- Email: [email protected]
Compound state resonances in molecular collisions: the integral elastic cross section for D2-Xe scattering
CHEMICAL
PHYSICS
LETTERS
COMPOUND THE
1 (19G7l 139 - 112. NORTH-HOLLAND
STATE
INTEGRAL
RESONANCES
ELASTIC
Theoretical Chemistv
A.
Compound
state
resonnnces
FOR
Dj-Xe
in molecular
COMPANY.
AMSTERDAM
COLLISIONS: SCATTERING
* *reJc
*
.’ -
MICHA
Instifute. Univevsitr of Wisconsin. Madison. Received
for the D2 -Xe
IN MOLECULAR
CROSS SECTION David
PUBLISHING
Wisconsin,
USA
5 May 1967
collisions
arc predicted,
and numerical resui:s
are presented
system.
The detailed behavior of scattering cross sections for molecular collisions studied, both theoretically and experimentally, during recent years. It is the to present quantitative arguments that predict the existence of compound state collisions. The isotropic part of the interaction between two molecules, or a molecule
has been extensively purpose of-this letier resonances in molecular
and an atom. is weLL described by a Van der Waals potential, attractive at large distances. In a collision process the relative kinetic energy in the molecular system may be transformed into internal (vibrational and rotational) excitation energy of one of the molecules, in which case the Van der Waals interaction between the excited molecule and its partner will lead to the formation of a pair described by quasi-stationary wave funetions, since the colliding particles may separate again. These quasi-stationary or compound states would be evidenced as resonance peaks in the scattering cross sections. To describe this process quantitatively it is necessary to consider the couplii:g between open and closed channels, i.e., between energetically accessible and inaccessible collision states, respectively. We concentrate, for simplicity, on neutral atom-diatomic molecule collisions. The scattering of an atom in the ?SO state with a molecule in the lx0 state may be described 2s the scattering of a StructureLess particle by a vibrating rotator with internal energy ~7ni (which defines the (n,j) channel), where II and j are the vibrational and rotational quantum numbers of BC. The interaction potential V(R, V,Y) is a function of the distance >-_be_tween B and C, the distance R behveen their center of mass and A, and the angIe Y defined by cos y = f?- r . For a total energy in the barycentric system E = A2knj /(2P 1 + ‘Qj
(11
the excitation to .a closed channel, with Iv,lj > E, leads to an imaginary wave number k,zj. The usual boundary conditibns of collision theory may then be used provided the choice 9m(kj,j) > 0 is made. Using the total angular’momentum representation (ZjJ~12),where 2 is the orbital angular momentum, the scattering matrix SAt XJ”, with X = fjzjl), is diagonal in Jand the total parity I?, independent of AIand symmetrical Cl]. The effective hamiltonian method for resonance collis:ons [Z] separates the part of this matrix connecting open channel states, $, in a term corresponding to direct scattering and mother corresponding to compound state scattering,
Hey ypJ isJa matrix connecting open channel states, which depends on the quasi-stationary state p, and For l?p and Ep. are the resonance width and energy for the A-BC Van der Waals pair, respectively. energies E smaller than the threshold energy for the first allowed (rotational) transition there is only one open channel, the elastic one, for n =j =O. The integral elastic cross section o(k), with k=RoO is a(k) * This research iKfty 1967
was supported
by the National
= ad(k)
+ UC(~)
Aeronautics 139
(3)
+ oi(a),
and S&e
Administration
Grant NsG-275-62.
D. A. MICHA
140
where the terms correspond by ad(k) = (4a/k2) c
J
compound state and interference
to direct,
contributions.
They are given
(2J+ 1) sir? qJ
o,(k) = (Cr/k2) I$ (2J+ 1) [F
i~z\ 2
(4a)
sin2
?$
+
2 pFD,
sin 7); sin $32e
Y$$,
exp i (n $- ,$)I
(4b)
and (2J+ 1) c sin # sin $%?e ~5 exp i (nJ$) . (4c) P The direct phase shifts 8-I correspond to scattering by a potential equal to the sum of the isotropic part of V averaged over the vibrational motion for n =j =O, plus the isotropic, non-local contribution from far resonances, i.e., from eoupliri, to compound states of high energy. The resonance phase shifts 5 are given by; of(k) = -(8r/k2) 7
(5) The previous equations have been used to describe the elastic The potential for this system may be approximated by v(R, ;,7) = Y’(R.
v)[l + B’P2(cos
scattering
of D2 with ?Z= j = 0 by Xe.
y)] + V”(R, r)[l + b”P2(cos
Y)] ,
primes and double primes refer to short range and long range respectively, (12, 6) potential may be used ior the isotropic part,
where
rjO(R)
= E(R/R,,,?~
with parameters R,,? = 3.9 ‘x 1O-8 cm and E = 1.2 x lo-l4 erg (*S- 8%) (deduced from experiments [3]), b” = 0.09 (*lo% [4] and b’ = 0.5 (*30%, which has only been estimated. The Van der Waals pair would have three bound states [5]. Since the parity of j is conserved during the coilision we may use eqs. (4a-c) for all E < Kk2. This situation is illustrated schematically in fig. l(a) for J = 0. The resonance phase shifts ?$J may be easily obtained using the following simplifications: (1) Retain only the lowest order terms in the anisotropy, in which case far resonances do not contribute to the direct collision and the resonances become nonoverlapping. (2) Calculate the integrals over scattering wave functions using the method of stationary phases, which leads to asymptotic expansions in (Rmk)- ‘. Since Rmk > 20 in the resoniulce region, it suffices to keep only the lowest order term. The results indicate that ad(k) is determined by the value of the potential around the classical turnIng point, and that the resonance widths depend primarily upon the short range anisotropy. Table 1 shows the computed resonance parameters TO< and E9J for 0 =z J C 5. While the values of the To” are highly sensitive to the pctential parameters, the EOJ (with errors ? 4%) are less so. The direct cross section Q(k) is given, to within a relative error of order b’2, by the scattering thrrzrgh uio(R) and may be written [S] as the sum
(6)
and a Lennard-Jones
- 2e(R/R,,,?,
ro
Rm
R-+
G 'C
40
5
-5
IO
E-E&y Fig.
1. (a.) The interacting
quasi-stationary
and scatter-
iw functions for an elastic coIIision with tl- j= I) and
J= 0 jschematic) (b) The direct (d,L._ comnound ~~~_ ~~~state (c\ . , and interference (i) contributions to the integral elastic cross section (heavy line) for J=O and p=O.
COMPOUND STATE RESON.ANCES IN MOLECULAR
COLLISIONS
241
Table 1 Resonance widths l$ and energies E$ for D2 +Xe (in ergs). J
lo”!
xE
J
0
l.ooO1014 x I?0
0
2.669
2.6 A 10-3
1 2
2.660 2.902
3.9 K 10-5 6.1 x 1O-4
3
2.932
7.2 x 1O-4
4
2.982
7.5 x 10-4
5
3.036
7.7 x 10-4
h -?
0 B b 0.990-J=O I o.g=-940
2
I III 9.45
3 II 950 IO-%
9.55 (cm set?)
960
9.65
9.7c
Fig. 2. Dependence tif the integral elastic cross secv in the region of resonances with p=OnndO CJ<5_
tion uon the velocity
of a smoothly
varying
part cd(h) plus a contribu;:on Q(k)
= 8.083
RL
from
the glory
(%Rmr_r/ti2)’
extrema
Aad(
where
(Rmk)-’
(8)
sin (2nm - $7)].
(9)
and
A&)
=
(4,/k2)h0/[(+.$
The maximum direct phase shift nm and n;;1= (d2n/dZ2)o at the glory angular momentum 10 are avaiIable from tables [?‘I. The cross section Crd(k) would also include narrow single particle “orbiting resonances”, but they would appear, for D2 + Xe, at energies appreciably smaller than the compound state resonance energies. The computation of uc and oi requires the knowledge of the direct phase shifts nJT which may also be obtained by interpolation from tables [8]. The three contributions to o(k), for Z=O ,wd artiund the energy E 00, are shown in fig. l(b). Fig. 2 shows the quantity o/@d versus the velocity 2 = fik//.f for a range that includes the resonances for p=O and 0 C J C 5. The resonance disturbances extend over several meters
per second in each case and the variations in the cross section are of the order of 1%. These variations might be detectable by means of recent experimental techniques [9]. The accuracy in the velocity selection of molecular beams should be sufficient to identify the position of each isolated resonance, although insufficient to give its shape. An experimental search for these compound state resonances should be of interest not onP;r in comiection with the description of molecular collisions, but also as a possible means of obtaining information on the interaction potential in eq. (6). Most molecular systems interact through potentials with capacity for one OF more bound states, and they should form compound states. A variety of situations may be ex-
pected, depending on the type of interaction and on the relation between l and the energy levels of the molecules. A detailed study of the atom-diatomic molecule case, including the molecular vibrational and rotational motions and a discussion of the computational method, as well as a few more numericaL examples, has been submitted for publication [lo]. The author thanks Professor
R. B. Bernstein for several
helpful discussions.
REFERENCES f l] (a) J. M. Blntt and L. C . Biedenharn, Rev.Mod.Phys.24 (London) 67 (1964) 1103; (b) A.hI.Arthurs and A.Dalgarno.
Proc.Roy.Soc.(London)
(1952) 258; see for corrections A256 (i960) 540.
R.Huby.
%oc.P::..s.xx.
2
D.A.
MICHA
Z] H.Feshbach, Ann.Phys.(N.Y.) 5 (1958) 357: 19 (1962) 287. 31 R. Helbing, Dissertation, Bonn University (1966). 4) J.O.Hirschfelder, C.F.Curtiss and R.B.Bird. Molecular Theory of Gases and Liquids (John Wiley ant! Sons, New York, 1964) ch. 13. 5] H. Harrison and R. B. Bernstein, J.Chem.Phys.38 (1963) 2135. S] See: R.B.Bernstein in Advances in Chemical Physics, ed. J-Ross. Vo1.X (Interscience Publisher;, New York, 1966) p-75. 2nd references there. 7] T.J. P.O’~rion. Tkzoretical Chemistr!; Institute, Univ. of Wisconsin, Report WIS-TCI-100G (1955). t;i R. B. Bernstein, J.Chem. Phys.33 (1960) 195. 91 (a) Fr.Von Busch, H.J.Strunck and C.Schlier, Z.Physik 199 (1967) 518; (b) See: R. B. Bernstein and J. T. Muckerman, Theoretical Chemistry Institute, Univ. of Wisconsin, Report \VI5-iCI-200 (1967) (to be published) for a review of experimental results. 0] David A.Micha, Theoretical Chemistry Institute, Univ. of Wisconsin, Report WIS-TCI-223 (1967).
PHYSICS
LETTERS
COMPOUND THE
1 (19G7l 139 - 112. NORTH-HOLLAND
STATE
INTEGRAL
RESONANCES
ELASTIC
Theoretical Chemistv
A.
Compound
state
resonnnces
FOR
Dj-Xe
in molecular
COMPANY.
AMSTERDAM
COLLISIONS: SCATTERING
* *reJc
*
.’ -
MICHA
Instifute. Univevsitr of Wisconsin. Madison. Received
for the D2 -Xe
IN MOLECULAR
CROSS SECTION David
PUBLISHING
Wisconsin,
USA
5 May 1967
collisions
arc predicted,
and numerical resui:s
are presented
system.
The detailed behavior of scattering cross sections for molecular collisions studied, both theoretically and experimentally, during recent years. It is the to present quantitative arguments that predict the existence of compound state collisions. The isotropic part of the interaction between two molecules, or a molecule
has been extensively purpose of-this letier resonances in molecular
and an atom. is weLL described by a Van der Waals potential, attractive at large distances. In a collision process the relative kinetic energy in the molecular system may be transformed into internal (vibrational and rotational) excitation energy of one of the molecules, in which case the Van der Waals interaction between the excited molecule and its partner will lead to the formation of a pair described by quasi-stationary wave funetions, since the colliding particles may separate again. These quasi-stationary or compound states would be evidenced as resonance peaks in the scattering cross sections. To describe this process quantitatively it is necessary to consider the couplii:g between open and closed channels, i.e., between energetically accessible and inaccessible collision states, respectively. We concentrate, for simplicity, on neutral atom-diatomic molecule collisions. The scattering of an atom in the ?SO state with a molecule in the lx0 state may be described 2s the scattering of a StructureLess particle by a vibrating rotator with internal energy ~7ni (which defines the (n,j) channel), where II and j are the vibrational and rotational quantum numbers of BC. The interaction potential V(R, V,Y) is a function of the distance >-_be_tween B and C, the distance R behveen their center of mass and A, and the angIe Y defined by cos y = f?- r . For a total energy in the barycentric system E = A2knj /(2P 1 + ‘Qj
(11
the excitation to .a closed channel, with Iv,lj > E, leads to an imaginary wave number k,zj. The usual boundary conditibns of collision theory may then be used provided the choice 9m(kj,j) > 0 is made. Using the total angular’momentum representation (ZjJ~12),where 2 is the orbital angular momentum, the scattering matrix SAt XJ”, with X = fjzjl), is diagonal in Jand the total parity I?, independent of AIand symmetrical Cl]. The effective hamiltonian method for resonance collis:ons [Z] separates the part of this matrix connecting open channel states, $, in a term corresponding to direct scattering and mother corresponding to compound state scattering,
Hey ypJ isJa matrix connecting open channel states, which depends on the quasi-stationary state p, and For l?p and Ep. are the resonance width and energy for the A-BC Van der Waals pair, respectively. energies E smaller than the threshold energy for the first allowed (rotational) transition there is only one open channel, the elastic one, for n =j =O. The integral elastic cross section o(k), with k=RoO is a(k) * This research iKfty 1967
was supported
by the National
= ad(k)
+ UC(~)
Aeronautics 139
(3)
+ oi(a),
and S&e
Administration
Grant NsG-275-62.
D. A. MICHA
140
where the terms correspond by ad(k) = (4a/k2) c
J
compound state and interference
to direct,
contributions.
They are given
(2J+ 1) sir? qJ
o,(k) = (Cr/k2) I$ (2J+ 1) [F
i~z\ 2
(4a)
sin2
?$
+
2 pFD,
sin 7); sin $32e
Y$$,
exp i (n $- ,$)I
(4b)
and (2J+ 1) c sin # sin $%?e ~5 exp i (nJ$) . (4c) P The direct phase shifts 8-I correspond to scattering by a potential equal to the sum of the isotropic part of V averaged over the vibrational motion for n =j =O, plus the isotropic, non-local contribution from far resonances, i.e., from eoupliri, to compound states of high energy. The resonance phase shifts 5 are given by; of(k) = -(8r/k2) 7
(5) The previous equations have been used to describe the elastic The potential for this system may be approximated by v(R, ;,7) = Y’(R.
v)[l + B’P2(cos
scattering
of D2 with ?Z= j = 0 by Xe.
y)] + V”(R, r)[l + b”P2(cos
Y)] ,
primes and double primes refer to short range and long range respectively, (12, 6) potential may be used ior the isotropic part,
where
rjO(R)
= E(R/R,,,?~
with parameters R,,? = 3.9 ‘x 1O-8 cm and E = 1.2 x lo-l4 erg (*S- 8%) (deduced from experiments [3]), b” = 0.09 (*lo% [4] and b’ = 0.5 (*30%, which has only been estimated. The Van der Waals pair would have three bound states [5]. Since the parity of j is conserved during the coilision we may use eqs. (4a-c) for all E < Kk2. This situation is illustrated schematically in fig. l(a) for J = 0. The resonance phase shifts ?$J may be easily obtained using the following simplifications: (1) Retain only the lowest order terms in the anisotropy, in which case far resonances do not contribute to the direct collision and the resonances become nonoverlapping. (2) Calculate the integrals over scattering wave functions using the method of stationary phases, which leads to asymptotic expansions in (Rmk)- ‘. Since Rmk > 20 in the resoniulce region, it suffices to keep only the lowest order term. The results indicate that ad(k) is determined by the value of the potential around the classical turnIng point, and that the resonance widths depend primarily upon the short range anisotropy. Table 1 shows the computed resonance parameters TO< and E9J for 0 =z J C 5. While the values of the To” are highly sensitive to the pctential parameters, the EOJ (with errors ? 4%) are less so. The direct cross section Q(k) is given, to within a relative error of order b’2, by the scattering thrrzrgh uio(R) and may be written [S] as the sum
(6)
and a Lennard-Jones
- 2e(R/R,,,?,
ro
Rm
R-+
G 'C
40
5
-5
IO
E-E&y Fig.
1. (a.) The interacting
quasi-stationary
and scatter-
iw functions for an elastic coIIision with tl- j= I) and
J= 0 jschematic) (b) The direct (d,L._ comnound ~~~_ ~~~state (c\ . , and interference (i) contributions to the integral elastic cross section (heavy line) for J=O and p=O.
COMPOUND STATE RESON.ANCES IN MOLECULAR
COLLISIONS
241
Table 1 Resonance widths l$ and energies E$ for D2 +Xe (in ergs). J
lo”!
xE
J
0
l.ooO1014 x I?0
0
2.669
2.6 A 10-3
1 2
2.660 2.902
3.9 K 10-5 6.1 x 1O-4
3
2.932
7.2 x 1O-4
4
2.982
7.5 x 10-4
5
3.036
7.7 x 10-4
h -?
0 B b 0.990-J=O I o.g=-940
2
I III 9.45
3 II 950 IO-%
9.55 (cm set?)
960
9.65
9.7c
Fig. 2. Dependence tif the integral elastic cross secv in the region of resonances with p=OnndO CJ<5_
tion uon the velocity
of a smoothly
varying
part cd(h) plus a contribu;:on Q(k)
= 8.083
RL
from
the glory
(%Rmr_r/ti2)’
extrema
Aad(
where
(Rmk)-’
(8)
sin (2nm - $7)].
(9)
and
A&)
=
(4,/k2)h0/[(+.$
The maximum direct phase shift nm and n;;1= (d2n/dZ2)o at the glory angular momentum 10 are avaiIable from tables [?‘I. The cross section Crd(k) would also include narrow single particle “orbiting resonances”, but they would appear, for D2 + Xe, at energies appreciably smaller than the compound state resonance energies. The computation of uc and oi requires the knowledge of the direct phase shifts nJT which may also be obtained by interpolation from tables [8]. The three contributions to o(k), for Z=O ,wd artiund the energy E 00, are shown in fig. l(b). Fig. 2 shows the quantity o/@d versus the velocity 2 = fik//.f for a range that includes the resonances for p=O and 0 C J C 5. The resonance disturbances extend over several meters
per second in each case and the variations in the cross section are of the order of 1%. These variations might be detectable by means of recent experimental techniques [9]. The accuracy in the velocity selection of molecular beams should be sufficient to identify the position of each isolated resonance, although insufficient to give its shape. An experimental search for these compound state resonances should be of interest not onP;r in comiection with the description of molecular collisions, but also as a possible means of obtaining information on the interaction potential in eq. (6). Most molecular systems interact through potentials with capacity for one OF more bound states, and they should form compound states. A variety of situations may be ex-
pected, depending on the type of interaction and on the relation between l and the energy levels of the molecules. A detailed study of the atom-diatomic molecule case, including the molecular vibrational and rotational motions and a discussion of the computational method, as well as a few more numericaL examples, has been submitted for publication [lo]. The author thanks Professor
R. B. Bernstein for several
helpful discussions.
REFERENCES f l] (a) J. M. Blntt and L. C . Biedenharn, Rev.Mod.Phys.24 (London) 67 (1964) 1103; (b) A.hI.Arthurs and A.Dalgarno.
Proc.Roy.Soc.(London)
(1952) 258; see for corrections A256 (i960) 540.
R.Huby.
%oc.P::..s.xx.
2
D.A.
MICHA
Z] H.Feshbach, Ann.Phys.(N.Y.) 5 (1958) 357: 19 (1962) 287. 31 R. Helbing, Dissertation, Bonn University (1966). 4) J.O.Hirschfelder, C.F.Curtiss and R.B.Bird. Molecular Theory of Gases and Liquids (John Wiley ant! Sons, New York, 1964) ch. 13. 5] H. Harrison and R. B. Bernstein, J.Chem.Phys.38 (1963) 2135. S] See: R.B.Bernstein in Advances in Chemical Physics, ed. J-Ross. Vo1.X (Interscience Publisher;, New York, 1966) p-75. 2nd references there. 7] T.J. P.O’~rion. Tkzoretical Chemistr!; Institute, Univ. of Wisconsin, Report WIS-TCI-100G (1955). t;i R. B. Bernstein, J.Chem. Phys.33 (1960) 195. 91 (a) Fr.Von Busch, H.J.Strunck and C.Schlier, Z.Physik 199 (1967) 518; (b) See: R. B. Bernstein and J. T. Muckerman, Theoretical Chemistry Institute, Univ. of Wisconsin, Report \VI5-iCI-200 (1967) (to be published) for a review of experimental results. 0] David A.Micha, Theoretical Chemistry Institute, Univ. of Wisconsin, Report WIS-TCI-223 (1967).