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Critical analysis and applications of a quasi-resonant theory of pressure broadening of linear molecules
JOURNAL
OF MOLECULAR
SPECTROSCOPY
2, 558-565(1958)
Critical Analysis and Applications of a Quasi-Resonant Theory of Pressure Broadening of Linear Molecules* L. J. KIEFFER? AND A. V. Department
of Physics,
Saint Louis
BUSHKOVITCH
University,
Saint Louis,
Missouri
A simplified Anderson theory of the pressure broadening of linear molecules has been critically analyzed and applied to five common types of molecular interactions. It has been shown that the simplified theory is equivalent to assuming that all partial cross sections of the exact Anderson theory are of the same order of magnitude as the resonant partial cross sections. Quantitative estimates of the errors introduced in the partial cross sections of the exact theory by making this simplifying assumption are given. INTRODUCTION
Because of the difficulty of applying the Anderson theory of pressure broadening (I), simplifications have been introduced (2). In a recent article by Perkins, Bushkovitch, and Kieffer (3) a simplified theory was outlined in detail and applied to three cases of foreign gas pressure broadening of OCS. The collision diameters calculated using this simplified theory were within 10 percent of the experimentally determined collision diameters. This investigation was undertaken to determine the relationship between the simplified theory as outlined by Perkins et al. (3) and the exact Anderson theory for linear molecules, and to examine the applicability of the simplified theory to various interactions between linear molecules. The two critical differences between the exact Anderson theory and the simplified theory are: (1) A classical average is substituted for a quantum mechanical average over the coordinates of the perturbing molecule. (2) The exponential factor is dropped in calculating Pa0 where for the exact theory 1 = PO,0 = z S_ exp(G90 m
(a I H(t) I P> dt.
* Supported in part by the Air Force Office of Scientific Research, AF 18(600)-1590. t Now at Argonne National Laboratory, Lemont, Illinois. 558
(1) under Contract
No.
THEORY
OF PRESSURE
BROADENING
559
LYand 0 represent all the quantum numbers necessary to specify the initial and final state of the system respectively where the transition (Y-+ 0 is caused by the collision. Here Wafl=
E, - EB n .
(2)
H(f) is the perturbing interaction and E, and Eb are the energies of the binary system before and after the collision, respectively. The partial cross section for linear molecules is CJ2 = 2r s * b&0, 0
Jx) db,
(3)
where Jz is the total angular momentum quantum number associated with the state of the perturbing molecule before collision and b is the distance of closest approach of the two molecules. The coordinate system used to describe the collision is the usual one with b, the distance of closest approach, taken as the polar axis. &(b, J2) may be considered to represent the probability of a complete interrupt)ion of the radiation with the perturbing molecule in the state characterized by Je and the distance of closest approach by b. &(b, Jz), not simplified, for linear molecules is1
Mb, Jz) = f [
c
Mi,MZ
(JiM$dYz
/ P2 1 J,MiJ2M,)
(2Ji + 1)(2Js + 1)
Mf,MZ
+
c
(J/M,JzM,
@Jf +
1 P’ 1 J&,JaM,) 1)(2Jz + 1)
1
(4)
Contributions to P for which AJi = AJ, = 0 must be left out in the calculation of P” when evaluating S, . Af represents the projection of the total angular momentum on a physically distinguishable axis. The subscripts i and .f indicate whether the radiating molecule is in the initial or final state of the radiative transition, respectively. The subscript 2 indicates that t.he quantum number refers to t*he perturbing molecule. The t,otal cross section for collisions between linear molecules using t*he nonsimplified dnderson theory is O-=
F
PJzflJz,
(5)
where pJ2 is the Boltzmann distribution function for the perturbing molewlc. A critical analysis of the simplified theory for linear molecules showed that it is equivalent to assuming that all partial cross sections of the exact hnderson 1This is Anderson’s SZ, outer . Terms corresponding to Anderson’s 82, middle have omitted in what follows as they either vanish or are small in most cases of interest.
been
560
KIEFFER
AND
BUSHKOVITCH
theory are quasi-resonant (in the sense explained below) and are equal to the quasi-resonant partial cross section in the limit Jz + 00. Any cross section calculated by setting wab = 0 will be called “quasi-resonant” so that it is obvious, keeping this definition in mind that all cross sections calculated using the simplified theory are “quasi-resonant” in this sense. The cross section calculated using the simplified theory for linear molecules is of the same order of magnitude as the resonant partial cross sections of the exact Anderson theory which is the reason for the use of the term quasi-resonant to describe them. A resonant partial cross section will be defined here as one which contains exactly resonant terms. Since the Anderson theory requires that all cont,ributing collisions be included in the calculation of any partial cross section a resonant partial cross section in our sense will contain some contributions due to collisions which are not resonant in the ordinary sense (that is the colliding molecles do not exchange a photon). In order to demonstrate that the cross section of the simplified theory is of the same order of magnitude as the resonant partial cross sections of the exact Anderson theory one must examine the equation for PO0 carefully. An exactly resonant contribution to SZ (one for which the molecules do exchange a photon) can be distinguished from a nonresonant contribution in two ways. First, the energy state of the perturbing molecule characterized by Jz is restricted by the radiative transition which is being observed. Second, ~~8 is identically zero. Two integrals appear as factors in the equation for Pa0 . One involves the time as the variable of integration and the other involves only the coordinates of the molecules. The contribution to Sz and to the partial cross section due to the integral which does not involve the time as the variable of integration is relatively constant and does not vary by an order of magnitude as Jz increases from 0 to CQ. This can easily be verified in any particular case by taking one of the matrix elements introduced by a term of the interaction, squaring and averaging over the magnetic substates as required to compute X2 . Since in the simplified theory it is assumed that w,~ = 0, which is the distinctive characteristic of the resonant contributions, it follows then that for linear molecules the cross section of the simplified theory is of the same order of magnitude as the resonant partial cross sections of the exact Anderson theory. For the case of linear molecules the following can be shown by using the correspondence principle lim C
C
I(JM If IJ’M’) I2= (fz),lass . ILv .
J-xa M J’.M’
(2J + 1) if f is a function of the coordinates only (4).
(6)
This shows that the substitution of a classical for a quantum mechanical average over the coordinates of the perturbing molecule in X2 is justified in the region of large Jz . It also justifies the statement that the quasi-resonant cross section
THEORY
OF PRESSURE
561
BROADENING
of the simplified theory is equal to the quasi-resonant partial cross section of the exact theory for large Jz since a classical instead of quantum mechanical average is used in the simplified theory. In actual practice the quantum mechanical average over the coordinates of the perturbing molecule does not differ from the classical average by more t,han 10 percent for J values of 4 or larger. The Boltzman factor weights more heavily states of relatively large J values and hence this error is masked. The simplified theory leads to a single cross section which does not depend upon the energy state of the perturbing molecule. This is equ~~ale~~t to assuming that all partial cross sections are equal. APPLICATION
OF THE SIMPLIFIED
The simplified theory has been used t)o calculat,e tions and the results are: (1) dipole-dipole
0= ;
[
F
THEORY
cross sections
for five interac-
1b(Ji) + dJdlKi2,
(7)
where g(J)
2J2 + 3J + 3 = 2J2+4J+:_j’
(2) dipole-quadrupole
(8) (3) quadrupoIe-quadrupole c
=
~[~~‘~[~~’
X Ig’(J3 +g’(J#‘“,
(9)
where 2
dJ) = ,;,‘,R, !_ ’ 1)
(4) dipole-induced-dipole s
=
2
[&]‘5 ids
x
[g’(Ji) + g’(J,)ll”,
(lo>
(5) dispersion
1 2/b
x Tg’(Ji> +
~‘(J_Al””
x Zr220.57) + ~41.05)
(11) + r12(0.62)1”‘,
562
KIEFFER
AND
BUSHKOVITCH
where y1 =
[a?
-
a,'][@!" -
yz =
a2'[u,N -
c&'J
a,'].
The subscripts 1 and 2 on the constants indicate that the quantities refer to the radiating and perturbing molecules, respectively. u as used in these equations and all equations in this paper is the mean relative velocity of the colliding molecules. or and Q are the electric dipole and quadrupole moment, respectively, in esu. a is the polarizability of a symmetric molecule. a’ is the polarizability of a linear molecule perpendicular to the internuclear axis and a” is the polarizability of a linear molecule parallel to the internuclear axis. I is the ionization energy of the molecule. The usual interpolation procedure to calculate the cross sections was used in which it is assumed that SZ is equal to 1 for all values of b for which &. > 1. Because of the complicated nature of the interactions some small terms were dropped from the interactions in the calculations made by Perkins et al. (3). No approximations in addition to those of the s~plified theory, such as dropping small terms, were made in the computation of the above cross sections. DISCUSSION
The dipole-dipole cross section calculated using the simplified theory, Eq. (7), has been applied to several cases of self-broadening and the results are listed in Table 1. It is immediately seen that the results are much larger than the measured values. Smith, Lackner, and Volkov (5) have already applied the exact Anderson theory t.o the self-broadening of OCS and BrCN using the dipoledipole interaction and the calculated cross sections were within 5 percent of the measured values for the OCS lines investigated. Comparison of the partial cross
EXPERIMENTAL MOl~Cl&
1. 2. 3. 4. 5. 6.
ocs BrCN ICl HCN CiCN ICN
AND
THEORETICAL
TABLE I COLLXS~ON DIAMETERSFOR SELF BROADENING
Line J=l-+Z J=2-+3 J=3--+4 J=O--+l J=l--+2 J=3+4
Measured8vebin A0
9.0 20.0 11.0 15.1 19.0 21.0
Quasi-resonant r, in A
13.6 65.1 16.1 41.5 54.0 90.4
a W. Gordy, W. V. Smith, and R. F. Trambarulo, “Microwave Spectroscopy,” Wiley, New York, 1953. b Q = * rc. c A is an angstrom unit.
p. 192.
THEORY
OF PRESSURE
BROADENING
563
sections calculated using the exact theory (5) and the quasi-resonant cross section Eq. (7), shows immediately the reason for the extremely large results obtained using the simplified theory. The partial cross sections calculated by Smith et al. (5) for the dipole-dipole interaction using the Anderson theory have :I resonance peak. The simplified theory on the contrary gives a total cross section which is equivalent to constant partial cross sections which are of the same order of magnitude as the resonant cross sections of the exact Anderson theory. This can easily be verified by comparing the cross section given by Eq. (7) with the resonant values of the part,ial cross sections as calculated by Smith et al. (6) for OCS and BrCN. The assumption of qasi-resonance, then, is a poor approximation for the partial cross sections in this case and Eq. (7) cannot be used to give an approximation to the exact Anderson theory for dipole-dipole collisions. The rapidly changing nature of the partial cross sections with increasing J:! for the dipole-dipole interaction is due to the exponentially varying Hankel functions which enter as factors in Sz through t,he time integrals in the elements of Pap (5). Since the time integrals which arise from t’he dipole-quadrupole and qundrupole-quadrupole interactions are evaluated in terms of the same Hank4 func*tions as those for the dipole-dipole interaction one would expect that the partial cross sections for the dipole-quadrupole and quadrupole-quadrupole interactions would have a Jp dependence similar to those for the dipole-dipole interaction. These two interactions have terms such as cos2 e2 for which one of the allowed transitions which the perturbing molecule can undergo is AJ, = 0 which meaus ma9will not be a function of Jz and hence neither will the Hankel function since its argument is 1~ = w&/u. This fact has been pointed out, by Smith (6) in connection with dipole-quadrupole interaction. Terms in Sf with this propert#y will be the major contribution to the partial cross sections of the exact theory for large JS . This means that the resonance peak in these two cases mill be much less sharp t’han that for the dipole-dipole interaction and hence quasi-resonance will be :I better approximation in these t’wo cases. If /; < 0.50 where only the trnnsitions of the radiator due to collision are considered then for both the dipolequadrupole and quadrupole-quadrupole interactions t,he cross section calculated with the simplified or quasi-resonant theory will be 30 percent larger at maximum than the partial cross sections calculated from t,he exact theory for large J2’s. In the c’ase of the dipole-induced-dipole and dispersion interactions if ;I symmt+rics pcrturber is assumed in both cases and I< < 1, t>hedifference betnccn t,he woss section calculated using the simplified theory and t,he partial cross scct,ions (~:~lculatcd using the exact L%nderson theory is 10 percent at the maximluu, the: cross sections calculated using the simplified theory being the larger. ‘I’hr>estimate of the errors in the above four cases was made by comparing the results of the time int>egrations when the exponential term is dropped in raa
561
KIEFFER
AND BUSHKOVITCH TABLE
ERRORS
II
IN QUASI-RESONANT
CROSS SECTIONS
Interaction
1. dipole-dipole 2. 3. 4. 5.
dipole-quadrupole quadrupole-quadrupole dipole-induced-dipole dispersion
The quasi-resonant cross section too large to be used in this case. 0.50 30 0.50 30 1.0 10 1.0 10
and when it is not, as required by the exact Anderson errors in the quasi-resonant cross sections as compared theory are summarized in Table II.
is
theory. Estimates of the with the exact Anderson
CONCLUSIONS
This critical analysis of a simplified Anderson theory for linear molecules has shown that a cross section calculated using this theory is equivalent to assuming that all the partial cross sections of the exact Anderson theory for linear molecules are quasi-resonant, that is of the same order of magnitude as the resonant partial cross sections; and are equal to the quasi-resonant partial cross section in the limit J2 - CU. It should be pointed out that this critical analysis of the simplified theory only compares it with the exact theory. One cannot decide the applicability of the theory itself to the broadening of the microwave spectral lines of linear molecules without an appeal to experimental findings. The cross section calculated for the dipole-dipole interaction using the quasiresonant theory cannot be used as an approximation to the exact theory. The dipole-quadrupole and quadrupole-quadrupole interactions although similar, as has been pointed out, do differ in the transitions caused by the collision. As a result of this there is a maximum error of 30 percent in the cross section calculated with the simplified theory as compared to the exact theory for these two interactions if k 6 0.50. The error in the cross section is ordinarily much less and hence the quasi-resonant, cross section can be used in a preliminary survey to determine the relative importance of interactions in causing broadening of the line being investigated. They can also be used to give estimat,es of quadrupole moments as proposed by Smith (6’). The cross sections calculated for the dipole-induced-dipole and dispersion interactions for symmetric perturbers are at a maximum 10 percent larger than the
THEORY
OF PRESSURE
BROADENING
565
corresponding exact theoretical cross sections for k < 1. Because of the complicated nature of these two interactions for nonsymmetric perturbers a more accurate estimate of the error in using the quasi-resonant theory is not given. It should be pointed out, however, that the dispersion interaction has terms for which one of the selection rules is AJ2 = 0. This would presumably lead to a similar analysis as that for the dipole-quadrupole and quadrupole-quadrupole interactions and as the perturber became more symmetric the cross section would approach the 10 percent error limit set by this analysis. In summary, then, one might say that although the quasi-resonant cross section for the dipole-dipole interaction cannot be used the other cross se&ions wlculnted using the simplified theory can. The approximate cross sections for the ot)her four interactions are useful because it is not possible in general to write closed expressions for these cross se&ions using the exact Anderson theory. This means that each change of the parameters requires a new calculation if the exact theory is used. These cross sections provide a convenient test of the relative importance of various interactions in the broadening of rotational lines of linear molecules and can be used to give approximate values of molecular const,nnt,s from experimentally measured line widths. REVEIVED:
May
7, 1958 REFERENCES
1. P. 2. W. 5. Ii. _J. D. 5. W. 6. W.
W. ANDERSON, Phys. Rev. 76, 647 (1949). V. SMITH AND R. HOWARD, Phys. Rev. 79, 132 (1950). L. PERKINS, A. V. BUSHKOVITCH, AND L. J. KIEFFER, J. Chem. Phys. 26, 779 (1957). BOHM, “Quantum Theory,” p. 185. Prentice-Hall, New York, 1951. V. SMITH, H. A. LACKNER, AND A. B. VOLKOV, J. Chew. Phys. 23, 389 (1955). V. SMITH, J. Chem. Phys. 26, 510 (1956).
OF MOLECULAR
SPECTROSCOPY
2, 558-565(1958)
Critical Analysis and Applications of a Quasi-Resonant Theory of Pressure Broadening of Linear Molecules* L. J. KIEFFER? AND A. V. Department
of Physics,
Saint Louis
BUSHKOVITCH
University,
Saint Louis,
Missouri
A simplified Anderson theory of the pressure broadening of linear molecules has been critically analyzed and applied to five common types of molecular interactions. It has been shown that the simplified theory is equivalent to assuming that all partial cross sections of the exact Anderson theory are of the same order of magnitude as the resonant partial cross sections. Quantitative estimates of the errors introduced in the partial cross sections of the exact theory by making this simplifying assumption are given. INTRODUCTION
Because of the difficulty of applying the Anderson theory of pressure broadening (I), simplifications have been introduced (2). In a recent article by Perkins, Bushkovitch, and Kieffer (3) a simplified theory was outlined in detail and applied to three cases of foreign gas pressure broadening of OCS. The collision diameters calculated using this simplified theory were within 10 percent of the experimentally determined collision diameters. This investigation was undertaken to determine the relationship between the simplified theory as outlined by Perkins et al. (3) and the exact Anderson theory for linear molecules, and to examine the applicability of the simplified theory to various interactions between linear molecules. The two critical differences between the exact Anderson theory and the simplified theory are: (1) A classical average is substituted for a quantum mechanical average over the coordinates of the perturbing molecule. (2) The exponential factor is dropped in calculating Pa0 where for the exact theory 1 = PO,0 = z S_ exp(G90 m
(a I H(t) I P> dt.
* Supported in part by the Air Force Office of Scientific Research, AF 18(600)-1590. t Now at Argonne National Laboratory, Lemont, Illinois. 558
(1) under Contract
No.
THEORY
OF PRESSURE
BROADENING
559
LYand 0 represent all the quantum numbers necessary to specify the initial and final state of the system respectively where the transition (Y-+ 0 is caused by the collision. Here Wafl=
E, - EB n .
(2)
H(f) is the perturbing interaction and E, and Eb are the energies of the binary system before and after the collision, respectively. The partial cross section for linear molecules is CJ2 = 2r s * b&0, 0
Jx) db,
(3)
where Jz is the total angular momentum quantum number associated with the state of the perturbing molecule before collision and b is the distance of closest approach of the two molecules. The coordinate system used to describe the collision is the usual one with b, the distance of closest approach, taken as the polar axis. &(b, J2) may be considered to represent the probability of a complete interrupt)ion of the radiation with the perturbing molecule in the state characterized by Je and the distance of closest approach by b. &(b, Jz), not simplified, for linear molecules is1
Mb, Jz) = f [
c
Mi,MZ
(JiM$dYz
/ P2 1 J,MiJ2M,)
(2Ji + 1)(2Js + 1)
Mf,MZ
+
c
(J/M,JzM,
@Jf +
1 P’ 1 J&,JaM,) 1)(2Jz + 1)
1
(4)
Contributions to P for which AJi = AJ, = 0 must be left out in the calculation of P” when evaluating S, . Af represents the projection of the total angular momentum on a physically distinguishable axis. The subscripts i and .f indicate whether the radiating molecule is in the initial or final state of the radiative transition, respectively. The subscript 2 indicates that t.he quantum number refers to t*he perturbing molecule. The t,otal cross section for collisions between linear molecules using t*he nonsimplified dnderson theory is O-=
F
PJzflJz,
(5)
where pJ2 is the Boltzmann distribution function for the perturbing molewlc. A critical analysis of the simplified theory for linear molecules showed that it is equivalent to assuming that all partial cross sections of the exact hnderson 1This is Anderson’s SZ, outer . Terms corresponding to Anderson’s 82, middle have omitted in what follows as they either vanish or are small in most cases of interest.
been
560
KIEFFER
AND
BUSHKOVITCH
theory are quasi-resonant (in the sense explained below) and are equal to the quasi-resonant partial cross section in the limit Jz + 00. Any cross section calculated by setting wab = 0 will be called “quasi-resonant” so that it is obvious, keeping this definition in mind that all cross sections calculated using the simplified theory are “quasi-resonant” in this sense. The cross section calculated using the simplified theory for linear molecules is of the same order of magnitude as the resonant partial cross sections of the exact Anderson theory which is the reason for the use of the term quasi-resonant to describe them. A resonant partial cross section will be defined here as one which contains exactly resonant terms. Since the Anderson theory requires that all cont,ributing collisions be included in the calculation of any partial cross section a resonant partial cross section in our sense will contain some contributions due to collisions which are not resonant in the ordinary sense (that is the colliding molecles do not exchange a photon). In order to demonstrate that the cross section of the simplified theory is of the same order of magnitude as the resonant partial cross sections of the exact Anderson theory one must examine the equation for PO0 carefully. An exactly resonant contribution to SZ (one for which the molecules do exchange a photon) can be distinguished from a nonresonant contribution in two ways. First, the energy state of the perturbing molecule characterized by Jz is restricted by the radiative transition which is being observed. Second, ~~8 is identically zero. Two integrals appear as factors in the equation for Pa0 . One involves the time as the variable of integration and the other involves only the coordinates of the molecules. The contribution to Sz and to the partial cross section due to the integral which does not involve the time as the variable of integration is relatively constant and does not vary by an order of magnitude as Jz increases from 0 to CQ. This can easily be verified in any particular case by taking one of the matrix elements introduced by a term of the interaction, squaring and averaging over the magnetic substates as required to compute X2 . Since in the simplified theory it is assumed that w,~ = 0, which is the distinctive characteristic of the resonant contributions, it follows then that for linear molecules the cross section of the simplified theory is of the same order of magnitude as the resonant partial cross sections of the exact Anderson theory. For the case of linear molecules the following can be shown by using the correspondence principle lim C
C
I(JM If IJ’M’) I2= (fz),lass . ILv .
J-xa M J’.M’
(2J + 1) if f is a function of the coordinates only (4).
(6)
This shows that the substitution of a classical for a quantum mechanical average over the coordinates of the perturbing molecule in X2 is justified in the region of large Jz . It also justifies the statement that the quasi-resonant cross section
THEORY
OF PRESSURE
561
BROADENING
of the simplified theory is equal to the quasi-resonant partial cross section of the exact theory for large Jz since a classical instead of quantum mechanical average is used in the simplified theory. In actual practice the quantum mechanical average over the coordinates of the perturbing molecule does not differ from the classical average by more t,han 10 percent for J values of 4 or larger. The Boltzman factor weights more heavily states of relatively large J values and hence this error is masked. The simplified theory leads to a single cross section which does not depend upon the energy state of the perturbing molecule. This is equ~~ale~~t to assuming that all partial cross sections are equal. APPLICATION
OF THE SIMPLIFIED
The simplified theory has been used t)o calculat,e tions and the results are: (1) dipole-dipole
0= ;
[
F
THEORY
cross sections
for five interac-
1b(Ji) + dJdlKi2,
(7)
where g(J)
2J2 + 3J + 3 = 2J2+4J+:_j’
(2) dipole-quadrupole
(8) (3) quadrupoIe-quadrupole c
=
~[~~‘~[~~’
X Ig’(J3 +g’(J#‘“,
(9)
where 2
dJ) = ,;,‘,R, !_ ’ 1)
(4) dipole-induced-dipole s
=
2
[&]‘5 ids
x
[g’(Ji) + g’(J,)ll”,
(lo>
(5) dispersion
1 2/b
x Tg’(Ji> +
~‘(J_Al””
x Zr220.57) + ~41.05)
(11) + r12(0.62)1”‘,
562
KIEFFER
AND
BUSHKOVITCH
where y1 =
[a?
-
a,'][@!" -
yz =
a2'[u,N -
c&'J
a,'].
The subscripts 1 and 2 on the constants indicate that the quantities refer to the radiating and perturbing molecules, respectively. u as used in these equations and all equations in this paper is the mean relative velocity of the colliding molecules. or and Q are the electric dipole and quadrupole moment, respectively, in esu. a is the polarizability of a symmetric molecule. a’ is the polarizability of a linear molecule perpendicular to the internuclear axis and a” is the polarizability of a linear molecule parallel to the internuclear axis. I is the ionization energy of the molecule. The usual interpolation procedure to calculate the cross sections was used in which it is assumed that SZ is equal to 1 for all values of b for which &. > 1. Because of the complicated nature of the interactions some small terms were dropped from the interactions in the calculations made by Perkins et al. (3). No approximations in addition to those of the s~plified theory, such as dropping small terms, were made in the computation of the above cross sections. DISCUSSION
The dipole-dipole cross section calculated using the simplified theory, Eq. (7), has been applied to several cases of self-broadening and the results are listed in Table 1. It is immediately seen that the results are much larger than the measured values. Smith, Lackner, and Volkov (5) have already applied the exact Anderson theory t.o the self-broadening of OCS and BrCN using the dipoledipole interaction and the calculated cross sections were within 5 percent of the measured values for the OCS lines investigated. Comparison of the partial cross
EXPERIMENTAL MOl~Cl&
1. 2. 3. 4. 5. 6.
ocs BrCN ICl HCN CiCN ICN
AND
THEORETICAL
TABLE I COLLXS~ON DIAMETERSFOR SELF BROADENING
Line J=l-+Z J=2-+3 J=3--+4 J=O--+l J=l--+2 J=3+4
Measured8vebin A0
9.0 20.0 11.0 15.1 19.0 21.0
Quasi-resonant r, in A
13.6 65.1 16.1 41.5 54.0 90.4
a W. Gordy, W. V. Smith, and R. F. Trambarulo, “Microwave Spectroscopy,” Wiley, New York, 1953. b Q = * rc. c A is an angstrom unit.
p. 192.
THEORY
OF PRESSURE
BROADENING
563
sections calculated using the exact theory (5) and the quasi-resonant cross section Eq. (7), shows immediately the reason for the extremely large results obtained using the simplified theory. The partial cross sections calculated by Smith et al. (5) for the dipole-dipole interaction using the Anderson theory have :I resonance peak. The simplified theory on the contrary gives a total cross section which is equivalent to constant partial cross sections which are of the same order of magnitude as the resonant cross sections of the exact Anderson theory. This can easily be verified by comparing the cross section given by Eq. (7) with the resonant values of the part,ial cross sections as calculated by Smith et al. (6) for OCS and BrCN. The assumption of qasi-resonance, then, is a poor approximation for the partial cross sections in this case and Eq. (7) cannot be used to give an approximation to the exact Anderson theory for dipole-dipole collisions. The rapidly changing nature of the partial cross sections with increasing J:! for the dipole-dipole interaction is due to the exponentially varying Hankel functions which enter as factors in Sz through t,he time integrals in the elements of Pap (5). Since the time integrals which arise from t’he dipole-quadrupole and qundrupole-quadrupole interactions are evaluated in terms of the same Hank4 func*tions as those for the dipole-dipole interaction one would expect that the partial cross sections for the dipole-quadrupole and quadrupole-quadrupole interactions would have a Jp dependence similar to those for the dipole-dipole interaction. These two interactions have terms such as cos2 e2 for which one of the allowed transitions which the perturbing molecule can undergo is AJ, = 0 which meaus ma9will not be a function of Jz and hence neither will the Hankel function since its argument is 1~ = w&/u. This fact has been pointed out, by Smith (6) in connection with dipole-quadrupole interaction. Terms in Sf with this propert#y will be the major contribution to the partial cross sections of the exact theory for large JS . This means that the resonance peak in these two cases mill be much less sharp t’han that for the dipole-dipole interaction and hence quasi-resonance will be :I better approximation in these t’wo cases. If /; < 0.50 where only the trnnsitions of the radiator due to collision are considered then for both the dipolequadrupole and quadrupole-quadrupole interactions t,he cross section calculated with the simplified or quasi-resonant theory will be 30 percent larger at maximum than the partial cross sections calculated from t,he exact theory for large J2’s. In the c’ase of the dipole-induced-dipole and dispersion interactions if ;I symmt+rics pcrturber is assumed in both cases and I< < 1, t>hedifference betnccn t,he woss section calculated using the simplified theory and t,he partial cross scct,ions (~:~lculatcd using the exact L%nderson theory is 10 percent at the maximluu, the: cross sections calculated using the simplified theory being the larger. ‘I’hr>estimate of the errors in the above four cases was made by comparing the results of the time int>egrations when the exponential term is dropped in raa
561
KIEFFER
AND BUSHKOVITCH TABLE
ERRORS
II
IN QUASI-RESONANT
CROSS SECTIONS
Interaction
1. dipole-dipole 2. 3. 4. 5.
dipole-quadrupole quadrupole-quadrupole dipole-induced-dipole dispersion
The quasi-resonant cross section too large to be used in this case. 0.50 30 0.50 30 1.0 10 1.0 10
and when it is not, as required by the exact Anderson errors in the quasi-resonant cross sections as compared theory are summarized in Table II.
is
theory. Estimates of the with the exact Anderson
CONCLUSIONS
This critical analysis of a simplified Anderson theory for linear molecules has shown that a cross section calculated using this theory is equivalent to assuming that all the partial cross sections of the exact Anderson theory for linear molecules are quasi-resonant, that is of the same order of magnitude as the resonant partial cross sections; and are equal to the quasi-resonant partial cross section in the limit J2 - CU. It should be pointed out that this critical analysis of the simplified theory only compares it with the exact theory. One cannot decide the applicability of the theory itself to the broadening of the microwave spectral lines of linear molecules without an appeal to experimental findings. The cross section calculated for the dipole-dipole interaction using the quasiresonant theory cannot be used as an approximation to the exact theory. The dipole-quadrupole and quadrupole-quadrupole interactions although similar, as has been pointed out, do differ in the transitions caused by the collision. As a result of this there is a maximum error of 30 percent in the cross section calculated with the simplified theory as compared to the exact theory for these two interactions if k 6 0.50. The error in the cross section is ordinarily much less and hence the quasi-resonant, cross section can be used in a preliminary survey to determine the relative importance of interactions in causing broadening of the line being investigated. They can also be used to give estimat,es of quadrupole moments as proposed by Smith (6’). The cross sections calculated for the dipole-induced-dipole and dispersion interactions for symmetric perturbers are at a maximum 10 percent larger than the
THEORY
OF PRESSURE
BROADENING
565
corresponding exact theoretical cross sections for k < 1. Because of the complicated nature of these two interactions for nonsymmetric perturbers a more accurate estimate of the error in using the quasi-resonant theory is not given. It should be pointed out, however, that the dispersion interaction has terms for which one of the selection rules is AJ2 = 0. This would presumably lead to a similar analysis as that for the dipole-quadrupole and quadrupole-quadrupole interactions and as the perturber became more symmetric the cross section would approach the 10 percent error limit set by this analysis. In summary, then, one might say that although the quasi-resonant cross section for the dipole-dipole interaction cannot be used the other cross se&ions wlculnted using the simplified theory can. The approximate cross sections for the ot)her four interactions are useful because it is not possible in general to write closed expressions for these cross se&ions using the exact Anderson theory. This means that each change of the parameters requires a new calculation if the exact theory is used. These cross sections provide a convenient test of the relative importance of various interactions in the broadening of rotational lines of linear molecules and can be used to give approximate values of molecular const,nnt,s from experimentally measured line widths. REVEIVED:
May
7, 1958 REFERENCES
1. P. 2. W. 5. Ii. _J. D. 5. W. 6. W.
W. ANDERSON, Phys. Rev. 76, 647 (1949). V. SMITH AND R. HOWARD, Phys. Rev. 79, 132 (1950). L. PERKINS, A. V. BUSHKOVITCH, AND L. J. KIEFFER, J. Chem. Phys. 26, 779 (1957). BOHM, “Quantum Theory,” p. 185. Prentice-Hall, New York, 1951. V. SMITH, H. A. LACKNER, AND A. B. VOLKOV, J. Chew. Phys. 23, 389 (1955). V. SMITH, J. Chem. Phys. 26, 510 (1956).