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Bound-state contributions to the triple-collision operator
Physica 106A (1981) 278-289 © North-Holland Publishing Co.
BOUND-STATE CONTRIBUTIONS TO THE TRIPLE-COLLISION OPERATOR James A. McLENNAN
Department of Physics, No. 16, Lehigh University, Bethlehem, PA 18015, USA
Abstract The kinetic theory for a quantum gas with Boltzmann statistics is analyzed for the case when bound pairs occur. The method used is the binary-collision expansion, applied to the triplecollision operator which occurs in the density-expansion of a Green-Kubo formula. The boundstate contributions are extracted with the aid of the Faddeev analysis of the three-body problem. The results take the form of a binary atom-molecule collision operator, in which the processes of molecular formation and breakup, rearrangement collisions, and elastic and inelastic atommolecule scattering each contribute a non-negative reaction rate. Reducible diagrams contribute the leading part to rearrangement collisions, and also a correlation correction to the Boltzmann collision operator. The fluxes in the Green-Kubo formula are assumed to be sums of singleparticle functions; the atom-molecule collision operator then acts on fluxes which are a sum of an atom term plus a two-particle term obtained by averaging over the molecular state.
R~sum~ La th6orie cin6tique d'un gaz quantique ob6issant h la statistique de Boltzmann est analys6e dans le cas off il existe des paires li6es. La m6thode utilis6e est le d6veloppement en collisions binaires appliqu6 h l'op6rateur de collision triple qui se pr6sente dans le d6veloppement densit6 d'une formule Green-Kubo. Les contributions d'6tats li6s sont extraites avec l'aide de l'analyse de Faddeev du probl6me h trois corps. Les r6sultats prennent la forme d'un op6rateur de collision binaire atom-mol6cule dans lequel les processus de formation et de rupture de mol6cules, les collisions de rdarrangement et la diffusion atome-moldcule, dlastique ou indlastique apportent chacun une contribution non-n6gative h la vitesse de r6action. Des diagrammes r6ductibles fournissent la contribution principale pour les collisions de r6arrangement, et aussi une correction de corr61ation h l'op6rateur de collision de Boltzmann. On suppose que les flux, dans la formule Green-Kubo sont des sommes de fonctions pour une seule particule; l'op6rateur de collision atome-mol~cule agit alors sur des flux qui sont une somme d'un terme atomique et d'un terme h deux particules obtenu en prenant la moyenne sur l'6tat mol6culaire.
I. Introduction
The Boltzmann equation provides a sound basis for the description of processes in low-density monatomic gases in which the particles interact by binary elastic collisions. Analysis of the effects of increasing density has led to a generalized Boltzmann equation, in which the binary collision operator is augmented by including the effects of triple, and in principle higher-order, collisions1). For a classical gas, particularly with repulsive interactions, this theory has been extensively developed and many interesting results obtained. 278
TRIPLE-COLLISION OPERATOR
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Some work has also been done on the triple-collision operator in quantum mechanics for the case that no bound states exist2). In this paper the contribution of two-body bound states to the triplecollision operator will be discussed. Extraction of the bound-state contributions provides a description of those molecular processes which can take place in a three-atom system. The analysis is quantum mechanical, but the particles are assumed to obey Boltzmann statistics. The three-body potential energy is assumed to be the sum of pair potentials. One kind of bound-state contribution is a correlation correction to the linearized Boltzmann collision operator, due to the presence of a third particle near the colliding pair. It is thus loosely analogous to the Enskog correction for a classical hard-sphere gas, but now the third particle is bound to one member of the colliding pair. This term is relatively simple to analyze because it involves only two-body dynamics. The remainder of the triple-collision operator involves the three-body dynamics in an essential way. The boundstate contributions to it are extracted by following Faddeev's 3) analysis of the three-body problem. The result has a gain-loss form with terms corresponding to the processes of formation and dissociation of diatomic molecules, elastic and inelastic scattering, and rearrangement collisions. An approach along the lines described here can provide a systematic derivation of kinetic equations, such as the Waldmann-Snider equation4), for polyatomic gases5).
2. Triple-collision operator We first define the triple-collision operator to be analyzed. Let a, b be single-particle functions of momentum. When it is necessary to distinguish different particles we will write a ( a ) = a(p~) where p~ is the momentum of the ath particle. Let AN, BN be sums of the single-particle functions, N
N
AN = ~ a(a),
BN = ~__ b(a).
~=1
=!
Let A2(t), A3(t) denote the time-dependent operators determined by
tr ~e-~n'aA 2(t )b = 12V0 tr 2e-~n2A 2[S2( t ) - 1]B2, trte-~H, aA3(t)b = ± 2 tr3 { e-~n3A3[S3(t)- liB3 3!Vo - ~ , e-~H*A3[Sv(t)- 1]B31. y
(1)
)
Here /3 = 1/kT where k is Boltzmann's constant and T the Kelvin tem-
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J.A. McLENNAN
perature, HN, N = 1, 2, 3, is the Hamiltonian operator for N particles moving in infinite space, and Hv is the Hamiltonian for a three-particle system in which only the pair ~/interacts. The S's are Heisenberg operators,
SN(t)BN = U~(t)BNUN(t),
UN(t) = exp( - iHNt/h),
Sv(t)B3 = U~t)B3Uv(t),
Uv(t ) = e x p ( - i H 4 / h ) ,
where h = h[2zr, h being Planck's constant, and the dagger denotes the adjoint. In addition in eq. (1) trN denotes the trace associated with the N-particle Hilbert space except that an overall momentum-conserving delta function is left out. Finally, V0 is given by
Vo = h3(2~mkT) -3/2, where m is the mass of a particle. The density expansion of a correlation function 1"6)leads to a triple-collision operator I3 defined by I3 = lim [p3/~2_ p2~3],
(2)
p-,0
where the tilde denotes the Laplace transform, p being the transform variable. We also note the formula for the linearized Boltzmann collision operator
12: 12 = --limp 2d2. p~O
(3)
Our purpose here is to obtain the bound-state contributions to the operator/3. It is well-known that the operators A2, A3 are singular for small p. Indeed eq. (3) indicates that for small p, A2 diverges as p-2. This implies a divergence as p-1 for the first term in eq. (2). However, it works out that A3 has a p-3 divergence such that the combination of terms in eq. (2) approaches a finite limit. The reason for these divergences is easily understood in classical terms. The operators Afft), A3(t) are determined by those regions of phase space for which a binary collision in the case of A2(t), or a connected sequence of collisions in the case of A3(t), occur in time t. The phase space volumes grow roughly as t for Afro and as t 2 for A3(t), and this leads to the stated small-p singularities. Now, if two of the particles are bound, one can expect a less singular behavior. For the two-particle problem the region of interaction is roughly the size of a molecule and does not grow with t. (Actually there is an oscillation, which however has an Abel limit.) For Afro there is a contraction to a two-body problem (an atom plus a diatomic molecule) so the phase space volume increases only linearly with t. Thus the bound-state contributions to A2 and .43 are less singular by one order in p. In particular the bound-state
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contributions to the terms in eq. (2) are individually convergent. Let J denote the bound-state contribution to 13. Then we divide J into two parts J = J~ +./2 with J~ = limp 3(./~2)bs, p~O
./2 = -lim p2(fit3)bs. p-.-0
(4)
3. Two-body term We now proceed to work out the operator Jl. This operator involves only two-body dynamics and so is relatively simple to analyze. The two-particle Hamiltonian is HE = H0 + v, w h e r e / 4 o is the kinetic energy and the potential v depends only on the distance between particles. We use momentum representation, in which case v becomes an integral operator, while H0 amounts to multiplication by the function
E~ = (p~ + p~)/2m. Here fi denotes the two momenta pl, p2. Let P denote the total momentum and K the center-of-mass energy; for a two-particle system they are given by
P =pl+p2,
K = p2/4m.
In addition the relative momentum q and the kinetic energy u of the relative motion are defined by q = ½(pl-p2),
u = qE/m.
We then have E~-- K + u. Let h2, h0 denote the total and non-interacting Hamiltonians in the center-of-mass frame, so hE = h0q-v where h0 is the operation of multiplication by u. The center-of-mass energy can be taken out in eq. (1) to yield
tr le-~n~aA E(t )b = ½Vo trEe-~K A E[U-~(t ), BE]UE(t )e -t3h2,
(5)
where UE(t) = e x p ( - i h E t / h ) , and brackets [ , ] denote the commutator. Let R2(z) denote the resolvent for h2 and Ro(z) the resolvent for h0. The last two factors in eq. (5) can be expressed in terms of RE(Z) by u2(t) exp( -/3h2) = -(2~ri)-' f dzRE( Z)exp[ - z([3 + it/h)].
(6)
C
Here C is a contour enclosing the spectrum of h2. We suppose h2 to have a continuous spectrum consisting of the positive real axis, plus a finite number of eigenvalues (corresponding to bound states) at - el with el > 0. Then C can
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J.A. M c L E N N A N
be taken to consist of two lines parallel to the real axis, say at Imz = E and Im z = -E', connected together on the left of all eigenvalues. We substitute eq. (6) into (5) and then take the Laplace transform, supposing ~ < hp so that the time integral can be taken inside the contour integral. This yields f
trle-O~'a3_2b = -(hVo/41r) / dze -~ tr2e-~rAz[Rz(z - ih), B2]Rz(z).
(7)
C
Here we have set h = tip. The transition operator T(z) is defined by
R2(z) = Ro(z) - Ro(z)T(z)Ro(z).
(8)
Like R:(z) and Ro(z), T(z) is an integral operator on functions of the relative momentum. Its kernel (or matrix elements) will be written T(z, q, q'):
(T(z)f)(q) = f dq'T(z, q, q')f(q'). If eq. (8) is substituted into (7) and the matrix elements written out, the result is
trle-t~aAzb
= (hVol4~) f dz e-~ f dP dq dq' e-~:a2(h)tB2(h) - B2(h')] C
x ro(z, q)ro(z - i h , q)ro(z, q')ro(z - i h , q')T(z - i h , q, q')T(z, q', q) where r0(z, q) = 1/(u - z). The bound states can be separated out in T by using
T(z, q, q') = ~ (ei + z + eiU)+ (ei u') Pi(q, q') + 7"(z, q, q'). Here T(z, q, q') is analytic except for a branch cut along the positive real axis, and has finite limiting values as the real axis is approached from above or below. The operator Pi is the projection onto the ith eigenfunction. The contour C can be deformed into two parts, Ca and C2, enclosing the positive real axis and the eigenvalues - el respectively. The integral over C2 then gives the bound-state contribution to fi,2. Without giving the details of the calculation, we remark that (fii2)bs is found to be singular as p-Z for small p, so it is appropriate to define an operator Y by Y = - l i m p (/i2)bs. p-,O
It is then straight-forward to show that
TRIPLE-COLLISION
f
OPERATOR
dp,cpaYb = (V~/2h 3) ~ e ~e' f dP e-~K[(A2B2), - (A2),{B2),].
283
(9)
Here ~ denotes the Maxwell-Boltzmann distribution function and the brackets < )i denote an average over the ith bound state,
(f)i =
f dqf(q)P~(q,q).
Of course in eq. (9), A2 and B2 are to be considered as functions of P, q rather than of p,, p2. The small-p behavior of fi-2is /~2 ~ - - p - 2 1 2 -- P -I
y + ...,
where the bound-state contribution is contained entirely in Y. The result obtained for J, is then J, = [12 Y + YI2]. This is a modification of the linearized Boltzmann collision operator due to the binding of a third particle to one m e m b e r of the colliding pair.
4. Three-body terms As preparation for the analysis of J2, we first introduce a set of variables which is more or less standard in discussions of the three-body problem. Modifying the notation of the previous section, we now l e t / t denote a set of three momenta pl, p2, p3, and define
E~ = (p~ + p2 + pE)[2m" The total m o m e n t u m P and center-of-mass energy K are now
P =Pl+p2+p3,
K =P2/6m.
A pair 3" can be labeled by the two particles in it or by the third particle. Thus for example 3' = 1, 2 and 3" = 3 both denote the pair 1, 2 as well as particle 3. Let q, to be the relative momentum of pair 3,, so for example q12 = q3 ---- 21(pl -- P2).
In addition define the variables k v by ku = ks = Zspa- ](pl + p2) = pa - 31P,etc.
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J.A. McLENNAN
We also define
u~ = q2/m,
K~ = 3k~/4m.
It follows that E~-K
= UI+KI=
U2q-K2----- U3+~:3.
The three m o m e n t a / , = p~, p:, p3 can be expressed in terms of P and any pair q,. kv; furthermore dp = dpl dp2 dp3 = d P dq v dk v. We will use q, k to denote any of the equivalent pairs qv, kv. To separate out the center-of-mass motion, define h3 -- H3 - K ,
h v = H v - K,
h0 =/40 - K.
The operators ha, h . h0 will be assumed to act on the Hilbert space for three particles with fixed total momentum (that is, the Hilbert space of functions of two vectors q, k). Let R3, Rv, R0 denote the associated resolvents. A straightforward calculation along the lines of the previous section shows that tr~e-~n~afi.ab = -(hV~/127r) f dz e-t~ZX(z),
(10)
C
where
X(z)=tr3e-~K A3{[R3(z-iA),B3]R3(z)- ~ [R~(z
iA),B3]R,(z) I.
(11)
3,
The binary-collision expansion for R3 is
R3 = Ro - ~ RoTvRo + ~ ' RoTvRoTxRo . . . . -f
,
"/A
where the prime on the summation means that no two consecutive pairs are to be the same. The operator Tv satisfies
Rv = R o - RoTvRo and its matrix elements are related to those of the operator T introduced previously by
Tv(z, kq, k'q') = T(z - K,, qv, q'v)8(Pv - P'v). Because of conservation of momentum P = P ' , which implies ~ ( p v - p ~ ) = ~(k~ - k~) and K~ = •r-I
TRIPLE-COLLISION OPERATOR
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If the binary-collision expansion is substituted into eq. (11), it is found that terms with two or fewer T's cancel out. The remaining terms are given by
I 0,z - [U(z - iA), B3]Ro(z) ~_~ Tv(z)Ro(z) + [U(z - iA), B3]U(z)), -/ where +
It can be shown that the first two terms in eq. (12) give no bound-state contributions to the collision operator. The reason is essentially that these terms involve the transition of a single pair between free and bound states, and such a transition is prohibited by conservation of energy and momentum. Hence for the present purposes we may replace X(z) by X'(z), given by
X'(z) = tr3e,~r A3[U (z -iA), B3]U (z). If the matrix elements are written out, this becomes
X'(z) = f dP dk dq dk' dq' e-~rA3(/0[B3(ff) - B3(/0] x U(z -iX, kq, k'q')U(z, k'q', kq). We will refer t o / , as the initial state and/~' as the final state. ' The next step is to combine eq. (10) and the second of eqs. (4). Along the lower half of the contour we keep Im z fixed, and then no contribution is obtained in the limit p - ~ 0 since the integrand remains bounded. Along the upper half we put z = E + i~, ~ = ½~, and obtain
trle-°~laJ2b = -(2 VE/3h ) lim E2 ~ dEe-~EXbs(E + i~), ~0
./
(14)
-a
where - a lies to the left of the spectrum of h3. In addition Xbs denotes the bound-state contribution to X', and is given by Xbs(E + ie) = f dP dk dq dk' dq' e-'KAa(/~)[B3(/~ ') - B3(/0] × IUbs(E +i~, k'q', kq)l 2.
(15)
A non-vanishing result for J2 is obtained only if the factor ~2 in eq. (14) is cancelled by singularities of U(z) as z approaches the real axis. The singular behavior of U(z) has been analyzed by Faddeev, and can be summarized as
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J.A. McLENNAN
follows: Write U in the form
(16)
U = -Ro ~, W~Ro,
where W ~ = - T~RoT~ (1 - 6 ~ ) + ~ ' T~RoT~RoT~ . . . .
.
a
Here the prime on the sum means that a is not equal to either /z or A; note however that in eq. (16) terms with /~ = h are to be included. A matrix element of U is U(z, kq, k'q') = - ( u + K - z) -Z ~ , W~,(z, kq, k ' q ' ) ( u ' + K' -- Z) -l.
(17)
~h
Here u + K means any of the three equal quantities, u~ + K~. Eq. (17) displays free-particle singularities at E = R e Z = U + K and E = U ' + K ' . These singularities are responsible for the three-particle scattering contribution to the triple-collision operator which has been analyzed by Resibois. If the potential supports bound states, then there are corresponding singularities in the transition operators T~ and T~ in W~. These singularities can be exhibited by writing W~ as a sum of four terms (called components by Faddeev) as follows: Wa~(z, kq, k' q') =" W°~(z, kq, k' q') + ~ (z + e~ - K~)-' W ~,,~(z, kq, k' q') !
W~,.~i(z, kq, k'q')(z + e i - K'A)-' + ~,~ (Z + ei -- K~) Z J
x W3im(z, kq, k'q')(z + es - K'A) ~.
q
(18)
Here the coefficients W °, W ~, W 2 and W 3 remain finite as z approaches the bound state singularities at z = - e l +K~ and z = - e i + K ] . The four terms above correspond respectively to no bound pairs in the initial or final states, pair /z being bound in the initial state, pair h being bound in the final state, and pair/~ being bound in the initial state together with the binding of pair h in the final state. Eq. (15) involves two U's, say U+, with arguments z -- E _+ie. Each U has a decomposition as given above, and one might expect the number of combinations of singularities to increase accordingly. However, detailed analysis shows that cross terms do not contribute to J2. Instead there are contributions only from "diagonal" terms: if a pair is bound in U+ (or U_) in the initial state, then in U- (or U+) the same pair must be bound, with the same binding energy, in the initial state, and similarly if there is a bound pair in the final
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state. Thus J2 has basically three kinds of terms: one with no bound pairs in the initial state and one in the final state (which is molecular formation), a second with one bound pair in the initial state and none in the final (which is dissociation), and the third with bound pairs in both initial and final states (which is atom-molecule scattering, including rearrangement collisions if the initial and final bound pairs are different, and inelastic collisions if the initial and final binding energies are different). Once the contributions to J2 are classified as above, the remainder of the analysis is straightforward. Here only the results will be stated. We write ./2 as the sum of three terms corresponding to molecular formation, dissociation, and atom-molecule scattering, J2 = JF + JD + Js. Then JF, JD and Js are found to be JFb : f dp2 dP3q~(2)~(3) f dk' dq'[B3(/0- B3(/*')] × ~ 8(u + K - K'A+ ej)RFxj(kq, k'q'), x, ej
Job = J dp2 dp3,~(2)q~(3) J dk' dq'[B3(p)
B3(p.')]
× ~ e~(U~+e')~(K.-- el - u , - K ,)R D ,i(kq, k'q'),
(19)
I~, e i
Jsb = f dp2 dp3q~(2)q~(3)f dk' dq'tB3(/0
B3(/~')]
× ~,, ,j e ~("~+")~ (K, - ei - ~ '~+ ei)R s ~J(kq, k' q'). The rate coefficients are given by, for example RSi, xi(kq, k' q') = 27r2hS(u, + ei)-2(u~ + ei) -2 x
3 K ~, - e~ - i 0 , kq, k'q') ~(ek, e,)~(e~, ej)W~,,~j(
,
(20)
with similar expressions for R E and R D. Here ~ ( e k , e i ) denotes the Kronecker delta. Clearly Jv and Jo correspond to molecular formation and dissociation. The terms in Js include elastic atom-molecule scattering (/~ = ~t, ei = ej), inelastic scattering (ix = A, e~~ ej), and rearrangement collisions (/~ # )t). The rate coefficients are non-negative, and as a consequence J2 satisfies the linearized form of the H-theorem. It also has the symmetry associated with detailed balance. The W-quantities have a simple dependence on the variable q for a bound pair. This makes it possible to give a more transparent form to the expressions (19). Consider for example the term in Js for which the bound pair is
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J.A. McLENNAN
p~ = h = 2, 3 in both initial and final states, with the initial and final binding energies being e~ and ej. Let this term be denoted by J~j: f
f
Ji,b = J dp2 dp3q~(2)q~(3) J dk'dq'[B3( /O - B3(/~')]e ~("23+') x ~5(K23- ei - K~3+ ej)RS3i,E3j(kq, k'q').
(21)
The dependence on q, q' of W (3) is
W~i.Aj(z, kq, k'q') = (el + u~)(ej + u~)6i(q,,)¢i(q~)A~j(z, k~, k'~), P
--
t
(22)
where 6~ is the eigenfunction for the ith bound state. The coefficients Ais are in principle determined by eq. (18) but their detailed form will not be considered here. The expression (22) is to be substituted into eq. (20), and the resulting form for R s then substituted into eq. (21). For simplicity we assume the energy levels e~ to be non-degenerate and a straightforward calculation then yields the result
Jijb = Ki f d p ~ ( p ) f g d~rij(p,, p ; p I, p')[b(p,) +/3i(p) - b(p ~) -/3i(P')]. Here K~ is the equilibrium constant K~ = 23/2V0 e ~e', while q~m is the Maxwellian for the molecule. In addition p = P2 + p3 denotes the total momentum for the molecule, g is the initial relative velocity (of the atom with respect to the molecule) and dtr~j is the differential cross section for the transition pj, p, e~ ~p~, p', ej. The quantity 13 is the flux for the bound pair, averaged over the molecular state:
/3i(p ) = f dq231t~i(q23)]2[b (p2) + b(p3)]. If the energy levels are degenerate then /3~, /3j are replaced by matrices, say /3~k, /3j~ and in place of the cross section there occurs a product of amplitudes for the transitions p 1, p, ~ ~ p I, p', ~i and p 1, p, Ck ~ p [, p', ~01.This situation is familiar in Waldmann-Snider theory4).
References 1) Cf. J.R. Dorfman and H. van Beijeren, in Statistical Mechanics, Pt. B, B.J. Berne, ed. (Plenum Press, New York, 1977),and other references given therein. 2) P. Resibois, Physica 31 (1965) 645; 32 (1966) 1473.
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3) L.D. Faddeev, Mathematical Aspects of the Three-Body Problem (Davey, New York, 1965). 4) Cf. H. Moraal, Phys. Rep. 17 (1975) 225, and other references given therein. 5) For a different approach, see D.K. Hoffman, D.J. Kouri and Z.H. Top, J. Chem. Phys. 70 (1979) 4640. 6) R. Zwanzig, Phys. Rev. 129 (1963) 486.
BOUND-STATE CONTRIBUTIONS TO THE TRIPLE-COLLISION OPERATOR James A. McLENNAN
Department of Physics, No. 16, Lehigh University, Bethlehem, PA 18015, USA
Abstract The kinetic theory for a quantum gas with Boltzmann statistics is analyzed for the case when bound pairs occur. The method used is the binary-collision expansion, applied to the triplecollision operator which occurs in the density-expansion of a Green-Kubo formula. The boundstate contributions are extracted with the aid of the Faddeev analysis of the three-body problem. The results take the form of a binary atom-molecule collision operator, in which the processes of molecular formation and breakup, rearrangement collisions, and elastic and inelastic atommolecule scattering each contribute a non-negative reaction rate. Reducible diagrams contribute the leading part to rearrangement collisions, and also a correlation correction to the Boltzmann collision operator. The fluxes in the Green-Kubo formula are assumed to be sums of singleparticle functions; the atom-molecule collision operator then acts on fluxes which are a sum of an atom term plus a two-particle term obtained by averaging over the molecular state.
R~sum~ La th6orie cin6tique d'un gaz quantique ob6issant h la statistique de Boltzmann est analys6e dans le cas off il existe des paires li6es. La m6thode utilis6e est le d6veloppement en collisions binaires appliqu6 h l'op6rateur de collision triple qui se pr6sente dans le d6veloppement densit6 d'une formule Green-Kubo. Les contributions d'6tats li6s sont extraites avec l'aide de l'analyse de Faddeev du probl6me h trois corps. Les r6sultats prennent la forme d'un op6rateur de collision binaire atom-mol6cule dans lequel les processus de formation et de rupture de mol6cules, les collisions de rdarrangement et la diffusion atome-moldcule, dlastique ou indlastique apportent chacun une contribution non-n6gative h la vitesse de r6action. Des diagrammes r6ductibles fournissent la contribution principale pour les collisions de r6arrangement, et aussi une correction de corr61ation h l'op6rateur de collision de Boltzmann. On suppose que les flux, dans la formule Green-Kubo sont des sommes de fonctions pour une seule particule; l'op6rateur de collision atome-mol~cule agit alors sur des flux qui sont une somme d'un terme atomique et d'un terme h deux particules obtenu en prenant la moyenne sur l'6tat mol6culaire.
I. Introduction
The Boltzmann equation provides a sound basis for the description of processes in low-density monatomic gases in which the particles interact by binary elastic collisions. Analysis of the effects of increasing density has led to a generalized Boltzmann equation, in which the binary collision operator is augmented by including the effects of triple, and in principle higher-order, collisions1). For a classical gas, particularly with repulsive interactions, this theory has been extensively developed and many interesting results obtained. 278
TRIPLE-COLLISION OPERATOR
279
Some work has also been done on the triple-collision operator in quantum mechanics for the case that no bound states exist2). In this paper the contribution of two-body bound states to the triplecollision operator will be discussed. Extraction of the bound-state contributions provides a description of those molecular processes which can take place in a three-atom system. The analysis is quantum mechanical, but the particles are assumed to obey Boltzmann statistics. The three-body potential energy is assumed to be the sum of pair potentials. One kind of bound-state contribution is a correlation correction to the linearized Boltzmann collision operator, due to the presence of a third particle near the colliding pair. It is thus loosely analogous to the Enskog correction for a classical hard-sphere gas, but now the third particle is bound to one member of the colliding pair. This term is relatively simple to analyze because it involves only two-body dynamics. The remainder of the triple-collision operator involves the three-body dynamics in an essential way. The boundstate contributions to it are extracted by following Faddeev's 3) analysis of the three-body problem. The result has a gain-loss form with terms corresponding to the processes of formation and dissociation of diatomic molecules, elastic and inelastic scattering, and rearrangement collisions. An approach along the lines described here can provide a systematic derivation of kinetic equations, such as the Waldmann-Snider equation4), for polyatomic gases5).
2. Triple-collision operator We first define the triple-collision operator to be analyzed. Let a, b be single-particle functions of momentum. When it is necessary to distinguish different particles we will write a ( a ) = a(p~) where p~ is the momentum of the ath particle. Let AN, BN be sums of the single-particle functions, N
N
AN = ~ a(a),
BN = ~__ b(a).
~=1
=!
Let A2(t), A3(t) denote the time-dependent operators determined by
tr ~e-~n'aA 2(t )b = 12V0 tr 2e-~n2A 2[S2( t ) - 1]B2, trte-~H, aA3(t)b = ± 2 tr3 { e-~n3A3[S3(t)- liB3 3!Vo - ~ , e-~H*A3[Sv(t)- 1]B31. y
(1)
)
Here /3 = 1/kT where k is Boltzmann's constant and T the Kelvin tem-
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J.A. McLENNAN
perature, HN, N = 1, 2, 3, is the Hamiltonian operator for N particles moving in infinite space, and Hv is the Hamiltonian for a three-particle system in which only the pair ~/interacts. The S's are Heisenberg operators,
SN(t)BN = U~(t)BNUN(t),
UN(t) = exp( - iHNt/h),
Sv(t)B3 = U~t)B3Uv(t),
Uv(t ) = e x p ( - i H 4 / h ) ,
where h = h[2zr, h being Planck's constant, and the dagger denotes the adjoint. In addition in eq. (1) trN denotes the trace associated with the N-particle Hilbert space except that an overall momentum-conserving delta function is left out. Finally, V0 is given by
Vo = h3(2~mkT) -3/2, where m is the mass of a particle. The density expansion of a correlation function 1"6)leads to a triple-collision operator I3 defined by I3 = lim [p3/~2_ p2~3],
(2)
p-,0
where the tilde denotes the Laplace transform, p being the transform variable. We also note the formula for the linearized Boltzmann collision operator
12: 12 = --limp 2d2. p~O
(3)
Our purpose here is to obtain the bound-state contributions to the operator/3. It is well-known that the operators A2, A3 are singular for small p. Indeed eq. (3) indicates that for small p, A2 diverges as p-2. This implies a divergence as p-1 for the first term in eq. (2). However, it works out that A3 has a p-3 divergence such that the combination of terms in eq. (2) approaches a finite limit. The reason for these divergences is easily understood in classical terms. The operators Afft), A3(t) are determined by those regions of phase space for which a binary collision in the case of A2(t), or a connected sequence of collisions in the case of A3(t), occur in time t. The phase space volumes grow roughly as t for Afro and as t 2 for A3(t), and this leads to the stated small-p singularities. Now, if two of the particles are bound, one can expect a less singular behavior. For the two-particle problem the region of interaction is roughly the size of a molecule and does not grow with t. (Actually there is an oscillation, which however has an Abel limit.) For Afro there is a contraction to a two-body problem (an atom plus a diatomic molecule) so the phase space volume increases only linearly with t. Thus the bound-state contributions to A2 and .43 are less singular by one order in p. In particular the bound-state
TRIPLE-COLLISION OPERATOR
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contributions to the terms in eq. (2) are individually convergent. Let J denote the bound-state contribution to 13. Then we divide J into two parts J = J~ +./2 with J~ = limp 3(./~2)bs, p~O
./2 = -lim p2(fit3)bs. p-.-0
(4)
3. Two-body term We now proceed to work out the operator Jl. This operator involves only two-body dynamics and so is relatively simple to analyze. The two-particle Hamiltonian is HE = H0 + v, w h e r e / 4 o is the kinetic energy and the potential v depends only on the distance between particles. We use momentum representation, in which case v becomes an integral operator, while H0 amounts to multiplication by the function
E~ = (p~ + p~)/2m. Here fi denotes the two momenta pl, p2. Let P denote the total momentum and K the center-of-mass energy; for a two-particle system they are given by
P =pl+p2,
K = p2/4m.
In addition the relative momentum q and the kinetic energy u of the relative motion are defined by q = ½(pl-p2),
u = qE/m.
We then have E~-- K + u. Let h2, h0 denote the total and non-interacting Hamiltonians in the center-of-mass frame, so hE = h0q-v where h0 is the operation of multiplication by u. The center-of-mass energy can be taken out in eq. (1) to yield
tr le-~n~aA E(t )b = ½Vo trEe-~K A E[U-~(t ), BE]UE(t )e -t3h2,
(5)
where UE(t) = e x p ( - i h E t / h ) , and brackets [ , ] denote the commutator. Let R2(z) denote the resolvent for h2 and Ro(z) the resolvent for h0. The last two factors in eq. (5) can be expressed in terms of RE(Z) by u2(t) exp( -/3h2) = -(2~ri)-' f dzRE( Z)exp[ - z([3 + it/h)].
(6)
C
Here C is a contour enclosing the spectrum of h2. We suppose h2 to have a continuous spectrum consisting of the positive real axis, plus a finite number of eigenvalues (corresponding to bound states) at - el with el > 0. Then C can
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J.A. M c L E N N A N
be taken to consist of two lines parallel to the real axis, say at Imz = E and Im z = -E', connected together on the left of all eigenvalues. We substitute eq. (6) into (5) and then take the Laplace transform, supposing ~ < hp so that the time integral can be taken inside the contour integral. This yields f
trle-O~'a3_2b = -(hVo/41r) / dze -~ tr2e-~rAz[Rz(z - ih), B2]Rz(z).
(7)
C
Here we have set h = tip. The transition operator T(z) is defined by
R2(z) = Ro(z) - Ro(z)T(z)Ro(z).
(8)
Like R:(z) and Ro(z), T(z) is an integral operator on functions of the relative momentum. Its kernel (or matrix elements) will be written T(z, q, q'):
(T(z)f)(q) = f dq'T(z, q, q')f(q'). If eq. (8) is substituted into (7) and the matrix elements written out, the result is
trle-t~aAzb
= (hVol4~) f dz e-~ f dP dq dq' e-~:a2(h)tB2(h) - B2(h')] C
x ro(z, q)ro(z - i h , q)ro(z, q')ro(z - i h , q')T(z - i h , q, q')T(z, q', q) where r0(z, q) = 1/(u - z). The bound states can be separated out in T by using
T(z, q, q') = ~ (ei + z + eiU)+ (ei u') Pi(q, q') + 7"(z, q, q'). Here T(z, q, q') is analytic except for a branch cut along the positive real axis, and has finite limiting values as the real axis is approached from above or below. The operator Pi is the projection onto the ith eigenfunction. The contour C can be deformed into two parts, Ca and C2, enclosing the positive real axis and the eigenvalues - el respectively. The integral over C2 then gives the bound-state contribution to fi,2. Without giving the details of the calculation, we remark that (fii2)bs is found to be singular as p-Z for small p, so it is appropriate to define an operator Y by Y = - l i m p (/i2)bs. p-,O
It is then straight-forward to show that
TRIPLE-COLLISION
f
OPERATOR
dp,cpaYb = (V~/2h 3) ~ e ~e' f dP e-~K[(A2B2), - (A2),{B2),].
283
(9)
Here ~ denotes the Maxwell-Boltzmann distribution function and the brackets < )i denote an average over the ith bound state,
(f)i =
f dqf(q)P~(q,q).
Of course in eq. (9), A2 and B2 are to be considered as functions of P, q rather than of p,, p2. The small-p behavior of fi-2is /~2 ~ - - p - 2 1 2 -- P -I
y + ...,
where the bound-state contribution is contained entirely in Y. The result obtained for J, is then J, = [12 Y + YI2]. This is a modification of the linearized Boltzmann collision operator due to the binding of a third particle to one m e m b e r of the colliding pair.
4. Three-body terms As preparation for the analysis of J2, we first introduce a set of variables which is more or less standard in discussions of the three-body problem. Modifying the notation of the previous section, we now l e t / t denote a set of three momenta pl, p2, p3, and define
E~ = (p~ + p2 + pE)[2m" The total m o m e n t u m P and center-of-mass energy K are now
P =Pl+p2+p3,
K =P2/6m.
A pair 3" can be labeled by the two particles in it or by the third particle. Thus for example 3' = 1, 2 and 3" = 3 both denote the pair 1, 2 as well as particle 3. Let q, to be the relative momentum of pair 3,, so for example q12 = q3 ---- 21(pl -- P2).
In addition define the variables k v by ku = ks = Zspa- ](pl + p2) = pa - 31P,etc.
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J.A. McLENNAN
We also define
u~ = q2/m,
K~ = 3k~/4m.
It follows that E~-K
= UI+KI=
U2q-K2----- U3+~:3.
The three m o m e n t a / , = p~, p:, p3 can be expressed in terms of P and any pair q,. kv; furthermore dp = dpl dp2 dp3 = d P dq v dk v. We will use q, k to denote any of the equivalent pairs qv, kv. To separate out the center-of-mass motion, define h3 -- H3 - K ,
h v = H v - K,
h0 =/40 - K.
The operators ha, h . h0 will be assumed to act on the Hilbert space for three particles with fixed total momentum (that is, the Hilbert space of functions of two vectors q, k). Let R3, Rv, R0 denote the associated resolvents. A straightforward calculation along the lines of the previous section shows that tr~e-~n~afi.ab = -(hV~/127r) f dz e-t~ZX(z),
(10)
C
where
X(z)=tr3e-~K A3{[R3(z-iA),B3]R3(z)- ~ [R~(z
iA),B3]R,(z) I.
(11)
3,
The binary-collision expansion for R3 is
R3 = Ro - ~ RoTvRo + ~ ' RoTvRoTxRo . . . . -f
,
"/A
where the prime on the summation means that no two consecutive pairs are to be the same. The operator Tv satisfies
Rv = R o - RoTvRo and its matrix elements are related to those of the operator T introduced previously by
Tv(z, kq, k'q') = T(z - K,, qv, q'v)8(Pv - P'v). Because of conservation of momentum P = P ' , which implies ~ ( p v - p ~ ) = ~(k~ - k~) and K~ = •r-I
TRIPLE-COLLISION OPERATOR
285
If the binary-collision expansion is substituted into eq. (11), it is found that terms with two or fewer T's cancel out. The remaining terms are given by
I 0,z - [U(z - iA), B3]Ro(z) ~_~ Tv(z)Ro(z) + [U(z - iA), B3]U(z)), -/ where +
It can be shown that the first two terms in eq. (12) give no bound-state contributions to the collision operator. The reason is essentially that these terms involve the transition of a single pair between free and bound states, and such a transition is prohibited by conservation of energy and momentum. Hence for the present purposes we may replace X(z) by X'(z), given by
X'(z) = tr3e,~r A3[U (z -iA), B3]U (z). If the matrix elements are written out, this becomes
X'(z) = f dP dk dq dk' dq' e-~rA3(/0[B3(ff) - B3(/0] x U(z -iX, kq, k'q')U(z, k'q', kq). We will refer t o / , as the initial state and/~' as the final state. ' The next step is to combine eq. (10) and the second of eqs. (4). Along the lower half of the contour we keep Im z fixed, and then no contribution is obtained in the limit p - ~ 0 since the integrand remains bounded. Along the upper half we put z = E + i~, ~ = ½~, and obtain
trle-°~laJ2b = -(2 VE/3h ) lim E2 ~ dEe-~EXbs(E + i~), ~0
./
(14)
-a
where - a lies to the left of the spectrum of h3. In addition Xbs denotes the bound-state contribution to X', and is given by Xbs(E + ie) = f dP dk dq dk' dq' e-'KAa(/~)[B3(/~ ') - B3(/0] × IUbs(E +i~, k'q', kq)l 2.
(15)
A non-vanishing result for J2 is obtained only if the factor ~2 in eq. (14) is cancelled by singularities of U(z) as z approaches the real axis. The singular behavior of U(z) has been analyzed by Faddeev, and can be summarized as
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J.A. McLENNAN
follows: Write U in the form
(16)
U = -Ro ~, W~Ro,
where W ~ = - T~RoT~ (1 - 6 ~ ) + ~ ' T~RoT~RoT~ . . . .
.
a
Here the prime on the sum means that a is not equal to either /z or A; note however that in eq. (16) terms with /~ = h are to be included. A matrix element of U is U(z, kq, k'q') = - ( u + K - z) -Z ~ , W~,(z, kq, k ' q ' ) ( u ' + K' -- Z) -l.
(17)
~h
Here u + K means any of the three equal quantities, u~ + K~. Eq. (17) displays free-particle singularities at E = R e Z = U + K and E = U ' + K ' . These singularities are responsible for the three-particle scattering contribution to the triple-collision operator which has been analyzed by Resibois. If the potential supports bound states, then there are corresponding singularities in the transition operators T~ and T~ in W~. These singularities can be exhibited by writing W~ as a sum of four terms (called components by Faddeev) as follows: Wa~(z, kq, k' q') =" W°~(z, kq, k' q') + ~ (z + e~ - K~)-' W ~,,~(z, kq, k' q') !
W~,.~i(z, kq, k'q')(z + e i - K'A)-' + ~,~ (Z + ei -- K~) Z J
x W3im(z, kq, k'q')(z + es - K'A) ~.
q
(18)
Here the coefficients W °, W ~, W 2 and W 3 remain finite as z approaches the bound state singularities at z = - e l +K~ and z = - e i + K ] . The four terms above correspond respectively to no bound pairs in the initial or final states, pair /z being bound in the initial state, pair h being bound in the final state, and pair/~ being bound in the initial state together with the binding of pair h in the final state. Eq. (15) involves two U's, say U+, with arguments z -- E _+ie. Each U has a decomposition as given above, and one might expect the number of combinations of singularities to increase accordingly. However, detailed analysis shows that cross terms do not contribute to J2. Instead there are contributions only from "diagonal" terms: if a pair is bound in U+ (or U_) in the initial state, then in U- (or U+) the same pair must be bound, with the same binding energy, in the initial state, and similarly if there is a bound pair in the final
TRIPLE-COLLISION OPERATOR
287
state. Thus J2 has basically three kinds of terms: one with no bound pairs in the initial state and one in the final state (which is molecular formation), a second with one bound pair in the initial state and none in the final (which is dissociation), and the third with bound pairs in both initial and final states (which is atom-molecule scattering, including rearrangement collisions if the initial and final bound pairs are different, and inelastic collisions if the initial and final binding energies are different). Once the contributions to J2 are classified as above, the remainder of the analysis is straightforward. Here only the results will be stated. We write ./2 as the sum of three terms corresponding to molecular formation, dissociation, and atom-molecule scattering, J2 = JF + JD + Js. Then JF, JD and Js are found to be JFb : f dp2 dP3q~(2)~(3) f dk' dq'[B3(/0- B3(/*')] × ~ 8(u + K - K'A+ ej)RFxj(kq, k'q'), x, ej
Job = J dp2 dp3,~(2)q~(3) J dk' dq'[B3(p)
B3(p.')]
× ~ e~(U~+e')~(K.-- el - u , - K ,)R D ,i(kq, k'q'),
(19)
I~, e i
Jsb = f dp2 dp3q~(2)q~(3)f dk' dq'tB3(/0
B3(/~')]
× ~,, ,j e ~("~+")~ (K, - ei - ~ '~+ ei)R s ~J(kq, k' q'). The rate coefficients are given by, for example RSi, xi(kq, k' q') = 27r2hS(u, + ei)-2(u~ + ei) -2 x
3 K ~, - e~ - i 0 , kq, k'q') ~(ek, e,)~(e~, ej)W~,,~j(
,
(20)
with similar expressions for R E and R D. Here ~ ( e k , e i ) denotes the Kronecker delta. Clearly Jv and Jo correspond to molecular formation and dissociation. The terms in Js include elastic atom-molecule scattering (/~ = ~t, ei = ej), inelastic scattering (ix = A, e~~ ej), and rearrangement collisions (/~ # )t). The rate coefficients are non-negative, and as a consequence J2 satisfies the linearized form of the H-theorem. It also has the symmetry associated with detailed balance. The W-quantities have a simple dependence on the variable q for a bound pair. This makes it possible to give a more transparent form to the expressions (19). Consider for example the term in Js for which the bound pair is
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J.A. McLENNAN
p~ = h = 2, 3 in both initial and final states, with the initial and final binding energies being e~ and ej. Let this term be denoted by J~j: f
f
Ji,b = J dp2 dp3q~(2)q~(3) J dk'dq'[B3( /O - B3(/~')]e ~("23+') x ~5(K23- ei - K~3+ ej)RS3i,E3j(kq, k'q').
(21)
The dependence on q, q' of W (3) is
W~i.Aj(z, kq, k'q') = (el + u~)(ej + u~)6i(q,,)¢i(q~)A~j(z, k~, k'~), P
--
t
(22)
where 6~ is the eigenfunction for the ith bound state. The coefficients Ais are in principle determined by eq. (18) but their detailed form will not be considered here. The expression (22) is to be substituted into eq. (20), and the resulting form for R s then substituted into eq. (21). For simplicity we assume the energy levels e~ to be non-degenerate and a straightforward calculation then yields the result
Jijb = Ki f d p ~ ( p ) f g d~rij(p,, p ; p I, p')[b(p,) +/3i(p) - b(p ~) -/3i(P')]. Here K~ is the equilibrium constant K~ = 23/2V0 e ~e', while q~m is the Maxwellian for the molecule. In addition p = P2 + p3 denotes the total momentum for the molecule, g is the initial relative velocity (of the atom with respect to the molecule) and dtr~j is the differential cross section for the transition pj, p, e~ ~p~, p', ej. The quantity 13 is the flux for the bound pair, averaged over the molecular state:
/3i(p ) = f dq231t~i(q23)]2[b (p2) + b(p3)]. If the energy levels are degenerate then /3~, /3j are replaced by matrices, say /3~k, /3j~ and in place of the cross section there occurs a product of amplitudes for the transitions p 1, p, ~ ~ p I, p', ~i and p 1, p, Ck ~ p [, p', ~01.This situation is familiar in Waldmann-Snider theory4).
References 1) Cf. J.R. Dorfman and H. van Beijeren, in Statistical Mechanics, Pt. B, B.J. Berne, ed. (Plenum Press, New York, 1977),and other references given therein. 2) P. Resibois, Physica 31 (1965) 645; 32 (1966) 1473.
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3) L.D. Faddeev, Mathematical Aspects of the Three-Body Problem (Davey, New York, 1965). 4) Cf. H. Moraal, Phys. Rep. 17 (1975) 225, and other references given therein. 5) For a different approach, see D.K. Hoffman, D.J. Kouri and Z.H. Top, J. Chem. Phys. 70 (1979) 4640. 6) R. Zwanzig, Phys. Rev. 129 (1963) 486.