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Theoretical formulation for a new interruption function in perturbation theory for molecular collisions
Volume 45, nuker
CHEMICAL PHYSICS LETTERS
i
15 January 1977
THEORETICAL FORMULATION FOR A NEW INTERRUPTiON FUNCTION IN PERTURBATION THEORY FOR MOLECULAR COLLISIONS G.K. JOHRI and S.L. SRIVASTAVA PhysicsDepartment, Allahabad Universiry, AIIahabad-.ZIIOOZ, India Received 16 August 1976
A new interruption function for the molecular collisions under Anderson-Tsao-Cumutte lated, which has the correct form and explains the observed width.
(ATC) theory has been fonnu-
I_ Introduction Recently Johri and Srivastava [l] proposed a new interruption function under Anderson’s theory [2] which includes the phase shifts produced by elastic collisions and of the interruption
of the radiation process by inelastic
collisions. Following Foley’s method [3] of averaging over the distribution of collisions and Anderson’s method of calculation [2] , the formulation for the proposed new interruption function in the perturbation theory [l] for molecular collisions has been obtained,
2. Theoretical fomulation
and is reported here.
of the collision interruption function
- The problem of the line shape can be solved by finding the Fourier transfotm of the correlation function C(t,t,,). Foley [3] and Anderson [2], in their interruption theory, evaluated this correlation function. Foley’s method demonstrates two limiting cases of non-adiabatic transitions, which are not always possible, whereas Anderson has carried out the calculations for the intermediate values. We have applied an averaging by a continued product l-la over a single collision of type (Tupon any level, when collisions are not correlated_
If V(t, fo) is the time development matrix and S(o) represents the effect of a single collision of type u upon the time deve!opment matrix upon any level, then for uncorrelated collisions
V(r, to) =
III S(0). u -
(1)
Since under the impact approximation the collision duration is infinitely short, each collision is binary and independent, therefore the total effect for n(u) collisions of type 0 occurring in the time interval t - to = 7 on the combined -kitial level i and foal levelfcan be represented by
The dot prod&t shows an average over all types of collisions and furthermore it is invariant with respect to the choice of the axis of quantization and spares us considerable manipulation. The mean free time between collisions taking place in the time interval 7 is then given by (7/7-~)n(U) exp(-r/rO);n(u)! 364
.
(3)
CHEMICAL PHYSICS LETTERS
Volume 45, number 2
15 January 1977
Multiplying eq. (2) and eq. (3) and summing over all values of n(a) yields an expression for the correlation tion as
(7/TuF)exp(-T/T,)
2
=
=
u
(d7,)n(“’eXp(-T/T,)
n
I-I[(ils(a)li)-(fiS(o)In*]“(u’
n(u)!
n(u)=0
n(u)!
(I
[~ils(o)lo~
Since the continued product in the above expression write
C(t, to) = exp [-
c
(l/To)
func-
[I
-
(4)
is equivalent to a summation with an exponential
term, we
-(fls(‘3)\f,*]].
(5)
a
We now suppose that collisions of different collision diameter and direction are designated by different values of G. The inverse of the average time between collisions within the type do is given by ‘/To
=
n(u)da,
(6)
where tz is the number of colliding molecules per unit volume, (u) is the mean collision velocity. Converting the summation over u as an integration over 0:
C(t,to) = exp - Jku) c
= exp
[
da[l
- CilS(u)li)
-
0 -n(u)
sdu[l
--
u
1 1
= exp(-rz(u)o),
(7)
where u = _f do]1 - (ilS(u)li>
-*]
(8)
is the complex cross section_ The integrand eq. (8) is rotation invariant, therefore, if we can compute the S(u) for one specific set of direction angles quantized along the impact parameter b, then,similarly for the final state as well. Since the directions of the collisions are averaged out to do = 27rb db, therefore
u=I2nb
db [l -*Q-lS(b)rr>“].
The collision interruption
function is
P(b)= [l - GlS(b)li)*(flS(b)If)*] From the perturbation
(9)
.
technique developed by Murphy and Boggs [4],
(10) for the initial state we can write
(11) Similarly for the final state f
365
CHEMICqL
Volume 45, number 2
PHYSICS
- 15 J&uary-1977
LETTERS
where [S2(b)]n+ji and ]Sz(~)]j~+~~ are Anderson’s cokhsion effic@ircy functions with the ji = ji and j; =jf terms I omitted. Substituting vaiues from eqs.(ll) and (12) in eq. (10) we obtain the re&.tired collision interruption function, p(b) = 1 - exp [-S(b)]
(13)
,
where (14)
Murphy and Boggs 41 neglected the last term on the rhs in eq. (14) which they calledthe phase term. In Ander(ATC) theory [S] the term in braces on the rhs is due to inelastic collisions and the phase son-Tsao-Curnutt shift contribution i 1 given by lI’~@b)ujl ji=ji
+
(1%
[s,(~)lj~=j~+sz(b),~,
which is the same as the last two terms in eq. (14). Further, and contributes to the line width, is Sz@)total
= Re s(b)
Tsao and Cumutte
= [s2(b)o,ili;:+ii
P(b) = 1 - exp [-This function
+ [S~@)O,~I,-~+~~+
[S] wrote the above expression
=S2(b)o,i +‘2(‘)uf+S2(b)mid’ ~2(%tal Therefore the collision interruption function,
the real part of S(b), which is the second order term
[‘z(‘)o,iIii=ii + 1$(5)~,J,--qf+S2(b)mid(16)
as (17)
reported
earlier [l] and contributing
to the linewidth
S2(b)totil-
(18)
has the correct form both for small and large values of b, since it couples the two regions smoothly.
Acknowledgement The authors thank Professor Krishnaji and Dr. V_ P&cash for many helpfiJI discussions.
References 111G-K. Johri and S-L. Srivastava, Chem. Phys. Letters 39 (1976) 579. PI P-W- Anderson, Phys Rev. 76 (1949) 647.
r31 H.M. Foley, Phys Rev. 69 (1946j 616. 141 J.S. Murphy and J.E. Boggs, J. Chem. Phys. 43 (1967) 691. iSI CJ. Tsao and B. Cumirtte, J. Quant. Spectry. Radiative Transfer 2 (1962) 41.
366
is
CHEMICAL PHYSICS LETTERS
i
15 January 1977
THEORETICAL FORMULATION FOR A NEW INTERRUPTiON FUNCTION IN PERTURBATION THEORY FOR MOLECULAR COLLISIONS G.K. JOHRI and S.L. SRIVASTAVA PhysicsDepartment, Allahabad Universiry, AIIahabad-.ZIIOOZ, India Received 16 August 1976
A new interruption function for the molecular collisions under Anderson-Tsao-Cumutte lated, which has the correct form and explains the observed width.
(ATC) theory has been fonnu-
I_ Introduction Recently Johri and Srivastava [l] proposed a new interruption function under Anderson’s theory [2] which includes the phase shifts produced by elastic collisions and of the interruption
of the radiation process by inelastic
collisions. Following Foley’s method [3] of averaging over the distribution of collisions and Anderson’s method of calculation [2] , the formulation for the proposed new interruption function in the perturbation theory [l] for molecular collisions has been obtained,
2. Theoretical fomulation
and is reported here.
of the collision interruption function
- The problem of the line shape can be solved by finding the Fourier transfotm of the correlation function C(t,t,,). Foley [3] and Anderson [2], in their interruption theory, evaluated this correlation function. Foley’s method demonstrates two limiting cases of non-adiabatic transitions, which are not always possible, whereas Anderson has carried out the calculations for the intermediate values. We have applied an averaging by a continued product l-la over a single collision of type (Tupon any level, when collisions are not correlated_
If V(t, fo) is the time development matrix and S(o) represents the effect of a single collision of type u upon the time deve!opment matrix upon any level, then for uncorrelated collisions
V(r, to) =
III S(0). u -
(1)
Since under the impact approximation the collision duration is infinitely short, each collision is binary and independent, therefore the total effect for n(u) collisions of type 0 occurring in the time interval t - to = 7 on the combined -kitial level i and foal levelfcan be represented by
The dot prod&t shows an average over all types of collisions and furthermore it is invariant with respect to the choice of the axis of quantization and spares us considerable manipulation. The mean free time between collisions taking place in the time interval 7 is then given by (7/7-~)n(U) exp(-r/rO);n(u)! 364
.
(3)
CHEMICAL PHYSICS LETTERS
Volume 45, number 2
15 January 1977
Multiplying eq. (2) and eq. (3) and summing over all values of n(a) yields an expression for the correlation tion as
(7/TuF)exp(-T/T,)
2
=
=
u
(d7,)n(“’eXp(-T/T,)
n
I-I[(ils(a)li)-(fiS(o)In*]“(u’
n(u)!
n(u)=0
n(u)!
(I
[~ils(o)lo~
Since the continued product in the above expression write
C(t, to) = exp [-
c
(l/To)
func-
[I
-
(4)
is equivalent to a summation with an exponential
term, we
-(fls(‘3)\f,*]].
(5)
a
We now suppose that collisions of different collision diameter and direction are designated by different values of G. The inverse of the average time between collisions within the type do is given by ‘/To
=
n(u)da,
(6)
where tz is the number of colliding molecules per unit volume, (u) is the mean collision velocity. Converting the summation over u as an integration over 0:
C(t,to) = exp - Jku) c
= exp
[
da[l
- CilS(u)li)
-
0 -n(u)
sdu[l
-
u
1 1
= exp(-rz(u)o),
(7)
where u = _f do]1 - (ilS(u)li>
-
(8)
is the complex cross section_ The integrand eq. (8) is rotation invariant, therefore, if we can compute the S(u) for one specific set of direction angles quantized along the impact parameter b, then
u=I2nb
db [l -
The collision interruption
function is
P(b)= [l - GlS(b)li)*(flS(b)If)*] From the perturbation
(9)
.
technique developed by Murphy and Boggs [4],
(10) for the initial state we can write
(11) Similarly for the final state f
365
CHEMICqL
Volume 45, number 2
PHYSICS
- 15 J&uary-1977
LETTERS
where [S2(b)]n+ji and ]Sz(~)]j~+~~ are Anderson’s cokhsion effic@ircy functions with the ji = ji and j; =jf terms I omitted. Substituting vaiues from eqs.(ll) and (12) in eq. (10) we obtain the re&.tired collision interruption function, p(b) = 1 - exp [-S(b)]
(13)
,
where (14)
Murphy and Boggs 41 neglected the last term on the rhs in eq. (14) which they calledthe phase term. In Ander(ATC) theory [S] the term in braces on the rhs is due to inelastic collisions and the phase son-Tsao-Curnutt shift contribution i 1 given by lI’~@b)ujl ji=ji
+
(1%
[s,(~)lj~=j~+sz(b),~,
which is the same as the last two terms in eq. (14). Further, and contributes to the line width, is Sz@)total
= Re s(b)
Tsao and Cumutte
= [s2(b)o,ili;:+ii
P(b) = 1 - exp [-This function
+ [S~@)O,~I,-~+~~+
[S] wrote the above expression
=S2(b)o,i +‘2(‘)uf+S2(b)mid’ ~2(%tal Therefore the collision interruption function,
the real part of S(b), which is the second order term
[‘z(‘)o,iIii=ii + 1$(5)~,J,--qf+S2(b)mid(16)
as (17)
reported
earlier [l] and contributing
to the linewidth
S2(b)totil-
(18)
has the correct form both for small and large values of b, since it couples the two regions smoothly.
Acknowledgement The authors thank Professor Krishnaji and Dr. V_ P&cash for many helpfiJI discussions.
References 111G-K. Johri and S-L. Srivastava, Chem. Phys. Letters 39 (1976) 579. PI P-W- Anderson, Phys Rev. 76 (1949) 647.
r31 H.M. Foley, Phys Rev. 69 (1946j 616. 141 J.S. Murphy and J.E. Boggs, J. Chem. Phys. 43 (1967) 691. iSI CJ. Tsao and B. Cumirtte, J. Quant. Spectry. Radiative Transfer 2 (1962) 41.
366
is