564
Progress of Theoretical Physics, Vol. 39, No.3, March 1968
Statistical Model for Vihrational=Rotational Population
Distribution of Diatomic Molecule in Splitting of
FouraAtomic Complex into Three Pieces
Sigeru '/If ATANABE, Takashi KASUGA* and Tadao HOl~IE**
Physics Dej)artmeJl t, Prefectural Uni'versity of Jl,die, Tsu
* Physics Department, Nara }IIedical College, Nara
**Faculty of Science, Osa/:z([ Uni'versity, Toyolla/:za
(Received October 12, 19(7)
A general expreSSlOl1 has been derived from a statistical model, which takes account of
the conservation ot angular momentum, in addition to energy and linear momentum, for the
initial non-equilibrium distribution of vibratiollal-rota'ciCllul populations of the diatomic species
AB resulting from the molecular splitting; [ABeD] --~AB + C+ D. Numerical calculations of
the expression under a reasonable assumption make it possible to know, for instance, the
rotational popul<"tion distribution of AB. The l10nthermal distribution curve obtained varies
considerably with the masses of the four atoms, the moment of inertia AB, and the spatial
volume of the intermediate complex.
Extremely abnormal populations of the rotational states of OH* ell, 21: i)
split from H:10 hm-e so far been rcyealed both by photon and by electron impact
experiments.1)~5)
It has also 11ee11 shOlvn that the same radical split from another
parent, H 2 0 2 , cxhibits quite a diHerent population distribution among the rotational
levels. 6),7) In order to account for these typical population distributions, statistical
treatments taking account of the consen-alion of angular momentum, in addition
to energy and lineal' momentum, have previollsly been proposed by Horie and
Kasuga with success. S),9)
Such kinds of abnormal population distributions among rotational or vibrational levels of diatomic species split frompolyatomic intermediates have recently
been drawing attentions of many authors, in particular those who are working
on upper atmosphere physics 10 ) and on chemical lasers.11)
In the present paper, the statistical model mentioned above will be extended
to a little more complicated case than discussed before in references 8) and 9).
The splitting of a complex compDscd of four atoms into one diatomic and two
monoatomic. species, for instance [ABCD] -> .1113 -I- C -+ D, will be brought up for
discussion.
To begin \vith, it will be ~)hOWi1 that em initial non-equilibrium vibrationalrotational distribution of the product molecule .11B is expressed in an integral
form. After that, the general expression ·will be applied to the rotational pop-
Statistical l\iodel for JJlolccztlar Sj;littiJlg
565
ulation distribution of AB resulting frorn the complex without rotation. In
addition, some of the distribution curves calculated numerically for non-equilibrium populations of the rotational levels of AB will be presented in comparison
with experimental plots available to date. Finally, qualitative discussions will
be done with respect to such a statistical theory approach.
§ 2.
Phase integral
The following assumption will be made. An excess energy, E, which is
much larger than the quantum of vibration of .1113, is initially possessed by the
complex [ABCD]. In addition, it 'will be distributed in random fashion among
all the degrees of freedom of the three fragments before they leave a small
vol ume. Only one of the fragments AB is diatomic, and the others mOlloatomic.
Let TJI (E; Ev, E T ) dEvdEr be the probability that the diatomic species will have
vibrational energies ranging Ev to Ev -1- dE and rotational energies ranging E? to
E?+dE n then we have
1"
7JI(E;
E l ), Er)
X
=const[J}jd3l"'i(PP'J~dqdj)~d,jr
0 (, E \
I.: P.
8
2
i
,,~l 2m'i
-
1
. 2
.. '
IC q2
1
2/l
-
:1
X()(C--
xo" ( E.,,·--
'C,
N2 )
.
21
8
L
I
j} -
()
(I.: p,J
8
t~l
~
lrixPi--N)()[(LlJlirJ/(LmJ]
l
1,
1
1
.2
1. j) 2) 0<, ( t:. r
. Icq-2
2/t
7,
-
N 2)
21
,
1
(1)
where C is initial angular momentum of [ABCD] , N angular momentum of
AB, q and j) vibrational coordinate and its conjugate momentum of AB, and IC, /.L
and I force constant, reduced mass, and moment of inertia of AB.*) Mass, spatial
coordinate, and linear momentum of ith fragment are denoted by mi, 71\ and Pi,
respectively. The spherical polar coordinates @ and (jj, and their conjugate
momenta P@ and PI/) are used as usual to describe the rotation of AB, where
d 4r stands for d@d(jjdP@dPI/), for brevity.
Let us change the variables 1"i and Pi (i== 1,2, 3) into Bo,R, R', and their
conjugate niomenta Po, P and P' as follows:
3
Ro =
:l
(I:
J71i1",J / (I: m,J,
z=]
.=1
*) It is assumed in the present paper that the moment of inertia I remains unchanged, in an
approximation, irrespective of the vibration and rotation of AB.
566
S. IVatallobc, T. Kasuga and T. IIorie
PO=PI +P2+ P S,
p= [(m2 -+ 17Z3) PI -
J7Z 1
(P2 + P3) ] / (m} + 7712 + 17Z3)
and
P' = (m2PS - 7n3P2) / (m2 + lJI3) ,
where Ro represents the coordinate of the center of mass of the whole system,
R the relative coordinate of fragment 1 with respect to the center of mass of
the rest and so on. Then we have, in place of (1),
7Jf(E;
Ev, E?")
=-----=const)d3Rd3R'd3l!10)::PPdSP'd3Po)dqdj)
1"2
21\1
X;; (C -
p/2
2M'
P02 \
2~loJ
N - R X P - H' >< P' - Ro X Po);; (Po)
" (R 0) 0"( E v - -21
X 0
where 1110 stands for
1Jl 1
r0-
ICC
+ 1JZ2 -+
1/ - j)"0) 0,_, ( Er - N:l )
2 1
-21 '
JJl
s, lJ1 for
JlZ I (m2
+ 1Jl3) /
(2)
(m i
+ 17Z2 + JJls),
and 1'11'
for m2llZ3/ (m2 + J7l3)'
After integrations over q, jJ, Ro and Po, "\ve have
><
"( ,
1<. ---
0
Eli -
N2
p:l
--__
21
21\1
- --
lJ)l2 \
_
)
211]'
(3)
As has been shown in Appendix A of reference 9), the integrals ) d 3Rd 3P···
and ~ (PR' d 3 P'··· can be transformed into con~t ~ dRdPRi d 3 L/L··· and const
X ~dR'dpR'~crL'/L! .. ·, respectin,ly, where L=
xP, P2=pR 2 +L 2/R 2 ;:-md so on.
Accordingly we have
X;;(C-N-L-L1);;(E?"--
N2) .
(4)
21
After integrations over the orientation of L and that of L', just
way as has been shown in Appendix B of reference 9), we have
111
the same
Statistical l.71Iodcl for 1\Ioleclllar
5'j~littinp;
567
XTC~NIO(E
X
0
(Er- -~;--) .
(5)
Quite similarly, ~ crr··· can be transformed into const ~ d 3 N / N .. ·, and further
into const(l/C) ~KdKldN2/1V .. ·, where K= IC-NI. Then, we have
?fI(E;
Eo, E1')
=const
.~.-
Cv
dPudP
Er (dR_dR'('
j
j
R'
~dK~ dLdL'
(6)
After integrations with respect to P R and P R ', \ve have the following multiple
integral with the integrand of unity:
?fI(E;
E l"
E).) =
constC);t-~dRdR/~dLdL' ~dK.
(7)
r
The integration limits concerning K, Land L' should be as follows:
C+N>K>IC NI,
(8a)
L'>L> IK--- L'I
(8b)
1( -+
and
(8c)
where N="/21E;, .1'=L//2Nl(E Ev--Er) and L'=cL'/../2A1'(E-Ev-Er).
On the other hand, as for the integration limits concerning Rand R', an
assumption will be made as follO\vs:
R
ROllt; and R' < R~ut.
(9a)
Needless to say, these maxima should have relation to the spatial volume of
the intermediate complex. , In addition, a minute volume defined by
(9b)
will be omitted from the core of the volume defined by (9a), in order to prevent
the three fragments from being too close to each other at the Game time. In
the present calculation, however, any two of the fragments _will, for simplicity,
be allowed to come close to one another while the rest one is distant from
them in the interior of the spatial volume.
Under the condition (8c), the area indicated by hatching in Fig. 1 IS
expressed by (RR' -..£ ./jFi=- ..£'2 -..£' JR? - '.1: 2 ) . In consequence, we have
568
S. 1VataJlabe, T. ](asup;a mzd T. Rorie
The triple integral with respect to L, L'
and K should be carried out under the
restrictions (8a), (8b) and (8c).
For the purpose of simplifying (10)
to some extent, the following dimensionless variables will be introduced:
£'r---~------------~----~
o
R
Fig. l.
c=CI V2J\;1R2(E-
The integral with respect to Rand
l~ =
R',
7l
=
Ev -
KI V21v1R2 (E ~ E
E r ),
v -
ET)'
NI V2lilR2 (E~ E~)=--E-?~),
I=LI v2 l.'vlR 2 (E--
E v -- E r ),
l' = L' I v2J1vi' R'2 (E - E1)~-E~)
and
r=
R'
R
/
~
/ 1\711'
(11)
]\,1
Then the inequalities, (8a), (8b) and (8c) can be rewritten as follows:
(12a)
c+JZ>7~>lc-JlI,
l~
+ rl'>l> II? - rl' I
(12b)
ana
(12c)
Finally, we have
dl?
c
(12d)
*)
In the present paper, it is defined that
[feR, R')]
for any function feR, R').
(RuUi R'Cou!)
,
=f(RoUi, R'out) -f(Rin, R'iu)
(Rin,
R1in)
569
Statistical l1.1odel for l1.1olecular Sj>liuillg
k < I
k <0'< I
kid
£1
klr
k
I
k
1< k < ?f
0
0
klY
k
1<;Y<k§~
klo
k
I
o
o
7<k< I
-r < I <
k '2=
I/T+f2
,------------------~-----
kid
Fig. 2. The double integral with TtSpect to 1
and l! is taken over the area indicated by
hatching, when
1.
r>
Fig. 3. The double integral with respect to 1
and l! is taken over the area indicated by
hatching, when r<1.
The area S over which the double integral \vith respect to land l' is taken
should be defined by (12b) and (12c). Accordingly, it changes considerably
depending upon /:: and r, as shown in Figs. 2 and 3. It should be noticed here
that the pair of parameters CR, R') is involved in all of the dimensionless variables, and, incidentally, the area S.
~ 3.
Rotationless co:mplex
The intermediate complex [ABeD] may be realized 111 various ways. For
instance, it may result from photon or electron impact of some tetratomic molecule at ground state. It may also come out in sticky collisions betvveen A and
BeD or between lie and ED. Especially in the latter cases, it may often be
S. '(\1atanabe, T. J\..asup.;a and T. I-Iorie
570
inadequate to neglect the angular momentum of the intermediate. However,
there may be many other cases 111 which it is possible to assume in good approximation that C = o.
In what follows, it will be shown that this assumption enables one to push
forward the above integration (12d) simply by means of elementary calculations.
If C tends to zero, then c tends to zero, and, incidentally, h tends to 7l, according
to (12a). In consequence, Eq. (12d) is reduced to
(Roub R~Ui)
(Rim R;n)
(13)
J=~~1.z:dy(l-xVl-y2-YVl-x2),
where
and where x 2 +y2'<1 and n-i-rx>y>IJl
- rxl. The area S over \vhich the double
integral J is taken is enclosed with a unit
circle and straight lines as sho\vn ill detail
in Fig. 4 only for r> 1 and Jl<l.
As each of the three terms in the integrand IS odd or even \vith respect to x
and y, \ve have
o
-I
0
x
-0
I
e i. __ _
lJ
1/'l-:1:2
~) dr: ely .Tvl-- y~ = ).<c d x ~ vl- :'1,2 d.l'.
S
II,
!l
~~ dx dy yVl-.x
2
=
I
tk-'Y.l;t
-I
1/'1--1/2
~ y dy)Jl-<L.
2
Fig. 4.
eLc
Ik-y! /'Y
S
The double integral I for 1'>1 and
n<l is takel10ver the area S indicated by hatching.
and
As seen in Figs. 2 and 3, there are six different cases concerning the relation
between rand 7Z or l~. The above expressions on the right-hand side, however,
remain unchanged even for any other cases than that shown in Fig. 4. By
carrying out these elementary integrations, we finally have
7T1'(E
':r
; E", E,. )
[R2RI2(EE,.) -'--'(
1
=const------..... E.,,-'1
r; n
VE
y
)J
I'
:
(R
j)
. ou,t R'
.ou
(Rim R;n)
.
(14)
The function F(r; n) 111 Eq. (14) results from the double integral J, and
polynomial expression as follows:
IS
a
571
Statistical l.Ivfodel for Jl,lolecular Sj)litting
F(r; n)
+
I ( . 1)
1 ) n2} (S'
=16r + ---;:- +-41 ( r + --ir- "- III
{
2
(r +_})
16
r
1
2
-1
b
-
S'
In
--1
Ia I)
(dla 1 + blel s -lald 3 -lelb 3 )
3
+6;2C:lb3-laI3) +!¥
(,2rd3-- leis)
+3;2- (1-;1 --1) +!;{ (I~I -1)
+-~- (r2 --~2--) (dlal- blel)
(15)
2r
where
1l"I_ a= ____
vi +
1+r
-
2
nr+
b = ---.------------v1+
----1-+ r 2
,
-n~
/1 +r 2 -n"
e -_ n-rv
--1+ r 2
'J
and
d=
§ 4.
n
+ IV1-+ r2-n 2
1 -+ 1'2
(16)
Rotational population distributions
The expression obtained above for the vibrational-rotational population distribution involves n which is a function of E1J and E
Accordingly, when E1J is
kept constant, rotational population distributions can be derived from Eqs. (14),
(15) and (16), if only the parameters (Rout, R;ut) and (Rill> Ri~') were known.
These parameters depend on geometrical features of the spatial volume of the
1••
572
s.
lVatanabe, T. I<:'asllga and T. I-Iorie
intermediate complex. Qualitatively, for instance, if long range interactions
take place among the three fragments, both ROllt and Ro'ut are expected to be
quite large. On the contrary, if strong but short-range interactions are predominant, the outer radii may both 1)13 small.
In the present case, these parameters are involved in r as \'veIl as 111 ll.
Accordingly, the matter seems more or less complicated. Fortunately, ho\vever,
Eq. (14) can be re'iHitten as follo'ws:
(17)
\vhere FouL refers to F(r; Jl) substituted by Rout and R:Llt for Rand R' involved
in rand 77, and quite similarly l~Jl to F(r; n) replaced by Rill and Rj~,. It may
be expected that Roul and R/ilt are both much larger than Rill and Ri~' respectively. As a result, the faclor, (R;nRi~J R01ilRo',ilY, becomes much smaller than
unity. Such being the casc, the second tcrm within the brace of Eq. (17) may
contribute only to a minor part of the population distribution, and it will be
neglected in what follc)"ws.
In order to see how the rotational population distribution actuany IS, let
us introduce constant 1V1'; and "\-ariable Y, as E- <:n=1Vrf/2I, and V=1V/1VI;. Then,
we have
(18)
where a = ) 1/ (,-n.11<C'~11)' and Y is angular rnomentum of the diatomic speCies 111
the unit of N B. Since <:1' becomes 1VB~Y~/21 instead of N 2 /'21, the rotational
population "\vith angular momenta ranging from
where
Y
to
Y
-+ dv
will be (j) (E,
E ,) ;
v) dy,
(19)
In Eq. (19), G (r, a; v) is obtained by substituting the expression (18) for
Jl
in the expression of F(r; n), where a and r should be taken to be )1/ (NJRo~ll)
and (R;',t! ReJlt!) ~/ Al' / ill, respectively.
Several of the rotational population distributions calculated for varied values
of rand 0:: are plotted against Y in Fig. 5. They are almost entirely different
from the rotational population distributions exhibited by the OH* (A, 2Z+) rac.hcal split by electron impact from both water (sec reference 8)) and hydrogen
peroxide (see reference 9)). For instance, the curve for r = 1/3, and a = 1/6
has a maximum near y=-=O.50, while that for 1'=3, and c[:c-=3 has a maximum
near y = 0.17. In particular the latter looks quite similar to a lvlaxwell-Boltzmann
type distribution. It sb.()uld be noticed here that the curve
a certain paIr
of r and a is quite the same i.l3 that for l/t and a/y, because it is readily seen
Statistical l\!Jodel for Molecular SjJliuill{J,'
c
o
o
::::J
0.
o _
0..
Fig. 5. H.otational population distributions are plotted against v. Each curve corresponds to a pair
of parameters (r, a) as follows: curve 1, (:3,3)
or (1/3, 1); curve 2, (1, 1); curve 3, (1/2, 1/2)
or (2, 1); curve 4, (1/3, 1/:3) or (3, 1); curve 5,
(1/6, 1/6) or (6, 1); and curve (i, (1/:3, 1/6) or
(3, 1/2).
that GeT? a; v) =G(l/r, all'; v).
In orclet to compare these
curves \vit h experimental plots,
the parameters r and (I:; should be
determined hy the masses of the
fragments, the moment of inertia of
the diatomic species, and the spatial
volume of the intermediate complex.
The formulation described above,
ho\vever, does not involve any
definite \\'ay of asslgnmg 17lr, 1712
and JJlg to the masses of the
three fragments. Accordingly, for
instance, one can take HZ 1 as the
mass of the diatomic molecule A.B,
without any loss of generality.
Then, the parameters I' and ex are
as follows:
I' :~=
+ m R + 17le +- JJl;;)(me+ 1nDY (JJlA + mn)
/ llZe In]) (lnA
'"
and
a=.1 JJlAlllll(mA+mn+mo+mj))
'V
(inA + JJllJY (nZe+1fZj))
where p means the internuclear distance of L1B,
~
5.
()
(20)
R'lIli
?llA
the mass of A. and so forth.
Comparison with experiment
So far as such a statistical model is available, Eg. (I 9) IS expected to be
applicable to triple fragmentations of four-atornic intermediates having initially
low rotational energies. However, there are as yet only a few examples of
experimental plots which allow one to check Eq. (19). In the first place, observations were made by Carrington and Broida 12 ) in a discharge through argon
containing about one-half mole percent water vapor at a total pressure of 0.05
mrnHg. The rotational line intensity distribution of the (0, 0) band of the
21,'1 ___ 2J] transition of the OH radical has been plotted by them for the Q2
branch, as shown by the filled circles in Fig. 6.
The intensity distribution observed resembles a Boltzmann distribution corresponding to a rotational temperature as high as 3340oK. It has been sug-
S. l11atanabe, T. Kasuga and T. Horie
574
gested that a possible excitation mechanism fOl: this Boltzmann distribution
as follows:
IS
(21)
where A' denotes a metastable argon atom.
In general, the relative intensity of a rotational line within a band is proportional to the transition probability for spontaneous emission, as well as to
the relative population of the initial rotational level. The former can be found
in the table of transition probabilities for the six main branches of the OH
band, which has already been presented by Learner. 13 ) The latter can be picked
out of the curves as shown in Fig. 5, as follows.
o
&
1.0.
zJ
Fig. (). Theoretical ,mel experimental rotational line intensity
distributions of the Q2 branch for OH* resulting from
the reaction, H 20 + A' -> H + 0 H* + A + 2.4ev.
,------_._-"--"
"-
•..""-.-.-~"~
"""-
.
------.~-.------."------
c
o
o
:::J
x
..... '-...,'\.
..-
0.
o
'-
///
0.
o
//
'-
X /"
o
c
o
o
/
/
"2
/
\
\
/
\
\
/
\
/
X
/
>
/
\
o
/
/
o
\
\
/
/
<V
\
/
0::
\
\
/
<V
,
\
/
/
'-
\
/
\
/
/
/
0.2
0.6
0.4
o.s
1.0.
v
Fig. 7. Experimental plots of the rotational distributions of CH*
resulting from electronically excited C2 H 2 (crosses) and
CH 20 (open circles). The full line curve is a theoretical
rotational distribution for [C 21-1 2J*-7H*+CH*+C, while the
clashed line curve is for [C 2H 2J *->CH* + CH.
Statistical J..VIodel for l\IIolecular 5'jJliUing
575
Let mA, mE, me and 177]) in (20) refer to the mass of an 0, H, A and H
atom respectively, then Rout is approximately equivalent to the maximum distance
. between 0 and A in the interior of the intermediate. Similarly, RO~ltcolTesponds
to that in the small volume between A and 11 which will soon flyaway separately. If it is tentatively assumed that Rout and
ut are both almost the same
as the internuclear distance of the OH radical, it can readily be seen with the
aid of (20) that the parameters r and a become equal to each other and to
about 1/4.
The theoretical intensity distribution is sho\vn by the full-line curve 111
Fig. 6, and this is also similar to a Boltzmann distribution corresponding to a
slightly higher rotational temperature than 33400K. This discrepancy may be
attributed in part to some doubt ill such a direct application of Eq. (19) to
the splitting of the intermediate which would result from the collision H 2 0 + A',
and in part to some effect due to the wall of the discharge tube which is likely
active on the rotational distribution considered. 12 ) Among others, the intermediate· complex in this case may be created by the collision partners of many angular
momenta with respect to the center of mass, contrary to the assumption included
in Eq. (19), that is, C = o.
In the seco~d place, the CH* CR, 3}; -) radical splits from a tetratomic molecule, for instance, C 2 H 2 bombarded by electron. H ) If it would result from such
a splitting as [C ZH2J *-"CH* + CH, its rotational population distribution would
be just the same as that of OH* (A, 217+) split by electron-impact from H 2 0 2 in
the breakup process [H 2 C\] *--"OH* + OH. In Fig. 7, two different curves are
presented, the dashed line curve and the full line one. The former is the initial
non-equilibrium rotational distribution of the product molecule CH* of the splitting process [C 2H2J *-->CH* + CII, and is reproduced from reference 9), while
the latter is that of the same product radical of another type of splitting as
follows:
R:
(22)
In Fig. 7, the experimental plots obtained. by a crossed-beam technique
with C 2H 2 are shown by the crosses together with the plots with CH 2 0 indicated
by the open circles. They are both much more close to the curve for the
triple fragmentation, (22), or
(23)
In good agreement with the spectroscopic evidences described before. 14 ) Incidentally, the difference betvveen the mass of the fragment C in (22) and that
of 0 in (23) has only a minor influence upon the shape of the calculated distribution curve.
Finally, according to the statistical model inGluding angular momentum
conservation, the OH or CH radical can exhibit Boltzmann-like distributions of
576
s.
1Vatanabe, T. ](asuga and T. l-Iorie
rotational population, at least ~when it results from such an asymmetric splitting
of a four-atomic complex into three pieces as shown by (21), (22) or (23).
~ 6.
Discussion
When a tetratomic molecule is electronically excited by electron-impact, it
usually dissociates into pieces. If one \\/ants to see the interaction among the
fragments before they fly out, it is necessary to have the potential energy surfaces
for the excited states of the molecule by means of the molecular orbital theory.
For that purpose, very many atomic orbitals with higher principal quantum
num bel's should be taken into consideration in addition to those contributing
to the valence states of the constituent atoms. As a result, a number of polydimensional potential energy surfaces "vith nearly the same energies may be
expected to come out. At the same time, partial crossings with each other
may take place in unaccessi ble manner, as has previously been shown by Niir( 15 )
for electronically excited states of \vater.
In treating this type of processes there has been the conventional perturbation theory approach. In that, however, a number of complicated hypersurfaces
entangled with each other are llSlWlly replaced by a single surface simplified
III some way.
On the contrary, in the statistical theory approach, the idea of
a small volume is introduced. The statistical description of the collision process
is probably as extreme although in the opposite direction as is the perturbation
theory approach. The actual state of fact may be somewhere III between the
two theories.
In the present paper, it IS assumed that the interaction among the three
fragments is so strong that a statistical equilibrium compatible with the conservation theorems o£ angular momentum, energy and linear momentum is attained
among all states. Jv10re precisely, the excess energy available in their centerof-mass system is partitioned among all of the degrees of freedom of the three
fragments before they leave a small volume. This basic assumption is an application to molecular dissociation of the statistical theory proposed by Fermi
for multiple meson production. 16 )--18)
Owing to angular momentum cOl1senT ation, it is impossible to separate the
spatial integral from· others, because the integrand of the phase integral has
cross terms of positional vectors 'with momentum ones. As a result, in order
to carry out the calculation, it is necessary to define the small volume, at least
the outer radii, Rout and R o'ut . In the above evaluation, the small volume is defined simply by (9a).
The spatial volume of the intermediate complex is usually taken to be a
sphere. The radius should be related to the outer radii given by (9a), and,
In addition, big enough to allow vibrationally excited states of the intermediate
to exist within the volume. If the radius is expressed by (j ao, where ao is the
Statistical i110del for }\dolecular 5j)littinl!,"
577
Bohr radius, then u is a dimensionless constant. The theoretical curve calculated
for (21) corresponds to u::::::4, and that for (22) and (23) to u::::::::3.
If one takes the process considered more seriously, then it is indispensable
to take into consideration the electron responsible for excitation of the molecule.
In the present paper, however, it is put aside whether the unstable complex is
produced by electron-impact or by any other means. ]\;1oreover, it is possible
for the complex [ABCD] to split into pieces in many other ways as A + BCD,
AC + BD, A + B + C + D and so on. The relative probabilities of these possible
processes of splitting are also out of the scope of the present paper.
Especially for the problem of branching ratios, a theoretical study has recently been made by Tannelnvald on the basis of a statistical description of an
intermediate complex. 19 ) On the contrary, the present study has entirely been
devoted to the relative populations of the rotational and vibrational states of
the diatomic speCIes resulting from such a particular type of splitting as
[£1BCD] -) AB + C + D.
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