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On the Senftleben-Beenakker effect on the thermal conductivity
Physica
102A (1980) 281-297 @ North-Holland
Publishing Co.
ON THE SENFTLEBEN-BEENAKKER EFFECT ON THE THERMAL CONDUCTIVITY B.J. THIJSSE,
G.W. ‘T HOOFT, H.F.P. KNAAP
Huygens Laborutorium
der Rijksuniversiteit,
and J.J.M. BEENAKKER
Leiden,
The Netherlands
Received 10 March 1980
Experimental results on the influence of a magnetic field on linear, spherical top and symmetric top molecules have been analyzed in terms of the kinetic theory based on the Waldmann-Snider equation. The traditional analysis on the basis of the two angular momentum polarizations W .B and W A .I is compared with an alternative analysis on the basis of the three irreducible tensors of WZ.
1. Introduction Recently a large number of data on the Senftleben-Beenakker effect for symmetric top molecules have become available’2). The complete body of data has by now grown to such an extent that a large-scale analysis of some general questions is feasible. In this paper we will study one of such problems, viz. one connected with the detailed interpretation of the behavior of the thermal conductivity in a magnetic field. The experimental results for the influence of a magnetic field on dilute gas transport coefficients are in excellent agreement with the predictions of the kinetic theory based on a Chapman-Enskog type approach that starts with the Waldmann-Snider equation (i.e., the quantum version of the Boltzmann equation for rotating molecules3*4)). The observed effects are explained in terms of contributions from one or more nonequilibrium polarizations, which describe the departure of the molecular distribution function f( W, J, W*, J*, _I/) from the Maxwellian fco)(W*,J*,J~*) in the presence of an external gradient (W = velocity, J = angular momentum, JY = projection of J on the figure axis). The general form of such a polarization, written as [W]@)[J](q) ( w*)qJ*p)(Jy) (‘), is an irreducible tensor of rank p in W and rank 4 in J(q 2 l), multiplied by polynomials of order r in W* and order s in J*, and a term in J/(t = 0 or 1) depending on the parity. Here we will only study tensorial aspects of the J dependent polarizations. The tensorial structure plays a special role. If one investigates transport coefficients with the magnetic field at different angles with the external gradients, the field dependence 281
282
B.J. THIJSSE
ET AL.
of the ratio of two such angular situations in the limit of infinite and zero fields is solely determined by the tensorial parameters p and 9. Much of the experimental work is focused at these aspects. Scalar effects, i.e., the W2 and 5’ dependence, only affect the intermediate field range. Their detailed properties cannot be disentangled with equal sensitivity, because the number of curve-fitting parameters would become unrealistically large. For the viscosity, the experimental situation is quite simple: the observed effects can be well explained on the basis of only one J dependent polarization,[J]‘*‘. Only the case of NH,and ND3 is an exception: for these molecules one has found that [ W]‘*‘J is the proper polarization’.‘). For the thermal conductivity, however, matters are more complicated. There is overwhelming evidence that W[J]‘*’ is the dominant J dependent polarization. But distinct deviations from the elementary W[J]‘*’ field dependence, i.e., the field effect stemming from the vector part W-x are found for all but the simplest diatomic molecules2*6). These deviations cannot be explained by incorporating additional scalar functions in the description. Hence, to explain the observed behavior, the addition of a second polarization is an obvious step. This is the traditional approach. The theory, however, offers yet another possibility for the explanation of the observed deviations from the simple W-2 behavior, one that does not need the introduction of a second polarization, but instead takes into account also the occurrence of the second and third rank tensor parts of W[J]‘*’ produced in a magnetic field’). Both approaches are treated in detail in this paper. Results are compared with each other and with the experimental data. It will be shown that the two methods yield almost identical results in virtually all cases. A comparison between experiment and theory alone will therefore not be powerful enough to completely discriminate between the two methods. This situation will be discussed; paths that are still open for reaching a solution will be indicated. Section 2 gives the theoretical expressions. Results are compared and discussed in section 3.
General expressions W[J](*’ polarization
2.
for the thermal
conductivity
in a magnetic
field. The
The field dependence of the Senftleben-Beenakker effect for the thermal conductivity A on the basis of the W[J]‘*’ polarization without simplifications has first been treated by Tip’). Subsequently, other authors have extensively studied the problem6”). A summary of the derivation is given in the appendix. One finds for the field induced change AA = h(B) - A(O),
SENFI’LEBEN-BEENAKKER
+jyl,*
- IKWG
EFFECT
ON THERMAL
+ K*)& + (419) Y(l0 Y - WM5:, 1 + (922 - 4 Y)& + 4 Y*&
L+‘y,*
K&25:2 1+
CONDUCTIVITY
WW2W,
+ 2K,)S:2’
’
283
(1)
(2)
where the superscripts I and 1 symbolize the angle between the magnetic field I3 and the temperature gradient: !@12and 5,* are given by9v6) I&*=!
&z,* =
6( ;;;o”)* 2E(101+ l)G,(12)
(3)
1 n(u)&0(12)
@-QIB fl
(4)
with E&(12) a weighted average of C%J,, G2 and &, G,,(12) =&(6G,(l2)+
10~2(12)+ 14&(12)),
(5)
and KL, Y and Z are defined by &
=
-5(12) GL(l2)’
Y = &4K,K2
L = 1,2,3,
(6)
+ 5K,K3 + K2K3),
(7)
1 2 = is(9Kl + 5K2 + 4K3).
(8)
Note that one always finds the combination KLf12
= --Gio(12)
&(12)
&@
h
1
n(u)&(l2)
L = 1,2,3,
(9)
in the formulae for AA* and Ah! This is independent of Go. The quantity Go has been introduced in eqs. (4) and (6) merely to obtain agreement for 5r2 with the formulation commonly used in literature. 6,(12), C%*(12)and &(12) are effective decay cross sections for the three irreducible tensor parts of W[J]‘*’ of rank 1, 2 and 3, respectively; B( ]!g,“) is the effective coupling cross section between the heat flow and the IV*%? polarization, while the decay cross section G(101 + 1) for the heat transport itself specifies the field free thermal conductivity’). The three Gi,(12)‘s and @r2determine the field behavior of AA. To reduce the number of parameters from four to two, many authors have used the so-called Spherical Approximation (SA)*e6), in which the collision operator acts separately on the directions of the velocities and those of the angular momenta (this is trivially true for a spherical potential, see also eq. (A.1 1)). This spherical approximation is the focal point of the present work.
284
B.J. THIJSSE ET AL.
Using the SA one has E,< 12) = G,,( 12) = G,( 12) = G’o(12),
(10)
and hence K,=Kz=K3=Y=Z=1,
(11)
from which it follows that Ah’ and AAn (eqs. (1) and (2)) reduce well-known expression@“)
to the
(12) (13) where !P,, is the natural limit of !@,*in the SA.
3. Extension
of the formalism
3.1. Theoretical
and discussion
of the experimental
data
calculations
As discussed in previous articles, the experimentally observed field dependence of AA* and AAMcan be described to a large extent on the basis of eqs. (12) and (13), i.e., the simple W:JJ\behavior using the SA’“*“). Note that only two adaptable parameters, ?P12 and Go(12) are involved in this case. In particular, the sign of the observed effect (AA < 0) is correctly predicted, and the theoretical value 3/2 for (AA’/AAl) B-tmis close to the experimental values for many molecules. A detailed inspection of the data, however, shows that for most gases systematic deviations from this elementary description occur. This is most clearly illustrated in table I where the experimental saturation values for AA’/AAl! often measured with less than 1% uncertainty, are given for a large number of molecules. A more sophisticated description is obviously needed. The first possibility is to drop the spherical approximation. No a priori theoretical arguments exist for the validity of the SA for molecules with a certain degree of nonsphericity. In fact, early model calculations have shown that deviations from the SA do already occur for such mildly nonspherical molecules as N2 and CH,“). The theoretical expressions for this case, eqs. (1) and (2), have four adaptable parameters: @r2, 6(12), e2(12) and g,(12). Another approach, usually followed in the literature, is to retain the SA but to invoke a second J dependent polarization. In view of the results of table I, where nearly all values are greater than 3/2, a term of odd rank in J is called
SENFTLEBEN-BEENAKKER
EFFECT ON THERMAL TABLE
The value of (Ah*/AA~)B, HCI DC1 HCN ocs CO2 NZ co 02 HD nH2 pH2 m 04
1.53 1.64 4.2 1.41 1.44 1.57 1.52 1.51 1.51 1.50 1.50 I .65 1.60
ref. ref. ref. ref. ref. ref. ref. ref. ref. ref. ref. ref. ref.
2 2 2 2 2 10 10 10 6 10 10 10 10
CONDUCTIVITY
285
I
for a number of gases (T = 300 K)
SF, CJA CD., CF, NH, ND3 NS CHFj CDS CH,F CDjF CH,CN
1.46 1.64 1.58 1.53 2.57 2.11 1.56 1.55 1.57 1.86 1.62 4.6
ref. ref. ref. ref. ref. ref. ref. ref. ref. ref. ref. ref.
10 10 10 10 2 2 12 2 2 2 2 2
for. Of the available terms the W A J polarization is the most likely candidate, as it has a very simple tensorial structure *, The expressions for this case, (14) +
= - P,,(27
8, 1+ 512
(15)
also have four adaptable parameters, viz. P12, &2, Yyll and &,; the parameters ?P,, and &, are defined analogous to VI2 and e12. An even more general description, in which both polarizations are considered without the SA (8 parameters), would be unworkable in the curve fitting procedure. An extension of the formalism by incorporating additional scalar functions in W2 and J2 has similar objections. However, as was mentioned earlier, contributions from these effects can be circumvented by concentrating on the high and low field limits of AA’/Ahq which are not affected by scalar properties of the polarizations. The best one can hope for is that one could discriminate between the two 4-parameter descriptions, one giving a much better agreement with the experiment than the other. One then would have obtained important information about the collision physics, being able to decide in favor of either the incorrectness of the property expressed in eq. (10) or the presence of an odd-in-J term, which is, in turn, related to (classically) the existence of collisions without an inverse. We will first give expressions for the two important limit quantities for the simple W *z (SA) description and in terms of the parameters for the W[J]‘2’ *Recently the use of WA J for the description of symmetric questioned by Bruev on the basis of first order perturbation theory”).
top molecules
has been
286
B.J. THIJSSE
ET AL.
(no SA) and W*55’+ W A J (SA) formalisms. Low field ratio
One derives:
for W *?&A),
(16)
for W[J](*)(no SA),
(17)
for W:JJ\+ W A J(SA).
(18)
for W *?$SA),
(19)
for W[Jlc2’(no SA),
(20)
for W:JJ’+ W A J(SA).
(21)
High field ratio
3 ;K, + ;K3 $K2 +;K3 > >( =2 Y
31-;$iL 12
=2
*-$ 12
After writing out the expressions for KL and Y (eqs. (3-o-()), one observes that the two more sophisticated descriptions each have two parameters that determine the difference with the simple description: G2(12)&(12) and &(12)/6,(12) in the case of W[J]‘2’ (no SA), and 1y11/q’12and G0(11)/G,(12) in the case of W-s+ W A J(SA). Before turning to the experiment, we will first analyze for what values of their parameters the two descriptions W[J]“’ (no SA) and W +‘JJ‘+ W A J(SA) coincide; then, of course, all discriminative power is lost. For that purpose we set out by comparing the predictions for the high and low field ratios, eqs. (17), (18) and (20), (21). For the low field ratio it follows from eq. (18) that, in the two-polarization description, (Ah’/Ahf)B4 2 9/2 for reasonable values of the parameters (negative values have never been observed in magnetic experiments and will not be considered). This corresponds to E2(12)/G1(12) 3 1 (eq. (17)), or to the region in fig. 1 that lies to the right of the vertical dashed line. Similarly, eq. (21) yields for the high field ratio: (AA*/AA~),,, 3 3/2 for q,,/ly,, c 1 (again, negative values have never been found). This corresponds (eq. (20)) to the area below the nearly horizontal dashed curve in fig. 1. In other words, values of 6;,(12)/&(12) and &(12)/~,(12) in the lower right
SENFTLEBEN-BEENAKKER
2
I
1
I
III
EFFECT
I
ON THERMAL
I
CONDUCTIVITY
I
I
.D
Other
I
287
III
symmetric
0.5
0.l
(3.5
Fig. 1. Description of the thermal conductivity change in a magnetic field on the basis of the W[J](*) polarization without the spherical approximation (no SA), eqs. (1) and (2). (%12)/%12) and G,(l2)/6,(12) are plotted on a double logarithmic scale. Both ratios are 1 when the spherical approximation is assumed, W:JJ’SA). In the hatched area the alternative description on the basis of W-wand WA J, using the SA, is equally good, eqs. (14) and (15). The correspondence between the two descriptions is only schematically indicated: the dotted line is the locus of SA functions with G,,(ll) = C&(12) for P,,/rY,, = 0 (top) to 0.7 (bottom); the dash/dot line represents functions with P,1/1y12= 0.25 for GO(l 1)&(12) = 2.5 (left) to 0.4 (right). Experimental data for a large number of gases are also given. Different symbols correspond to molecules of different structure; open symbols denote mixtures with Ar. See table II for identification of the molecules and for numerical results.
hand part of fig. 1 give W[J]“’ (no SA) field curves for AA*, Ah” that coincide with the W*‘JJ’+ W A J(SA) curves in the high and low field limits of AALlAh! Values outside the dashed boundaries yield W[J](2) (no SA) curves that cannot be constructed in the two-polarization model. These statements are true irrespective of scalar extensions to the polarization tensors. A drawback to the above method for determining the region of coincidence of the two descriptions is that equal weight is given to the high and low field
Molecule and number in fig. 1 see fig. 1
G(12) E,(12)
E,(12) 6,(12)
no SA
SA
no SA, SA (*)
“The experimental AA data for OCS, CO* and SF6 have been analyzed in terms of W %?+.I_$, (SA), instead of in terms of W:JJ‘+ W A J(SA). Accordingly, numbers refer to PO,/tu,, and G(Ol)/G(12). b Cross section values for the mixtures are, in fact, values for the total W :JJ\decay, taking into account collisions between both like and unlike particles “*‘s). They are calculated from the experimental AA curves in exactly the same way as for the case of the pure polyatomic component.
TABLE II Curve fitting results for the thermal conductivity change in a magnetic field for various molecules at T: 300 K; no SA = description on the basis of W[J]‘*’ without the spherical approximation, eqs. (1) and (2). The table gives values for G,,(12)/GI(12) and Gs(12)/G,(12); see also fig. 1. SA = (traditional) description on the basis of W*%+ W A J with the spherical approximation, eqs. (14) and (15); values are listed for YII/Y12 and GO(l 1)/%,,(12). Data are also included for the “overall” shift between the SA and no SA curves, ~sA/ty~~ and G,(12) noSA/Ga(12)SA.D = scaled standard deviation for the no SA fit, eq. (22). An asterisk denotes that the D value for the SA fit has also been determined. In all cases the two D values correspond within 10e3.
SF, CH., CF,
NH3 NHa(0.75)-Ar(0.25) NH,(0.50kAr(0.50) NH,(0.26)-Ar(0.74) ND3
CHF, CDF> CHaF CD,F CH,CN CH,CN(0.67kAr(0.33) CH,CN(0.48)-Ar(0.52) CHJCN(0.25)-Ar(0.75)
9 10 11
12 12a 12b 12c 13
14 15 16 17 18 18a 18b 18c
0
HD
N2
CD,
HCl DC1 HCN ocs
0.87 0.95 1.09 1.02 4.40 1.28 1.08 0.93
1.13 1.77 1.90 2.06 1.05
0.83 0.97 0.87
0.79 0.74 2.07 0.82 0.87 0.86 1.07 1.04
0.87 0.87 0.64 0.79 0.16 0.41 0.68 0.77
0.35 0.33 0.48 0.40 0.47
1.03 0.79 0.95
0.89 0.74 0.15 0.98 1.07 0.88 0.99 0.80
0.05 0.W 0.28 0.11 0.75 0.41 0.21 0.12
0.52 0.57 0.42 0.51 0.38
[0.08]8’ 0.12 0.03
0.04 0.12 0.73 [OMI [O.lll 0.07 0.01 0.13
1.67 1.66 0.83 0.99 0.90 1.25
1.28 1.16 0.90 0.97 1.37
1.59
1.00 r0.3p [0.3la)
1.01 0.99 0.88 0.92 0.69 0.85 0.94 0.98
0.85 0.80 0.83 0.80 0.90
0.98 0.97 0.97
1.02 1.10 0.74 1.00 0.97 1.04 1.00 0.97
0.98 0.98 1.15 1.19 0.82 0.92”’ 0 %b’ 0:97b’
0.90 0.9Ob’ 0.94b’ 0 94b’ 0:94
1.09 0.96 1.02
O.% 0.95 0.88 1.11 1.06 1.02 0.99 1.00
2.0* 1.3* 2.0* 1.1* 2.7* 2.6* 2.7* 1.7*
2.3* 1.9 3.8* 5.8* 2.0
3.9 1.2* 2.0
3.1 3.7* 2.5 1.2 0.3 0.2 1.9 3.6
B.J. THIJSSE ET AL.
290
behavior. Experimental results at low fields, however, are normally considerably less accurate than at high fields since Ah’ and Ah’ become extremely small. To bring the comparison of the theoretical curves into accordance with the experimental situation, we will slightly refine the above method by comparing the calculated curves over a whole range of field values (unweighted), instead of only at the extremes. For that purpose the W[J]‘*’ (no SA) functions of eqs. (1) and (2) are fitted to W -3‘ W A J(SA) curves which were generated with various sets [Wll/W12; GO(l 1)/C&(12)]. In the computations the range of the field parameter is 0.1 < &G 10, being the experimental region of interest. The curve fitting procedure yields the corresponding sets @&(12)/G,(12); (%i;3(12)/Gj1(12)]for the best overlapping W[J]‘*‘(no SA) functions. In addition, values are obtained for the quantities @i;;;SA/P~? and 6,(12)“SA /C%,(12)SA,which represent the necessary adjustment in the “underlying” basis functions W IJJ\(SA) for getting an optimum fit. The quality of the fit, i.e., the distinguishability of the two sets of curves, is the crucial parameter for this discussion. As a measure of discrimination, D, we use the standard deviation of the fit, scaled by the saturation value of Ah’/,+:
(22) PoSA and FSA are the field functions of eqs. (1,2) and (14,15), respectively. The probing points 512,iare equally spaced on a logarithmic scale between 0.1 and 10, and the summation is taken along both the Ah’ and the AA’ curves: N =42. The results of the calculations are shown in fig. 1. In the finger-shaped region the discrimination D between the two descriptions is less than 0.01: the AA curves are practically identical. It is found that the region of coincidence is now much smaller than the earlier found “south-eastern” quadrant. On the other hand, a new region, previously inaccessible, has appeared. We will next see how this relates to the experimental situation. 3.2. Experimental
data
The data points in fig. 1 represent the curve fitting results of eqs. (1) and (2) to the experimental AA field dependence. Table II contains the numerical results and provides the identification of the data points. One first observes the concentration of data points around the origin. This again illustrates that the simple description on the basis of W-3 using the SA, is good enough for many molecules. Another important result is that the majority of the data falls inside the finger-shaped area. The D values for the adaption of the theoretical curves to the experimental data is around 0.02 (see
SENFTLEBEN-BEENAKKER
EFFECT
ON THERMAL
CONDUCTIVITY
291
table II). Hence on the basis of our D = 0.01 discrimination level both theoretical descriptions, W[J]’ (no SA) and W-w+ W A J(SA), are equally good for these molecules. One cannot decide in favor of either of the two. Only five gases, HCl, DCl, OCS, SF6 and CO2 fall outside the finger-shaped region. All five show only slight deviations from the simple W(JJ’(SA) behavior. For the first two gases the experimental D-value is too high ( = 0.035) for any rigorous conclusions to be possible. The other three molecules are special cases: (Ah’/Ah~)B,, is less than 3/2. Obviously, in these cases WA J is not the correct choice for the second polarizations; accordingly, these molecules are less relevant for the present discussion. As we have seen that no direct choice between the two descriptions can be made on the basis of the quality of the curve-fits, one might try to analyze the results quantitatively. Perhaps one can distinguish between physical and mere numerical effects in the generated values for the adaptation parameters. As for the We%‘+ WA J(SA) data, the values found for !@,,/!?‘12 and 5,(11)/6,(12) certainly do not look suspicious. In particular -as was shown elsewhere2)- the relative size of the W A J polarization bears a certain correlation with the size of the rotational level splitting. The cross section ratios, always around 1, are also quite realistic. The same, in fact, is true for the W[Jlc2’ (no SA) set of data. Most of the cross sections 6,, @i2and 6, are found to be within a factor 2 in size (with ammonia and the cyanides as exceptions), indicating that the three tensor parts of W[Jlc2’ do not behave widely different. Whether the observed trend 1.4 G2 > 6, > G3 has a special meaning, is not certain. The general behavior of the collision integrals in this respect deserves further theoretical study. On the whole, no signs of unexpected results are found in either of the two sets of data. Next, let us look upon this matter from a different angle. It is recalled that the spherical approximation implies a certain degree of independence between the decay of velocities and the decay of angular momenta. Hence a study of the effective cross section Q(12) (subscript and bar are left out on purpose), in particular in relation to 6(02), the cross section for purely angular momentum reorientation, seems very appropriate. Each GL(12) -and hence also E&(12& has a trivial contribution equal to @~(02)‘*~),which will be subtracted in the analysis. In fig. 2, 60(12) - @(02) is plotted versus G(O2) for various molecules. Data are taken from W:JJ\+ W A J curve-fitting results2). In spite of the experimental scatter a clear correlated behavior is discovered. Actually, a linear relation G)o(12)- fG(02) = Cl + c&(02), with cl = 15.8 A2 and c 2 = 0.55, is certainly
(23) a not unrealistic
description.
This
B.J. THIJSSE ET AL.
292 90
A2
60
?D4
l
N2,z ,02
co
bH, 0 0
I
6lO2, _ 3o
I
I
60
90
1213
I\2
Fig. 2. @*(12) - @(02) versus G(O2) for various molecules: n = linear, 0 = spherical top, A = symmetric top. The quantity on the vertical axis is the nontrivial part of the decay cross section for W *%f? the horizontal axis displays the reorientation cross section. The solid line illustrates the linear dependence (CH& and NH3 are possible exceptions). Data are taken from W!JJ‘+ W A J(SA) curve-fitting results*).
60 co2
CfiF,
C;
N;i,
t-4 2
30 -
no2
~~osn(12)
-;
/
GlO2 7% I
OL O
I
G[O21
3o
I
60
I
90
A
120
Fig 3. GFs”(12) -:G(OZ) versus G(O2) for various molecules. The figure is similar to fig. 2, but G&&2) values are obtained from the W[5]‘*) curve-fitting results and eq. (5).
SENFTLEBEN-BEENAKKEREFFECTONTHERMALCONDUCTIVITY
293
indicates that GO(12) - @(02) can be interpreted as a sum of two independent terms: cl, an elastic contribution involving only W; and c+G(O2), a reorientational contribution involving only J. This summational independence is even stronger than is required by the spherical approximation. Hence it is tempting to conclude that fig. 2 supports the validity of the spherical approximation. One can only say, however, that the results are consistent with the assumption of uncoupled W and J behavior, made in the (SA) data analysis. Let us do a complementary test as well. Fig. 3 shows the same quantities as fig. 2, but this time the E&(12) data are obtained from the W[J]‘*’ (no SA) analysis (table II) and eq. (5). It is found that the linear and spherical top molecules still fall on a straight line, a result not necessarily to be expected, since the SA has now been dropped from the data analysis. At the other hand, one must realize that these molecules are not far from simple !V-%?(SA) behavior, and as long as the collision integrals are only weakly influenced by the physical background of the spherical approximation, one would not expect strongly different trends in the results of the two methods of analysis. The symmetric top molecules, then, do indeed show sizeable deviations from the straight line. But again, this is not inconsistent with any pre-made assumptions. For a somewhat sharper analysis to be possible, more Q(02) data for symmetric top molecules would be needed. We’d like to add that the general behavior of GrsA (12) versus G(O2) also shows up in the individual cross sections G,, g2 and & In conclusion, there is no ground for a discrimination between the two descriptions on the basis of this analysis. There are two additional factors to the present issue. In electric field measurements of AA, positive effects have been found”j*“). This would definitely prove the existence of a second, odd-in-J polarization (eqs. (1) and (2) never yield positive values) for strongly polar prolate molecules, such as CH3F and CH$ZN. However, considerable discrepancies exist between these measurements and the corresponding magnetic field results, which makes further conclusions rather doubtful 2*‘5V’8). These discrepancies, complicated by differences between the electric and magnetic Liouville operators, are not yet fully understood. Recently, also the thermal diffusion’s23) and the diffusion23Z4) have been investigated in a magnetic field. A preliminary joint analysis of the results of these experiments and those of the thermal conductivity for rather simple systems such as N2-noble gas mixtures suggests the need for including an odd-in-J contribution. The results are still qualitative. To summarize, we have found that it is impossible to make a clearcut choice between the W[J]‘*’ (no SA) and W*‘JTT‘+W A J(SA) descriptions of the Senftleben-Beenakker effect on the thermal conductivity on the basis of the available data from magnetic experiments. There are indications from external sources for the presence of a term odd in J in some cases, but no
294
B.J. THIJSSE ET AL.
conclusive evidence is found. In view of the results obtained in this work we have arrived at the following situation: awaiting further developments, one should approach AA results given only in terms of the W iJj\+ W A J(SA) formalism with certain caution. There is certainly room for the alternative explanation in terms of W[J]” (no SA). A mixture of the two descriptions, though in practice unworkable, seems to be even closer to the truth. A way out of this situation would be provided either by sophisticated model calculations of the relevant effective cross sections, or by high precision measurements. In the near future model calculations seem to be the more promising of the two.
Acknowledgements
This work is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)” and has been made possible by financial support from the “Nederlandse Organsiatie voor Zuiver-Wetenschappelijk Onderzoek (ZWO)“.
Appendix
Starting from the Waldmann-Snider equation one can write down an expression for the thermal conductivity in terms of matrix elements of the collision operator. Using an inverse operator technique (see, e.g., ref. 19) one obtains for A: A_2kBT
w m
(
w2 (
-;+
&~W(W2-;+~-_(~)o))0.
c-(E)0
(A.11
>
Here 5? is the Waldmann-Snider collision operator and 8 is the Liouville operator, representing the effect of external electric and magnetic fields. First discussing the situation without external fields, one realizes that, since the collision operator is isotropic, the vector W( W2 - 5/2 + E - (E)~) can only couple to polar vectors built up from the vectors in the problem, the velocity W and the angular momentum J. Some examples: W -3 and W A J. Concentrating on the term W-w, which has been shown in many experiments to be the dominant angular momentum term, one finds’)
A=gSkBT -(1+r2) 2m(uh
with
(g3.( Kll&l)2
l-1 l 36(101+ Gj(101+ 1) i
l)EJ,(12) I
-I
64.2)
SENFT’LEBEN-BEENAKKER
EFFECT ON THERMAL
295
CONDUCTIVITY
and (u). = fy*. G(lO1 + 1) is the decay cross section for the total heat transport, and G(!%,“> describes the coupling between heat transport and the W-‘jj\ polarization. 6,(12) is related to the correlation function for the W -%? term,
([@‘*];%[@‘*]~‘)0 = (u)&&,(l2).
(A.3)
Here @l* is defined as W[ J]‘*’ @I2 = {(l/ls)J*(J* -3>>“’
(A.4)
and [@‘*]A represents the M component, with A4 = 0, +l, of the totally irreducible first rank (upper index 1) tensor part of W[J]‘*‘. Note that the spherical notation is used8,25). In the presence of a magnetic field the polarization [@‘*I$ is, through the action of the Liouville operator, mixed with the second and third rank tensors [@‘*I&and [@‘*I,&that can be constructed from W and JJ. The .%-operator is diagonal in these polarizations, ([@‘*l~~[W]~‘)0
= S,,&49(u)&(12),
(A.9
(L = 1, 2, 3), while the Liouville operator is not. All three G,,(l2)‘s are positive, but not necessarily equal, although they usually are on the same order of magnitude. The field dependence of the Senftleben-Beenakker effect for the thermal conductivity in this case has first been treated by Tip’). His results are given in the eqs. (1) through (8). In the literature also another scheme for the treatment of the thermal conductivity is being used. Rather than direct coupling W and J (eq. (A.S)), one couples velocities on both sides of the collision operator and likewise the angular momenta26). Again three cross sections are obtained: E&,(12), C&(12) and G2(12). These are defined according to (@:;9@:?~,.) = (V)oc n(r1 l)“*R(/22)“*( - I)‘+a+fl’+‘(Ia;, ‘,)(2, ;, QEl,(l2), ,m (A.@ where (Y, p, (Y’, p’ are spherical tensor indices, a(. . .) is a numerical factor defined in refs. 11 and 12 and (:I:) is a 3j-symbo12~25).G)0(12) is always positive while Gi(l2) and G,(l2) can have either sign and are normally much smaller.
2%
B.J. THIJSSE ET AL.
The three relatior&):
G/(12) and the three
G;,(12) are connected
by the following
G,(12) = G0(12)+36,(12)+&B*(12),
(A.7)
C&(12)= G0(12)+;G,(12)+~(12),
(A.@
6412) = GlJ(12) - G,(12) ++*(12).
(A.9)
While the effect of the collision operator in the description of eq. (A.6) is more complicated than in eq. (AS), the expression involving the Liouville operator is in this case very simple as it is diagonal in the indices: (A. 10)
(@$$?@:$‘)o = - s,,M3,,,$
with o the Larmor frequency. In the so-called spherical approximation (see section 2) one has 6, = Gj2= G3 = G,, and G, = CZz= 0, and one obtains the expression (@$(S
+ i6p)@&~)0= 6,,3,~
(
(v)&(12)-
i?),
(A.ll)
which is now diagonal in CX,/3 for both W and R
References 1) PG. van Ditzhuyzen, 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
B.J. Thijsse, L.K. van der Meij, L.J.F. Hermans and H.F.P. Knaap, Physica EllA (1977) 53. B.J. Thijsse, W.A.P. Denissen, L.J.F. Hermans, H.F.P. Knaap and J.J.M. Beenakker, Physica 97A (1979) 467. L. Waldmann, Z. Naturforsch. 12a (1957) 660. R.F. Snider, J. Chem. Phys. 32 (1960) 1051. P.G. van Ditzhuyzen, L.J.F. Hermans and J.F.P. Knaap, Physica 88A (1977) 452. J.P.J. Heemskerk, F.G. van Kuik, H.F.P. Knaap and J.J.M. Beenakker, Physica 71 (1974) 484. A. Tip, Physica 37 (1%7) 82. W.E. Kahler and H.F.P. Knaap, Z. Naturforsch. 3la (1976) 1485. B.J. Thijsse, G.W. ‘t Hooft, D.A. Coombe, H.F.P. Knaap and J.J.M. Beenakker, Physica 98A (1979) 307. L.J.F. Hermans, J.M. Koks, A.F. Hengeveld and H.F.P. Knaap, Physica SO (1970) 410. E.R. Cooper and D.K. Hoffman, J. Chem. Phys. 53 (1970) 1100. J.J. de Groot, L.J.F. Hermans, C.J.N. van den Meijdenberg and J.J.M. Beenakker, Phys. Lett. 31A (1970) 304. A.S. Bruev, Sov. Phys. JETP 47 (1978) 1054.
SENFTLEBEN-BEENAKKER
EFFECT ON THERMAL CONDUCTIVITY
297
14) B.J. Thijsse, W.A.P. Denissen, H.F.P. Knaap and J.J.M. Beenakker, Physica lO2A (1980) 305. 15) B.J. Thijsse, W.A.P. Denissen, J.J.M. Beenakker and H.F.P. Knaap, Physica lO2A (1980) 298. 16) F. Tommasini, A.C. Levi, G. Stoles, J.J. de Groot, J.W. van den Broeke, C.J.N. van den Meijdenberg and J.J.M. Beenakker, Physica 49 (1970) 299. 17) J.J. de Groot, J.W. van den Broeke, H.J. Martinius and C.J.N. van den Meijdenberg, Physica 49 (1970) 342. 18) V.D. Borman, B.I. Nikolaev and V.I. Troyan, Inzh. Fiz. Zh. (J. Eng. Phys.) 27 (1974) 640. 19) G.E.J. Eggermont, H. Vestner and H.F.P. Knaap, Physica 82A (1976) 23. 20) E. Mazur, G.W. ‘t Hooft and L.J.F. Hermans, Phys. Lett. 64A (1977) 35. 21) G.W. ‘t Hooft, E. Mazur, J.M. Bienfait, L.J.F. Hermans, H.F.P. Knaap and J.J.M. Beenakker, Physica 98A (1979) 41. 22) E. Mazur, G.W. ‘t Hooft, L.J.F. Hermans and H.F.P. Knaap, Physica 98A (1979) 87. 23) H.F.P. Knaap, G.W. ‘t Hooft, E. Mazur and L.J.F. Hermans, Eleventh International Symp. on Rarefied Gas Dynamics, Cannes (1978). 24) E. Mazur et al., to be published. 25) A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, l%O, 2nd. ed. 1974). 26) F.M. Chen, H. Moraal and R.F. Snider, J. Chem. Phys. 57 (1972) 542.
102A (1980) 281-297 @ North-Holland
Publishing Co.
ON THE SENFTLEBEN-BEENAKKER EFFECT ON THE THERMAL CONDUCTIVITY B.J. THIJSSE,
G.W. ‘T HOOFT, H.F.P. KNAAP
Huygens Laborutorium
der Rijksuniversiteit,
and J.J.M. BEENAKKER
Leiden,
The Netherlands
Received 10 March 1980
Experimental results on the influence of a magnetic field on linear, spherical top and symmetric top molecules have been analyzed in terms of the kinetic theory based on the Waldmann-Snider equation. The traditional analysis on the basis of the two angular momentum polarizations W .B and W A .I is compared with an alternative analysis on the basis of the three irreducible tensors of WZ.
1. Introduction Recently a large number of data on the Senftleben-Beenakker effect for symmetric top molecules have become available’2). The complete body of data has by now grown to such an extent that a large-scale analysis of some general questions is feasible. In this paper we will study one of such problems, viz. one connected with the detailed interpretation of the behavior of the thermal conductivity in a magnetic field. The experimental results for the influence of a magnetic field on dilute gas transport coefficients are in excellent agreement with the predictions of the kinetic theory based on a Chapman-Enskog type approach that starts with the Waldmann-Snider equation (i.e., the quantum version of the Boltzmann equation for rotating molecules3*4)). The observed effects are explained in terms of contributions from one or more nonequilibrium polarizations, which describe the departure of the molecular distribution function f( W, J, W*, J*, _I/) from the Maxwellian fco)(W*,J*,J~*) in the presence of an external gradient (W = velocity, J = angular momentum, JY = projection of J on the figure axis). The general form of such a polarization, written as [W]@)[J](q) ( w*)qJ*p)(Jy) (‘), is an irreducible tensor of rank p in W and rank 4 in J(q 2 l), multiplied by polynomials of order r in W* and order s in J*, and a term in J/(t = 0 or 1) depending on the parity. Here we will only study tensorial aspects of the J dependent polarizations. The tensorial structure plays a special role. If one investigates transport coefficients with the magnetic field at different angles with the external gradients, the field dependence 281
282
B.J. THIJSSE
ET AL.
of the ratio of two such angular situations in the limit of infinite and zero fields is solely determined by the tensorial parameters p and 9. Much of the experimental work is focused at these aspects. Scalar effects, i.e., the W2 and 5’ dependence, only affect the intermediate field range. Their detailed properties cannot be disentangled with equal sensitivity, because the number of curve-fitting parameters would become unrealistically large. For the viscosity, the experimental situation is quite simple: the observed effects can be well explained on the basis of only one J dependent polarization,[J]‘*‘. Only the case of NH,and ND3 is an exception: for these molecules one has found that [ W]‘*‘J is the proper polarization’.‘). For the thermal conductivity, however, matters are more complicated. There is overwhelming evidence that W[J]‘*’ is the dominant J dependent polarization. But distinct deviations from the elementary W[J]‘*’ field dependence, i.e., the field effect stemming from the vector part W-x are found for all but the simplest diatomic molecules2*6). These deviations cannot be explained by incorporating additional scalar functions in the description. Hence, to explain the observed behavior, the addition of a second polarization is an obvious step. This is the traditional approach. The theory, however, offers yet another possibility for the explanation of the observed deviations from the simple W-2 behavior, one that does not need the introduction of a second polarization, but instead takes into account also the occurrence of the second and third rank tensor parts of W[J]‘*’ produced in a magnetic field’). Both approaches are treated in detail in this paper. Results are compared with each other and with the experimental data. It will be shown that the two methods yield almost identical results in virtually all cases. A comparison between experiment and theory alone will therefore not be powerful enough to completely discriminate between the two methods. This situation will be discussed; paths that are still open for reaching a solution will be indicated. Section 2 gives the theoretical expressions. Results are compared and discussed in section 3.
General expressions W[J](*’ polarization
2.
for the thermal
conductivity
in a magnetic
field. The
The field dependence of the Senftleben-Beenakker effect for the thermal conductivity A on the basis of the W[J]‘*’ polarization without simplifications has first been treated by Tip’). Subsequently, other authors have extensively studied the problem6”). A summary of the derivation is given in the appendix. One finds for the field induced change AA = h(B) - A(O),
SENFI’LEBEN-BEENAKKER
+jyl,*
- IKWG
EFFECT
ON THERMAL
+ K*)& + (419) Y(l0 Y - WM5:, 1 + (922 - 4 Y)& + 4 Y*&
L+‘y,*
K&25:2 1+
CONDUCTIVITY
WW2W,
+ 2K,)S:2’
’
283
(1)
(2)
where the superscripts I and 1 symbolize the angle between the magnetic field I3 and the temperature gradient: !@12and 5,* are given by9v6) I&*=!
&z,* =
6( ;;;o”)* 2E(101+ l)G,(12)
(3)
1 n(u)&0(12)
@-QIB fl
(4)
with E&(12) a weighted average of C%J,, G2 and &, G,,(12) =&(6G,(l2)+
10~2(12)+ 14&(12)),
(5)
and KL, Y and Z are defined by &
=
-5(12) GL(l2)’
Y = &4K,K2
L = 1,2,3,
(6)
+ 5K,K3 + K2K3),
(7)
1 2 = is(9Kl + 5K2 + 4K3).
(8)
Note that one always finds the combination KLf12
= --Gio(12)
&(12)
&@
h
1
n(u)&(l2)
L = 1,2,3,
(9)
in the formulae for AA* and Ah! This is independent of Go. The quantity Go has been introduced in eqs. (4) and (6) merely to obtain agreement for 5r2 with the formulation commonly used in literature. 6,(12), C%*(12)and &(12) are effective decay cross sections for the three irreducible tensor parts of W[J]‘*’ of rank 1, 2 and 3, respectively; B( ]!g,“) is the effective coupling cross section between the heat flow and the IV*%? polarization, while the decay cross section G(101 + 1) for the heat transport itself specifies the field free thermal conductivity’). The three Gi,(12)‘s and @r2determine the field behavior of AA. To reduce the number of parameters from four to two, many authors have used the so-called Spherical Approximation (SA)*e6), in which the collision operator acts separately on the directions of the velocities and those of the angular momenta (this is trivially true for a spherical potential, see also eq. (A.1 1)). This spherical approximation is the focal point of the present work.
284
B.J. THIJSSE ET AL.
Using the SA one has E,< 12) = G,,( 12) = G,( 12) = G’o(12),
(10)
and hence K,=Kz=K3=Y=Z=1,
(11)
from which it follows that Ah’ and AAn (eqs. (1) and (2)) reduce well-known expression@“)
to the
(12) (13) where !P,, is the natural limit of !@,*in the SA.
3. Extension
of the formalism
3.1. Theoretical
and discussion
of the experimental
data
calculations
As discussed in previous articles, the experimentally observed field dependence of AA* and AAMcan be described to a large extent on the basis of eqs. (12) and (13), i.e., the simple W:JJ\behavior using the SA’“*“). Note that only two adaptable parameters, ?P12 and Go(12) are involved in this case. In particular, the sign of the observed effect (AA < 0) is correctly predicted, and the theoretical value 3/2 for (AA’/AAl) B-tmis close to the experimental values for many molecules. A detailed inspection of the data, however, shows that for most gases systematic deviations from this elementary description occur. This is most clearly illustrated in table I where the experimental saturation values for AA’/AAl! often measured with less than 1% uncertainty, are given for a large number of molecules. A more sophisticated description is obviously needed. The first possibility is to drop the spherical approximation. No a priori theoretical arguments exist for the validity of the SA for molecules with a certain degree of nonsphericity. In fact, early model calculations have shown that deviations from the SA do already occur for such mildly nonspherical molecules as N2 and CH,“). The theoretical expressions for this case, eqs. (1) and (2), have four adaptable parameters: @r2, 6(12), e2(12) and g,(12). Another approach, usually followed in the literature, is to retain the SA but to invoke a second J dependent polarization. In view of the results of table I, where nearly all values are greater than 3/2, a term of odd rank in J is called
SENFTLEBEN-BEENAKKER
EFFECT ON THERMAL TABLE
The value of (Ah*/AA~)B, HCI DC1 HCN ocs CO2 NZ co 02 HD nH2 pH2 m 04
1.53 1.64 4.2 1.41 1.44 1.57 1.52 1.51 1.51 1.50 1.50 I .65 1.60
ref. ref. ref. ref. ref. ref. ref. ref. ref. ref. ref. ref. ref.
2 2 2 2 2 10 10 10 6 10 10 10 10
CONDUCTIVITY
285
I
for a number of gases (T = 300 K)
SF, CJA CD., CF, NH, ND3 NS CHFj CDS CH,F CDjF CH,CN
1.46 1.64 1.58 1.53 2.57 2.11 1.56 1.55 1.57 1.86 1.62 4.6
ref. ref. ref. ref. ref. ref. ref. ref. ref. ref. ref. ref.
10 10 10 10 2 2 12 2 2 2 2 2
for. Of the available terms the W A J polarization is the most likely candidate, as it has a very simple tensorial structure *, The expressions for this case, (14) +
= - P,,(27
8, 1+ 512
(15)
also have four adaptable parameters, viz. P12, &2, Yyll and &,; the parameters ?P,, and &, are defined analogous to VI2 and e12. An even more general description, in which both polarizations are considered without the SA (8 parameters), would be unworkable in the curve fitting procedure. An extension of the formalism by incorporating additional scalar functions in W2 and J2 has similar objections. However, as was mentioned earlier, contributions from these effects can be circumvented by concentrating on the high and low field limits of AA’/Ahq which are not affected by scalar properties of the polarizations. The best one can hope for is that one could discriminate between the two 4-parameter descriptions, one giving a much better agreement with the experiment than the other. One then would have obtained important information about the collision physics, being able to decide in favor of either the incorrectness of the property expressed in eq. (10) or the presence of an odd-in-J term, which is, in turn, related to (classically) the existence of collisions without an inverse. We will first give expressions for the two important limit quantities for the simple W *z (SA) description and in terms of the parameters for the W[J]‘2’ *Recently the use of WA J for the description of symmetric questioned by Bruev on the basis of first order perturbation theory”).
top molecules
has been
286
B.J. THIJSSE
ET AL.
(no SA) and W*55’+ W A J (SA) formalisms. Low field ratio
One derives:
for W *?&A),
(16)
for W[J](*)(no SA),
(17)
for W:JJ\+ W A J(SA).
(18)
for W *?$SA),
(19)
for W[Jlc2’(no SA),
(20)
for W:JJ’+ W A J(SA).
(21)
High field ratio
3 ;K, + ;K3 $K2 +;K3 > >( =2 Y
31-;$iL 12
=2
*-$ 12
After writing out the expressions for KL and Y (eqs. (3-o-()), one observes that the two more sophisticated descriptions each have two parameters that determine the difference with the simple description: G2(12)&(12) and &(12)/6,(12) in the case of W[J]‘2’ (no SA), and 1y11/q’12and G0(11)/G,(12) in the case of W-s+ W A J(SA). Before turning to the experiment, we will first analyze for what values of their parameters the two descriptions W[J]“’ (no SA) and W +‘JJ‘+ W A J(SA) coincide; then, of course, all discriminative power is lost. For that purpose we set out by comparing the predictions for the high and low field ratios, eqs. (17), (18) and (20), (21). For the low field ratio it follows from eq. (18) that, in the two-polarization description, (Ah’/Ahf)B4 2 9/2 for reasonable values of the parameters (negative values have never been observed in magnetic experiments and will not be considered). This corresponds to E2(12)/G1(12) 3 1 (eq. (17)), or to the region in fig. 1 that lies to the right of the vertical dashed line. Similarly, eq. (21) yields for the high field ratio: (AA*/AA~),,, 3 3/2 for q,,/ly,, c 1 (again, negative values have never been found). This corresponds (eq. (20)) to the area below the nearly horizontal dashed curve in fig. 1. In other words, values of 6;,(12)/&(12) and &(12)/~,(12) in the lower right
SENFTLEBEN-BEENAKKER
2
I
1
I
III
EFFECT
I
ON THERMAL
I
CONDUCTIVITY
I
I
.D
Other
I
287
III
symmetric
0.5
0.l
(3.5
Fig. 1. Description of the thermal conductivity change in a magnetic field on the basis of the W[J](*) polarization without the spherical approximation (no SA), eqs. (1) and (2). (%12)/%12) and G,(l2)/6,(12) are plotted on a double logarithmic scale. Both ratios are 1 when the spherical approximation is assumed, W:JJ’SA). In the hatched area the alternative description on the basis of W-wand WA J, using the SA, is equally good, eqs. (14) and (15). The correspondence between the two descriptions is only schematically indicated: the dotted line is the locus of SA functions with G,,(ll) = C&(12) for P,,/rY,, = 0 (top) to 0.7 (bottom); the dash/dot line represents functions with P,1/1y12= 0.25 for GO(l 1)&(12) = 2.5 (left) to 0.4 (right). Experimental data for a large number of gases are also given. Different symbols correspond to molecules of different structure; open symbols denote mixtures with Ar. See table II for identification of the molecules and for numerical results.
hand part of fig. 1 give W[J]“’ (no SA) field curves for AA*, Ah” that coincide with the W*‘JJ’+ W A J(SA) curves in the high and low field limits of AALlAh! Values outside the dashed boundaries yield W[J](2) (no SA) curves that cannot be constructed in the two-polarization model. These statements are true irrespective of scalar extensions to the polarization tensors. A drawback to the above method for determining the region of coincidence of the two descriptions is that equal weight is given to the high and low field
Molecule and number in fig. 1 see fig. 1
G(12) E,(12)
E,(12) 6,(12)
no SA
SA
no SA, SA (*)
“The experimental AA data for OCS, CO* and SF6 have been analyzed in terms of W %?+.I_$, (SA), instead of in terms of W:JJ‘+ W A J(SA). Accordingly, numbers refer to PO,/tu,, and G(Ol)/G(12). b Cross section values for the mixtures are, in fact, values for the total W :JJ\decay, taking into account collisions between both like and unlike particles “*‘s). They are calculated from the experimental AA curves in exactly the same way as for the case of the pure polyatomic component.
TABLE II Curve fitting results for the thermal conductivity change in a magnetic field for various molecules at T: 300 K; no SA = description on the basis of W[J]‘*’ without the spherical approximation, eqs. (1) and (2). The table gives values for G,,(12)/GI(12) and Gs(12)/G,(12); see also fig. 1. SA = (traditional) description on the basis of W*%+ W A J with the spherical approximation, eqs. (14) and (15); values are listed for YII/Y12 and GO(l 1)/%,,(12). Data are also included for the “overall” shift between the SA and no SA curves, ~sA/ty~~ and G,(12) noSA/Ga(12)SA.D = scaled standard deviation for the no SA fit, eq. (22). An asterisk denotes that the D value for the SA fit has also been determined. In all cases the two D values correspond within 10e3.
SF, CH., CF,
NH3 NHa(0.75)-Ar(0.25) NH,(0.50kAr(0.50) NH,(0.26)-Ar(0.74) ND3
CHF, CDF> CHaF CD,F CH,CN CH,CN(0.67kAr(0.33) CH,CN(0.48)-Ar(0.52) CHJCN(0.25)-Ar(0.75)
9 10 11
12 12a 12b 12c 13
14 15 16 17 18 18a 18b 18c
0
HD
N2
CD,
HCl DC1 HCN ocs
0.87 0.95 1.09 1.02 4.40 1.28 1.08 0.93
1.13 1.77 1.90 2.06 1.05
0.83 0.97 0.87
0.79 0.74 2.07 0.82 0.87 0.86 1.07 1.04
0.87 0.87 0.64 0.79 0.16 0.41 0.68 0.77
0.35 0.33 0.48 0.40 0.47
1.03 0.79 0.95
0.89 0.74 0.15 0.98 1.07 0.88 0.99 0.80
0.05 0.W 0.28 0.11 0.75 0.41 0.21 0.12
0.52 0.57 0.42 0.51 0.38
[0.08]8’ 0.12 0.03
0.04 0.12 0.73 [OMI [O.lll 0.07 0.01 0.13
1.67 1.66 0.83 0.99 0.90 1.25
1.28 1.16 0.90 0.97 1.37
1.59
1.00 r0.3p [0.3la)
1.01 0.99 0.88 0.92 0.69 0.85 0.94 0.98
0.85 0.80 0.83 0.80 0.90
0.98 0.97 0.97
1.02 1.10 0.74 1.00 0.97 1.04 1.00 0.97
0.98 0.98 1.15 1.19 0.82 0.92”’ 0 %b’ 0:97b’
0.90 0.9Ob’ 0.94b’ 0 94b’ 0:94
1.09 0.96 1.02
O.% 0.95 0.88 1.11 1.06 1.02 0.99 1.00
2.0* 1.3* 2.0* 1.1* 2.7* 2.6* 2.7* 1.7*
2.3* 1.9 3.8* 5.8* 2.0
3.9 1.2* 2.0
3.1 3.7* 2.5 1.2 0.3 0.2 1.9 3.6
B.J. THIJSSE ET AL.
290
behavior. Experimental results at low fields, however, are normally considerably less accurate than at high fields since Ah’ and Ah’ become extremely small. To bring the comparison of the theoretical curves into accordance with the experimental situation, we will slightly refine the above method by comparing the calculated curves over a whole range of field values (unweighted), instead of only at the extremes. For that purpose the W[J]‘*’ (no SA) functions of eqs. (1) and (2) are fitted to W -3‘ W A J(SA) curves which were generated with various sets [Wll/W12; GO(l 1)/C&(12)]. In the computations the range of the field parameter is 0.1 < &G 10, being the experimental region of interest. The curve fitting procedure yields the corresponding sets @&(12)/G,(12); (%i;3(12)/Gj1(12)]for the best overlapping W[J]‘*‘(no SA) functions. In addition, values are obtained for the quantities @i;;;SA/P~? and 6,(12)“SA /C%,(12)SA,which represent the necessary adjustment in the “underlying” basis functions W IJJ\(SA) for getting an optimum fit. The quality of the fit, i.e., the distinguishability of the two sets of curves, is the crucial parameter for this discussion. As a measure of discrimination, D, we use the standard deviation of the fit, scaled by the saturation value of Ah’/,+:
(22) PoSA and FSA are the field functions of eqs. (1,2) and (14,15), respectively. The probing points 512,iare equally spaced on a logarithmic scale between 0.1 and 10, and the summation is taken along both the Ah’ and the AA’ curves: N =42. The results of the calculations are shown in fig. 1. In the finger-shaped region the discrimination D between the two descriptions is less than 0.01: the AA curves are practically identical. It is found that the region of coincidence is now much smaller than the earlier found “south-eastern” quadrant. On the other hand, a new region, previously inaccessible, has appeared. We will next see how this relates to the experimental situation. 3.2. Experimental
data
The data points in fig. 1 represent the curve fitting results of eqs. (1) and (2) to the experimental AA field dependence. Table II contains the numerical results and provides the identification of the data points. One first observes the concentration of data points around the origin. This again illustrates that the simple description on the basis of W-3 using the SA, is good enough for many molecules. Another important result is that the majority of the data falls inside the finger-shaped area. The D values for the adaption of the theoretical curves to the experimental data is around 0.02 (see
SENFTLEBEN-BEENAKKER
EFFECT
ON THERMAL
CONDUCTIVITY
291
table II). Hence on the basis of our D = 0.01 discrimination level both theoretical descriptions, W[J]’ (no SA) and W-w+ W A J(SA), are equally good for these molecules. One cannot decide in favor of either of the two. Only five gases, HCl, DCl, OCS, SF6 and CO2 fall outside the finger-shaped region. All five show only slight deviations from the simple W(JJ’(SA) behavior. For the first two gases the experimental D-value is too high ( = 0.035) for any rigorous conclusions to be possible. The other three molecules are special cases: (Ah’/Ah~)B,, is less than 3/2. Obviously, in these cases WA J is not the correct choice for the second polarizations; accordingly, these molecules are less relevant for the present discussion. As we have seen that no direct choice between the two descriptions can be made on the basis of the quality of the curve-fits, one might try to analyze the results quantitatively. Perhaps one can distinguish between physical and mere numerical effects in the generated values for the adaptation parameters. As for the We%‘+ WA J(SA) data, the values found for !@,,/!?‘12 and 5,(11)/6,(12) certainly do not look suspicious. In particular -as was shown elsewhere2)- the relative size of the W A J polarization bears a certain correlation with the size of the rotational level splitting. The cross section ratios, always around 1, are also quite realistic. The same, in fact, is true for the W[Jlc2’ (no SA) set of data. Most of the cross sections 6,, @i2and 6, are found to be within a factor 2 in size (with ammonia and the cyanides as exceptions), indicating that the three tensor parts of W[Jlc2’ do not behave widely different. Whether the observed trend 1.4 G2 > 6, > G3 has a special meaning, is not certain. The general behavior of the collision integrals in this respect deserves further theoretical study. On the whole, no signs of unexpected results are found in either of the two sets of data. Next, let us look upon this matter from a different angle. It is recalled that the spherical approximation implies a certain degree of independence between the decay of velocities and the decay of angular momenta. Hence a study of the effective cross section Q(12) (subscript and bar are left out on purpose), in particular in relation to 6(02), the cross section for purely angular momentum reorientation, seems very appropriate. Each GL(12) -and hence also E&(12& has a trivial contribution equal to @~(02)‘*~),which will be subtracted in the analysis. In fig. 2, 60(12) - @(02) is plotted versus G(O2) for various molecules. Data are taken from W:JJ\+ W A J curve-fitting results2). In spite of the experimental scatter a clear correlated behavior is discovered. Actually, a linear relation G)o(12)- fG(02) = Cl + c&(02), with cl = 15.8 A2 and c 2 = 0.55, is certainly
(23) a not unrealistic
description.
This
B.J. THIJSSE ET AL.
292 90
A2
60
?D4
l
N2,z ,02
co
bH, 0 0
I
6lO2, _ 3o
I
I
60
90
1213
I\2
Fig. 2. @*(12) - @(02) versus G(O2) for various molecules: n = linear, 0 = spherical top, A = symmetric top. The quantity on the vertical axis is the nontrivial part of the decay cross section for W *%f? the horizontal axis displays the reorientation cross section. The solid line illustrates the linear dependence (CH& and NH3 are possible exceptions). Data are taken from W!JJ‘+ W A J(SA) curve-fitting results*).
60 co2
CfiF,
C;
N;i,
t-4 2
30 -
no2
~~osn(12)
-;
/
GlO2 7% I
OL O
I
G[O21
3o
I
60
I
90
A
120
Fig 3. GFs”(12) -:G(OZ) versus G(O2) for various molecules. The figure is similar to fig. 2, but G&&2) values are obtained from the W[5]‘*) curve-fitting results and eq. (5).
SENFTLEBEN-BEENAKKEREFFECTONTHERMALCONDUCTIVITY
293
indicates that GO(12) - @(02) can be interpreted as a sum of two independent terms: cl, an elastic contribution involving only W; and c+G(O2), a reorientational contribution involving only J. This summational independence is even stronger than is required by the spherical approximation. Hence it is tempting to conclude that fig. 2 supports the validity of the spherical approximation. One can only say, however, that the results are consistent with the assumption of uncoupled W and J behavior, made in the (SA) data analysis. Let us do a complementary test as well. Fig. 3 shows the same quantities as fig. 2, but this time the E&(12) data are obtained from the W[J]‘*’ (no SA) analysis (table II) and eq. (5). It is found that the linear and spherical top molecules still fall on a straight line, a result not necessarily to be expected, since the SA has now been dropped from the data analysis. At the other hand, one must realize that these molecules are not far from simple !V-%?(SA) behavior, and as long as the collision integrals are only weakly influenced by the physical background of the spherical approximation, one would not expect strongly different trends in the results of the two methods of analysis. The symmetric top molecules, then, do indeed show sizeable deviations from the straight line. But again, this is not inconsistent with any pre-made assumptions. For a somewhat sharper analysis to be possible, more Q(02) data for symmetric top molecules would be needed. We’d like to add that the general behavior of GrsA (12) versus G(O2) also shows up in the individual cross sections G,, g2 and & In conclusion, there is no ground for a discrimination between the two descriptions on the basis of this analysis. There are two additional factors to the present issue. In electric field measurements of AA, positive effects have been found”j*“). This would definitely prove the existence of a second, odd-in-J polarization (eqs. (1) and (2) never yield positive values) for strongly polar prolate molecules, such as CH3F and CH$ZN. However, considerable discrepancies exist between these measurements and the corresponding magnetic field results, which makes further conclusions rather doubtful 2*‘5V’8). These discrepancies, complicated by differences between the electric and magnetic Liouville operators, are not yet fully understood. Recently, also the thermal diffusion’s23) and the diffusion23Z4) have been investigated in a magnetic field. A preliminary joint analysis of the results of these experiments and those of the thermal conductivity for rather simple systems such as N2-noble gas mixtures suggests the need for including an odd-in-J contribution. The results are still qualitative. To summarize, we have found that it is impossible to make a clearcut choice between the W[J]‘*’ (no SA) and W*‘JTT‘+W A J(SA) descriptions of the Senftleben-Beenakker effect on the thermal conductivity on the basis of the available data from magnetic experiments. There are indications from external sources for the presence of a term odd in J in some cases, but no
294
B.J. THIJSSE ET AL.
conclusive evidence is found. In view of the results obtained in this work we have arrived at the following situation: awaiting further developments, one should approach AA results given only in terms of the W iJj\+ W A J(SA) formalism with certain caution. There is certainly room for the alternative explanation in terms of W[J]” (no SA). A mixture of the two descriptions, though in practice unworkable, seems to be even closer to the truth. A way out of this situation would be provided either by sophisticated model calculations of the relevant effective cross sections, or by high precision measurements. In the near future model calculations seem to be the more promising of the two.
Acknowledgements
This work is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)” and has been made possible by financial support from the “Nederlandse Organsiatie voor Zuiver-Wetenschappelijk Onderzoek (ZWO)“.
Appendix
Starting from the Waldmann-Snider equation one can write down an expression for the thermal conductivity in terms of matrix elements of the collision operator. Using an inverse operator technique (see, e.g., ref. 19) one obtains for A: A_2kBT
w m
(
w2 (
-;+
&~W(W2-;+~-_(~)o))0.
c-(E)0
(A.11
>
Here 5? is the Waldmann-Snider collision operator and 8 is the Liouville operator, representing the effect of external electric and magnetic fields. First discussing the situation without external fields, one realizes that, since the collision operator is isotropic, the vector W( W2 - 5/2 + E - (E)~) can only couple to polar vectors built up from the vectors in the problem, the velocity W and the angular momentum J. Some examples: W -3 and W A J. Concentrating on the term W-w, which has been shown in many experiments to be the dominant angular momentum term, one finds’)
A=gSkBT -(1+r2) 2m(uh
with
(g3.( Kll&l)2
l-1 l 36(101+ Gj(101+ 1) i
l)EJ,(12) I
-I
64.2)
SENFT’LEBEN-BEENAKKER
EFFECT ON THERMAL
295
CONDUCTIVITY
and (u). = fy*. G(lO1 + 1) is the decay cross section for the total heat transport, and G(!%,“> describes the coupling between heat transport and the W-‘jj\ polarization. 6,(12) is related to the correlation function for the W -%? term,
([@‘*];%[@‘*]~‘)0 = (u)&&,(l2).
(A.3)
Here @l* is defined as W[ J]‘*’ @I2 = {(l/ls)J*(J* -3>>“’
(A.4)
and [@‘*]A represents the M component, with A4 = 0, +l, of the totally irreducible first rank (upper index 1) tensor part of W[J]‘*‘. Note that the spherical notation is used8,25). In the presence of a magnetic field the polarization [@‘*I$ is, through the action of the Liouville operator, mixed with the second and third rank tensors [@‘*I&and [@‘*I,&that can be constructed from W and JJ. The .%-operator is diagonal in these polarizations, ([@‘*l~~[W]~‘)0
= S,,&49(u)&(12),
(A.9
(L = 1, 2, 3), while the Liouville operator is not. All three G,,(l2)‘s are positive, but not necessarily equal, although they usually are on the same order of magnitude. The field dependence of the Senftleben-Beenakker effect for the thermal conductivity in this case has first been treated by Tip’). His results are given in the eqs. (1) through (8). In the literature also another scheme for the treatment of the thermal conductivity is being used. Rather than direct coupling W and J (eq. (A.S)), one couples velocities on both sides of the collision operator and likewise the angular momenta26). Again three cross sections are obtained: E&,(12), C&(12) and G2(12). These are defined according to (@:;9@:?~,.) = (V)oc n(r1 l)“*R(/22)“*( - I)‘+a+fl’+‘(Ia;, ‘,)(2, ;, QEl,(l2), ,m (A.@ where (Y, p, (Y’, p’ are spherical tensor indices, a(. . .) is a numerical factor defined in refs. 11 and 12 and (:I:) is a 3j-symbo12~25).G)0(12) is always positive while Gi(l2) and G,(l2) can have either sign and are normally much smaller.
2%
B.J. THIJSSE ET AL.
The three relatior&):
G/(12) and the three
G;,(12) are connected
by the following
G,(12) = G0(12)+36,(12)+&B*(12),
(A.7)
C&(12)= G0(12)+;G,(12)+~(12),
(A.@
6412) = GlJ(12) - G,(12) ++*(12).
(A.9)
While the effect of the collision operator in the description of eq. (A.6) is more complicated than in eq. (AS), the expression involving the Liouville operator is in this case very simple as it is diagonal in the indices: (A. 10)
(@$$?@:$‘)o = - s,,M3,,,$
with o the Larmor frequency. In the so-called spherical approximation (see section 2) one has 6, = Gj2= G3 = G,, and G, = CZz= 0, and one obtains the expression (@$(S
+ i6p)@&~)0= 6,,3,~
(
(v)&(12)-
i?),
(A.ll)
which is now diagonal in CX,/3 for both W and R
References 1) PG. van Ditzhuyzen, 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)
B.J. Thijsse, L.K. van der Meij, L.J.F. Hermans and H.F.P. Knaap, Physica EllA (1977) 53. B.J. Thijsse, W.A.P. Denissen, L.J.F. Hermans, H.F.P. Knaap and J.J.M. Beenakker, Physica 97A (1979) 467. L. Waldmann, Z. Naturforsch. 12a (1957) 660. R.F. Snider, J. Chem. Phys. 32 (1960) 1051. P.G. van Ditzhuyzen, L.J.F. Hermans and J.F.P. Knaap, Physica 88A (1977) 452. J.P.J. Heemskerk, F.G. van Kuik, H.F.P. Knaap and J.J.M. Beenakker, Physica 71 (1974) 484. A. Tip, Physica 37 (1%7) 82. W.E. Kahler and H.F.P. Knaap, Z. Naturforsch. 3la (1976) 1485. B.J. Thijsse, G.W. ‘t Hooft, D.A. Coombe, H.F.P. Knaap and J.J.M. Beenakker, Physica 98A (1979) 307. L.J.F. Hermans, J.M. Koks, A.F. Hengeveld and H.F.P. Knaap, Physica SO (1970) 410. E.R. Cooper and D.K. Hoffman, J. Chem. Phys. 53 (1970) 1100. J.J. de Groot, L.J.F. Hermans, C.J.N. van den Meijdenberg and J.J.M. Beenakker, Phys. Lett. 31A (1970) 304. A.S. Bruev, Sov. Phys. JETP 47 (1978) 1054.
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EFFECT ON THERMAL CONDUCTIVITY
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14) B.J. Thijsse, W.A.P. Denissen, H.F.P. Knaap and J.J.M. Beenakker, Physica lO2A (1980) 305. 15) B.J. Thijsse, W.A.P. Denissen, J.J.M. Beenakker and H.F.P. Knaap, Physica lO2A (1980) 298. 16) F. Tommasini, A.C. Levi, G. Stoles, J.J. de Groot, J.W. van den Broeke, C.J.N. van den Meijdenberg and J.J.M. Beenakker, Physica 49 (1970) 299. 17) J.J. de Groot, J.W. van den Broeke, H.J. Martinius and C.J.N. van den Meijdenberg, Physica 49 (1970) 342. 18) V.D. Borman, B.I. Nikolaev and V.I. Troyan, Inzh. Fiz. Zh. (J. Eng. Phys.) 27 (1974) 640. 19) G.E.J. Eggermont, H. Vestner and H.F.P. Knaap, Physica 82A (1976) 23. 20) E. Mazur, G.W. ‘t Hooft and L.J.F. Hermans, Phys. Lett. 64A (1977) 35. 21) G.W. ‘t Hooft, E. Mazur, J.M. Bienfait, L.J.F. Hermans, H.F.P. Knaap and J.J.M. Beenakker, Physica 98A (1979) 41. 22) E. Mazur, G.W. ‘t Hooft, L.J.F. Hermans and H.F.P. Knaap, Physica 98A (1979) 87. 23) H.F.P. Knaap, G.W. ‘t Hooft, E. Mazur and L.J.F. Hermans, Eleventh International Symp. on Rarefied Gas Dynamics, Cannes (1978). 24) E. Mazur et al., to be published. 25) A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, l%O, 2nd. ed. 1974). 26) F.M. Chen, H. Moraal and R.F. Snider, J. Chem. Phys. 57 (1972) 542.