, reduced to straightforward numerical integrations. The models of paper I,  of [11], and of the present work are among the few exactly soluble but nontrivial models found in spectral line-shape studies.The extension of the induced dipole moment model to dipole moments which are proportional to an arbitrary power of the integrated intermolecular force shows that the interference dip is partially filled in for any disproportionality between induced dipole moment and integrated induced dipole moment. In this paper, the calculation is given for the three-dimensional case. For a realistic value of the power the infilling is slight, being about 0.6% of spectral maximum for the three-dimensional case.



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*Source: 561697-2009-07-21.xml*