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On the single relaxation time approximation (SRTA) in the kinetic theory of gas phase NMR
Volume 44, number 2
CHEMICAL PHYSICS LETTERS
ON THE SINGLE RELAXATION TIME APPROXIMATION IN THE KINETIC THEORY OF GAS PHASE NMR*
1 December 1976
(SRTA)
Wing-Ki LIU and F.R. MCCOURT Chemistry Department, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI Received 16 August 1976
The use of single-relaxation-time approximations (SRTA) in describing nuclear magnetic relaxation (NMR) is discussed and the proper identification of the relaxation parameters with collision cross sections is addressed. Two different SRTA curves giving the variation of the longitudinal relaxation time T, with number density R are evaluated using the exact numerical closccoupled results of Shafer and Gordon for molecular hydrogen infmitely dilute in helium and both SRTA curves are compared with that obtained from the full matrix inversion procedure.
Recently, the application of kinetic theory to gas phase NMR via the memory function approach has been a subject of interest [l-5] . The longitudinal and transverse relaxation times T1 and Tz can be expressed under certain well-justified assumptions as the Laplace transform of the appropriate tensorial components of a kernel which includes all collision dynamics. To simplify further the resulting expressions, the single relaxation time approximation (SRTA) has been introduced [l-S]. This amounts to representing the collisional dynamics by a single parameter for each relaxation mechanism, and attempts have been made to extract these parameters from experimental results [3, 41. However, the detailed nature of the SRTA has received little attention, and it is the nurpose of this letter to examine the applicability of the SRTA and the proper identification of the relaxation parameters with various collision cross sections. In light of the availability of an exact (close-coupled) calculation of various collision cross sections for the system H2-He by Shafer and Gordon [6], a numerical test of the SRTA is performed for this system. For hydrogen gas, the nuclear spins can rela.. via the intramolecular spin-rotation and dipolar interactions. The spin-rotation relaxation will be considered * Supported in part by a National Research Council of Canada grant in aid of research.
first. From eq. (22) of ref. [I], T, is given by &)-I
= go2 3 sr s = dt(J 9, exp (-C’t]J>
cosAwt,
(I)
0
where Aw = wJ - 01 is the difference between the rotational and nuclear spin Armor frequencies of the molecule, I+ is the spin-rotation coupiing constant and C’ is the self-part of the Waidmann-Snider collision operator [ 11. For H2, wr = 13.4 X LO7 cad/s [7] , wf = 2.12 X lo7 rad/s [8] and w,, = 0.71567 X LO6 rad/s [8] . The usual statement of the SRTA [I -41 is that (i) (J 0, exp{-e’t) I) can be repIaced by exp(--n(u)o,,t). hence yielding for Tr the well-known formula (1 /T,gRTA= 2 2 3%r(J2)o
tdU>U,, (Awj’ f (n<~>o,,>~
(2)
and (ii) by comparing the expansion of the exponential operator in the kernel of eq. (1) with that of exp(-n(u)u,,t), the parameter osr can be identified witl an appropriate coliision bracket [i-4] as
l
usr = (J .C?J)/(J2>,,(~1,
(?I
where (u) = (8kgT/np)1’2 is 2 mean thermal relative speed with ~1the reduced mass of the colhding pair and n is the number density of the mixture. For a system consisting of an active molecule inti289
CHEMICALPHYSICS LETTERS
Volume 44, number 2
nitely dilute in a bath ofstructureless perturbers , such as H2-He, the collision bracket can be related 191 to the relaxation matrix of the theory of Fano [lOI and Ben-Reuven [I I]. The integration over t in eq. (1) can be carried out, yielding, in the notation of Shafer and Gordon, (4)
(1 D-1 )sr =$&m[d,ro(Awl-
irz(v)~(~))-~
-P
ldsr]
,
where 1 is the unit matrix d,, is a vector whose components for thej-rotational level equals [jo + l)] “‘, P is a diagonal population matrix whose diagonal element for the j-level is + 1) exp{-Bj(j
Q-‘(2j
+ l)/kgT}
1
Comparing with the discussion in arriving at eq. (2), it is clear that the first aspect of the SRTA consists in replacing the relazzation matrix by a single cross section usr such that (l/T1 ),, is given by eq. (2). It is noted in passing that V”,,
= d,;
P dsr .
The second aspect of the SRTA, the identification a,, by eq. (3), becomes usr = d,,*o(l) P l dsr/dsrP .dsr. l
Ha- He
(6)
Now, accurate experimental
data can be obtained easily onlv in the high density region where (nC~>o,,)~ 9 (Au)~. In this region, eq. (2) reduces to (nW7s,)-*,
(7)
while eq. (4) yields (I&),,
= +.$d,, +hh.r(i)]--l* P l dsr ,
(8)
Comparing eqs. (7) and (8), osr [called air below to distinguish it from that given by eq. (6)3 can alternatively be identified by ozr = CJ2>o/dsr~ [&)]-'* P d,, . l
osr = 0.45 A2,
(9)
ozr = 0.28 A2.
(10)
These two values of usr are then substituted into eq. (2) and the resulting Tl versus n curves are plotted in fig. 1 along with the exact result of eq. (4). ‘Ihe case of dipolar reIaxation can be treated similarly. The “exact” expression for TI in this case is
1
t E
I t-
(IIT1 >d = &&
Q2
as
1
2
, 5
n tamagats) Fig. 1. The contribution to T1 due to the spin-rotation re-
laxation mechanism evaIuated using the close-coupled resultsof Shafer and Gordon [6 ] : solid curve, full matrix inversion or “exact” result; dashed curve, SRTA result using eq. (9) to define asr; dotted curve SRTA result using eq. (6) to define usr.
290
l
of
In general, ozr in eq. (9) will not be equal to usr in eq. (6) except at low enough temperature such that only the Iowest rotational level is significantly populated. Using the results of Shafer and Gordon 163 for Hz-He at 298 K, it is found that
10
9
(5)
l
(liTl)sr SRTA = ~&J2)(J
(B and Q are the rotational constant and rotational partition function, respectively, of the active molecule), and t#) is the relaxation matrix cross section oftensorial rank 1 [lo]. The numerical results for 0(l) given by Shafer and Gordon [6] are employed to calculate T1 as a function of n at 298 K and the res’ult is plotted in fig. I. For ortho-Hz at 298 K, only the j = 1 and 3 levels are signii?cantly populated [6] and a 2 X 2 matrix suffices in the calculation.
December 1976
t d= (Awl -in
Im [4d* (2Awl - in(u) 6(2))-1 lP*d Wa(2))-’
*P-d] ,
(11)
where ad = 5.4336 X IO6 rad/s [8] is the dipolar coupling constant for Hz, d is a vector whose corn onent for thej-level is -[i(j + 1)/(2j + 3)(2j - l)] ‘/‘and m(2) is the relaxation matrix cross section of iensorial rank 2. As before, the first aspect of the SRTA assumes that eq. (11) can be replaced by
Volume 44, number 2
(Wl)d sRTA=$,&J2/(4J2
- 3)>,
nG_Jhd
4n(vhQ +
X
am
(Au)~ + (n(~)o~)~ As for the second Cation
of od,
aspect
the USUd
1 December 1976
CHEMICAL PHYSICS LETTERS
+ (~(v>o,)~
1-(12)
of the SRTA, i.e. the identifi-
expression
[ 1,2]
that corresponds
to eq. (3) or eq. (6) is od=d-a
‘2)-P-d/d-P*d.
(13)
Here it is noted that
- 3)),
=d - P
l
d .
On the other hand, if expressions
(14) (11)
and (12)
are
compared in the high density regime, an alternative identification of od which corresponds to eq. (9) for the spin-rotation case is l_r(;=d - P ld/d - [a(2)] -l - P -d _
(15)
When the results of Shafer and Gordon [6] are used in eqs. (13) and (15), it is found that for H2-He at 298 K, o,j = 0.42 w2,
0;; = 0.41 82.
(16)
The resulting Tl versus iz curves calculated from eqs. (11) and (12) with cr: and od given in (16) are plotted in fig. 2. The total Tl due to both spin-rotation and dipolar interaction can be found from l/T1 = (l/Tl)sr+(l/Tl)d
-
(17)
Curves of Tl versus n for the exact result as well as the SRTA results are also plotted in fig. 2. From fig. 1, it is observed that for the case of spinrotation rekation, the exact curve differs from either of the SRTA curves. The SRTA curve using a& as parameter fits the high density region well (as expected from the definition of uzr), but its minimum occurs at too high a density. The SRTA curve with usr as parameter has its minimum at about the right density, but would not fit the exact curve except at very low densities. Both SRTA curves give too low a value of T1 at the minimum. For the case of the dipolar interaction, on the other hand, both SRTA curves fit the exact curve very well, as all three curves are indistinguishable in fig. 2. In fig. 2, when both the spin-rotation and dipolar interactions are taken into account, the agreement between the exact curve and the SRTA curves is fairly good. The SRTA curve with uzr and uz as param-
n camagatst Fig. 2. The contirbution to Tr due to the dipok rehrxation mechanism and the total Tt as given by eq. (17) evaluated using the results of ref. [6] : dashed curve, dipok contnbution (the “exact” and both SRTA curves are indistinguishable on the scale of the figure); solid curve, full matrix inversion as given by eqs. (41, (11) and (17); dotted curve, SRTA result obtained from eqs. (2), (12) and (17) together with eqs. (9) and (15); dotdash curve, SRTA result obtained from eqs. (2), (12) and (17) together with eqs. (6) and (13).
eters is preferred since, (a) it fits the exact curve in the region of high density where accurate experimental measurement can be performed and, (b) its minimum occurs at about the same density as that for the exact curve. The value of Tl minimum predicted by this SRTA curve is about 11% lower than the exact valueThus, if it is insisted that a one-parameter fit to the Tl versus n curve according to eqs. (2), (12) and (17) be made, then the defmitions of eqs. (9) and (15) for the effective collision cross sections are to be preferred. Finally, it is noted that Neilson and Gordon [12] had pointed out that the matrix inversion in eq. (4) can be carried out explicitly and expressed in terms of the eigenvalues and eigenvectors of the reIa..&ion matrix. Thus if the matrk U diagonalizes u ,i.e.,if
(18) where the hi are the eigenvahxes of&l, d,, - [Aol
- in(v) &I]-1
- P
l
then
d sr
(1% 291
Volume 44,
number 2
CHEMICAL PHYSICS LETTERS
less dependent on the specific rotational levels occupied by the molecule, and it can be expected that the Xjs are not very different, rendering the SRTA more valid
with the Wi given as
and pk being the kk element of the diagonal popuiation matrix P. Eqs. (6) and (7) can then be written as
(22) where the obvious result Zi Wi = dsr* P d,, = (.12), has been used. It may now be asked when the SRTA is a good approximation. A trivial case will be that when only one Xi predominates. This is so for one-level system mentioned earlier. Another case of interest would be if all the eigenvalues of c(l) are approximately equal, i.e., Xi = X for all i. Eq. (19) then becomes l
When the results of (20), (21) and (23) are substituted into eq. (4), the SRTA result of eq. (2) is recovered, with X in place of usr. Also, from eqs. (23) and (24), usr and ozr can be shown to be equal. The same consideration can be applied to the exact expression for l/T, of eq. (11). Thus it may be concluded that if the eigenvalucs of the relaxation matrix are about equal, the SRTA will be a good approximation, in which case the u and u* expressions given by eqs. (6) and (9) [or by eqs. (13) and (15)] are equivalent. The expressions for u [eqs. (6) and (13)] are more useful since they are the kinetic cross sections for which analytic expressions within the distorted wave Born approximntion(DWBA) are available 191. For molecules having larger moments of inertia such as HCi. collision transition & mobabilities are
292
1 December 1976
for such systems. Preliminary analysis on the system HCI-HCi appears to indicate that the SRTA is sufficient to fit the experimental data 1131. From the results of Shafer and Gordon [6], the eigenvalues of the 2 X 2 matriv #) are found to be 0.71 A2 and 0.33 A2, while those of m(2) are 0.43 A2 and 0.33 A2. Hence it is not surprising that the SRTA works well for dipolar relaxation but not for spin-rotation relaxation in molecular hydrogen infinitely dilute in helium. References [ 1] F.R. McCourt, in: NMR basis principles and progress, Vol. 13, ed. M.M. Pintar (Springer, Berlin, 1976) p. 55. [2] F.R. McCourt, T-E. Raidy, R. Festa and A.C. Levi, Can. J. Phys. 53 (1975) 2449 [3 ] F.R. McCourt, T.E. Raidy, T. Rudensky and A.C. Levi, Can. J. Phys. 53 (1975) 2463. [4 ] T.E. Raidy and F-R. McCourt, Chem. Phys. Letters 38
(1976) 300. [S] G. Tenti and F.R. McCourt, J. Chem. Phys. 65 (1976) 623. [6] R. Shafer and R.G. Gordon, J. Chem. Phys. 58 (1973) 5422. [7] A. Carrington and A.D. McLaughlin, Introduction to
magnetic resonance (Harper and Row, London, 1967). [8J N-J. Harrick, R.G. Barnes, P.G. Bray and N.F. Ramsey, Phys. Rev. 90 (1953) 260. [9] WK. Liu and F-R. McCourt, Chem. Phys., to be submitted for publication.
[lo] U. Fano, Phys. Rev. 131 (1963) 259. [I 1] A.Ben-Reuven, Phys. Rev. 145 (1966) 7. [ 121 W.B. Ncilson and R.G. Gordon, J. Chem. Phys. 58 (1973) 4131. [ 131 R-L. Armstrong, private communication; T.E. Raidy, Ph.D. Thesis, University of Waterloo (1976), unpublished.
CHEMICAL PHYSICS LETTERS
ON THE SINGLE RELAXATION TIME APPROXIMATION IN THE KINETIC THEORY OF GAS PHASE NMR*
1 December 1976
(SRTA)
Wing-Ki LIU and F.R. MCCOURT Chemistry Department, University of Waterloo, Waterloo, Ontario, Canada N2L 3GI Received 16 August 1976
The use of single-relaxation-time approximations (SRTA) in describing nuclear magnetic relaxation (NMR) is discussed and the proper identification of the relaxation parameters with collision cross sections is addressed. Two different SRTA curves giving the variation of the longitudinal relaxation time T, with number density R are evaluated using the exact numerical closccoupled results of Shafer and Gordon for molecular hydrogen infmitely dilute in helium and both SRTA curves are compared with that obtained from the full matrix inversion procedure.
Recently, the application of kinetic theory to gas phase NMR via the memory function approach has been a subject of interest [l-5] . The longitudinal and transverse relaxation times T1 and Tz can be expressed under certain well-justified assumptions as the Laplace transform of the appropriate tensorial components of a kernel which includes all collision dynamics. To simplify further the resulting expressions, the single relaxation time approximation (SRTA) has been introduced [l-S]. This amounts to representing the collisional dynamics by a single parameter for each relaxation mechanism, and attempts have been made to extract these parameters from experimental results [3, 41. However, the detailed nature of the SRTA has received little attention, and it is the nurpose of this letter to examine the applicability of the SRTA and the proper identification of the relaxation parameters with various collision cross sections. In light of the availability of an exact (close-coupled) calculation of various collision cross sections for the system H2-He by Shafer and Gordon [6], a numerical test of the SRTA is performed for this system. For hydrogen gas, the nuclear spins can rela.. via the intramolecular spin-rotation and dipolar interactions. The spin-rotation relaxation will be considered * Supported in part by a National Research Council of Canada grant in aid of research.
first. From eq. (22) of ref. [I], T, is given by &)-I
= go2 3 sr s = dt(J 9, exp (-C’t]J>
cosAwt,
(I)
0
where Aw = wJ - 01 is the difference between the rotational and nuclear spin Armor frequencies of the molecule, I+ is the spin-rotation coupiing constant and C’ is the self-part of the Waidmann-Snider collision operator [ 11. For H2, wr = 13.4 X LO7 cad/s [7] , wf = 2.12 X lo7 rad/s [8] and w,, = 0.71567 X LO6 rad/s [8] . The usual statement of the SRTA [I -41 is that (i) (J 0, exp{-e’t) I) can be repIaced by exp(--n(u)o,,t). hence yielding for Tr the well-known formula (1 /T,gRTA= 2 2 3%r(J2)o
tdU>U,, (Awj’ f (n<~>o,,>~
(2)
and (ii) by comparing the expansion of the exponential operator in the kernel of eq. (1) with that of exp(-n(u)u,,t), the parameter osr can be identified witl an appropriate coliision bracket [i-4] as
l
usr = (J .C?J)/(J2>,,(~1,
(?I
where (u) = (8kgT/np)1’2 is 2 mean thermal relative speed with ~1the reduced mass of the colhding pair and n is the number density of the mixture. For a system consisting of an active molecule inti289
CHEMICALPHYSICS LETTERS
Volume 44, number 2
nitely dilute in a bath ofstructureless perturbers , such as H2-He, the collision bracket can be related 191 to the relaxation matrix of the theory of Fano [lOI and Ben-Reuven [I I]. The integration over t in eq. (1) can be carried out, yielding, in the notation of Shafer and Gordon, (4)
(1 D-1 )sr =$&m[d,ro(Awl-
irz(v)~(~))-~
-P
ldsr]
,
where 1 is the unit matrix d,, is a vector whose components for thej-rotational level equals [jo + l)] “‘, P is a diagonal population matrix whose diagonal element for the j-level is + 1) exp{-Bj(j
Q-‘(2j
+ l)/kgT}
1
Comparing with the discussion in arriving at eq. (2), it is clear that the first aspect of the SRTA consists in replacing the relazzation matrix by a single cross section usr such that (l/T1 ),, is given by eq. (2). It is noted in passing that V”,,
= d,;
P dsr .
The second aspect of the SRTA, the identification a,, by eq. (3), becomes usr = d,,*o(l) P l dsr/dsrP .dsr. l
Ha- He
(6)
Now, accurate experimental
data can be obtained easily onlv in the high density region where (nC~>o,,)~ 9 (Au)~. In this region, eq. (2) reduces to (nW7s,)-*,
(7)
while eq. (4) yields (I&),,
= +.$d,, +hh.r(i)]--l* P l dsr ,
(8)
Comparing eqs. (7) and (8), osr [called air below to distinguish it from that given by eq. (6)3 can alternatively be identified by ozr = CJ2>o/dsr~ [&)]-'* P d,, . l
osr = 0.45 A2,
(9)
ozr = 0.28 A2.
(10)
These two values of usr are then substituted into eq. (2) and the resulting Tl versus n curves are plotted in fig. 1 along with the exact result of eq. (4). ‘Ihe case of dipolar reIaxation can be treated similarly. The “exact” expression for TI in this case is
1
t E
I t-
(IIT1 >d = &&
Q2
as
1
2
, 5
n tamagats) Fig. 1. The contribution to T1 due to the spin-rotation re-
laxation mechanism evaIuated using the close-coupled resultsof Shafer and Gordon [6 ] : solid curve, full matrix inversion or “exact” result; dashed curve, SRTA result using eq. (9) to define asr; dotted curve SRTA result using eq. (6) to define usr.
290
l
of
In general, ozr in eq. (9) will not be equal to usr in eq. (6) except at low enough temperature such that only the Iowest rotational level is significantly populated. Using the results of Shafer and Gordon 163 for Hz-He at 298 K, it is found that
10
9
(5)
l
(liTl)sr SRTA = ~&J2)(J
(B and Q are the rotational constant and rotational partition function, respectively, of the active molecule), and t#) is the relaxation matrix cross section oftensorial rank 1 [lo]. The numerical results for 0(l) given by Shafer and Gordon [6] are employed to calculate T1 as a function of n at 298 K and the res’ult is plotted in fig. I. For ortho-Hz at 298 K, only the j = 1 and 3 levels are signii?cantly populated [6] and a 2 X 2 matrix suffices in the calculation.
December 1976
t d= (Awl -in
Im [4d* (2Awl - in(u) 6(2))-1 lP*d Wa(2))-’
*P-d] ,
(11)
where ad = 5.4336 X IO6 rad/s [8] is the dipolar coupling constant for Hz, d is a vector whose corn onent for thej-level is -[i(j + 1)/(2j + 3)(2j - l)] ‘/‘and m(2) is the relaxation matrix cross section of iensorial rank 2. As before, the first aspect of the SRTA assumes that eq. (11) can be replaced by
Volume 44, number 2
(Wl)d sRTA=$,&J2/(4J2
- 3)>,
nG_Jhd
4n(vhQ +
X
am
(Au)~ + (n(~)o~)~ As for the second Cation
of od,
aspect
the USUd
1 December 1976
CHEMICAL PHYSICS LETTERS
+ (~(v>o,)~
1-(12)
of the SRTA, i.e. the identifi-
expression
[ 1,2]
that corresponds
to eq. (3) or eq. (6) is od=d-a
‘2)-P-d/d-P*d.
(13)
Here it is noted that
- 3)),
=d - P
l
d .
On the other hand, if expressions
(14) (11)
and (12)
are
compared in the high density regime, an alternative identification of od which corresponds to eq. (9) for the spin-rotation case is l_r(;=d - P ld/d - [a(2)] -l - P -d _
(15)
When the results of Shafer and Gordon [6] are used in eqs. (13) and (15), it is found that for H2-He at 298 K, o,j = 0.42 w2,
0;; = 0.41 82.
(16)
The resulting Tl versus iz curves calculated from eqs. (11) and (12) with cr: and od given in (16) are plotted in fig. 2. The total Tl due to both spin-rotation and dipolar interaction can be found from l/T1 = (l/Tl)sr+(l/Tl)d
-
(17)
Curves of Tl versus n for the exact result as well as the SRTA results are also plotted in fig. 2. From fig. 1, it is observed that for the case of spinrotation rekation, the exact curve differs from either of the SRTA curves. The SRTA curve using a& as parameter fits the high density region well (as expected from the definition of uzr), but its minimum occurs at too high a density. The SRTA curve with usr as parameter has its minimum at about the right density, but would not fit the exact curve except at very low densities. Both SRTA curves give too low a value of T1 at the minimum. For the case of the dipolar interaction, on the other hand, both SRTA curves fit the exact curve very well, as all three curves are indistinguishable in fig. 2. In fig. 2, when both the spin-rotation and dipolar interactions are taken into account, the agreement between the exact curve and the SRTA curves is fairly good. The SRTA curve with uzr and uz as param-
n camagatst Fig. 2. The contirbution to Tr due to the dipok rehrxation mechanism and the total Tt as given by eq. (17) evaluated using the results of ref. [6] : dashed curve, dipok contnbution (the “exact” and both SRTA curves are indistinguishable on the scale of the figure); solid curve, full matrix inversion as given by eqs. (41, (11) and (17); dotted curve, SRTA result obtained from eqs. (2), (12) and (17) together with eqs. (9) and (15); dotdash curve, SRTA result obtained from eqs. (2), (12) and (17) together with eqs. (6) and (13).
eters is preferred since, (a) it fits the exact curve in the region of high density where accurate experimental measurement can be performed and, (b) its minimum occurs at about the same density as that for the exact curve. The value of Tl minimum predicted by this SRTA curve is about 11% lower than the exact valueThus, if it is insisted that a one-parameter fit to the Tl versus n curve according to eqs. (2), (12) and (17) be made, then the defmitions of eqs. (9) and (15) for the effective collision cross sections are to be preferred. Finally, it is noted that Neilson and Gordon [12] had pointed out that the matrix inversion in eq. (4) can be carried out explicitly and expressed in terms of the eigenvalues and eigenvectors of the reIa..&ion matrix. Thus if the matrk U diagonalizes u ,i.e.,if
(18) where the hi are the eigenvahxes of&l, d,, - [Aol
- in(v) &I]-1
- P
l
then
d sr
(1% 291
Volume 44,
number 2
CHEMICAL PHYSICS LETTERS
less dependent on the specific rotational levels occupied by the molecule, and it can be expected that the Xjs are not very different, rendering the SRTA more valid
with the Wi given as
and pk being the kk element of the diagonal popuiation matrix P. Eqs. (6) and (7) can then be written as
(22) where the obvious result Zi Wi = dsr* P d,, = (.12), has been used. It may now be asked when the SRTA is a good approximation. A trivial case will be that when only one Xi predominates. This is so for one-level system mentioned earlier. Another case of interest would be if all the eigenvalues of c(l) are approximately equal, i.e., Xi = X for all i. Eq. (19) then becomes l
When the results of (20), (21) and (23) are substituted into eq. (4), the SRTA result of eq. (2) is recovered, with X in place of usr. Also, from eqs. (23) and (24), usr and ozr can be shown to be equal. The same consideration can be applied to the exact expression for l/T, of eq. (11). Thus it may be concluded that if the eigenvalucs of the relaxation matrix are about equal, the SRTA will be a good approximation, in which case the u and u* expressions given by eqs. (6) and (9) [or by eqs. (13) and (15)] are equivalent. The expressions for u [eqs. (6) and (13)] are more useful since they are the kinetic cross sections for which analytic expressions within the distorted wave Born approximntion(DWBA) are available 191. For molecules having larger moments of inertia such as HCi. collision transition & mobabilities are
292
1 December 1976
for such systems. Preliminary analysis on the system HCI-HCi appears to indicate that the SRTA is sufficient to fit the experimental data 1131. From the results of Shafer and Gordon [6], the eigenvalues of the 2 X 2 matriv #) are found to be 0.71 A2 and 0.33 A2, while those of m(2) are 0.43 A2 and 0.33 A2. Hence it is not surprising that the SRTA works well for dipolar relaxation but not for spin-rotation relaxation in molecular hydrogen infinitely dilute in helium. References [ 1] F.R. McCourt, in: NMR basis principles and progress, Vol. 13, ed. M.M. Pintar (Springer, Berlin, 1976) p. 55. [2] F.R. McCourt, T-E. Raidy, R. Festa and A.C. Levi, Can. J. Phys. 53 (1975) 2449 [3 ] F.R. McCourt, T.E. Raidy, T. Rudensky and A.C. Levi, Can. J. Phys. 53 (1975) 2463. [4 ] T.E. Raidy and F-R. McCourt, Chem. Phys. Letters 38
(1976) 300. [S] G. Tenti and F.R. McCourt, J. Chem. Phys. 65 (1976) 623. [6] R. Shafer and R.G. Gordon, J. Chem. Phys. 58 (1973) 5422. [7] A. Carrington and A.D. McLaughlin, Introduction to
magnetic resonance (Harper and Row, London, 1967). [8J N-J. Harrick, R.G. Barnes, P.G. Bray and N.F. Ramsey, Phys. Rev. 90 (1953) 260. [9] WK. Liu and F-R. McCourt, Chem. Phys., to be submitted for publication.
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