13591
J. Phys. Chem. 1995,99, 13591-13596
Molecular Axis Orientation by the “Brute Force” Method J. C. Loison, A. Durand, G. Bazalgette, R. White, E. Audouard, and J. ViguC* Laboratoire Collisions Agrkgats Rkactiuitt?, IRSAMC, Uniuersitt? Paul Sabatier et CNRS URA 770, 118, route de Narbonne, 31062 Toulouse Cedex, France Received: March 8, 1995; In Final Form: June 9, 1 9 9 9
Using a strong electric field, it is possible to orient polar molecules in their lowest rotational states. This “brute force” technique, introduced by Loesch, is very powerful and has many applications. In this paper, we recall the general theory of the effect, Le., the properties of the pendular states introduced by Friedrich and Herschbach. We then discuss the effects of hyperfine structure on the orientation process. Finally, we discuss the optimization of an oriented molecular beam of ICl.
1. Introduction Important progress has recently been made in producing molecules with an oriented molecular axis, which may be used for scattering or spectroscopic studies. Loesch and Remscheid’ have introduced the “brute force” technique which can be applied to almost any polar molecule: the combination of a very low rotational temperature and a strong Stark mixing of rotational levels induces the orientation effect. Friedrich and H e r ~ c h b a c h have ~ . ~ discussed the properties of the “pendular states” which are produced by this Stark mixing, and they have started the spectroscopic studies of these states. After these initial works, several experiments have used these ideas to study reaction dynamics4 and inelastic collisions? to observe pendular motion spectroscopically,6 and to measure the sign of dipole moment^.^,'-'^ Pendular states have also been generalized to the case of the_Zeem_aneffect, alone or combined with the Stark effect (with E and B parallel).” Finally, R. E. Miller has shown that the photopredissociation of oriented van der Waals complexes is very informative.I2 In this paper, we will discuss this orientation effect in detail and consider the effect of hyperfine structure. In the experiments performed by Loesch and c o - w o r k e r ~ ,the ~ . ~orientation is weak: the Orientation is measured by the average value of cos 8 (wheie 8 is the angle betwecn the molecular axis carrying the dipole d and the electric field E ) and its value is small ((cos 8) FZ 0.01-0.02). Many experiments would be feasible only if this orientation value were typically 10 times larger. We commence by recalling the theory of pendular states and their orientation. We then consider the effect of hyperfine structure on the orientation process in the case of the molecule IC1. Finally, we discuss the optimization of an oriented IC1 molecular beam.
2. Pendular States of Diatomic Molecules: A Brief Theoretical Description of the Orientation Effect
TABLE 1: Values of the Dipole Moment d (in Debye) and of the Rotational Constant B (in cm-’) for a Set of Typical Molecules. The w Value Corresponding to a Field E = 10 MVlm Is Also Calculated molecule d B w
(1)
The interesting effects appear mostly for low J levels, so that @
Abstract published in Advance ACS Abstracts, August 1, 1995.
0022-365419512099-13591$09.0010
~ ~ 1 1 7
HFI7
(HCNh6
1.24 0.114 18.3
10.24 0.129 133.3
1.874 20.96 0.15
10.6 0.0156 1142
neglecting centrifugal distortion is a good approximation. If we use the rotational constant B as the energy unit, the problem depends on only one parameter, w = dElB
The study of this Hamiltonian is as old as quantum mechanic^.'^-^^ If the quantization axis is chosen parallel to the electric field E, in the J,M basis set, M remains a good quantum number. The matrix element of cos 8 is simply (J,Mlcos O l J + 1,M) =
[
+ 1)2- M2 (U + 1 ) ( U + 3) (J
(3)
For most molecules, the w value cannot be very large. This quantity is evaluated for a few typical molecules in Table 1, which also presents the exceptional case of (HCN)3 studied in ref 6. Similar information for many other molecules is collected in ref 16. Usually, w will not exceed 100, and an efficient procedure’ is to project eq 3 in a truncated basis set. Convergence is easily obtained. Figure 1 presents the energy levels thus calculated. This procedure gives little physical insight, and it is interesting to make approximate calculations to understand the behavior of the levels. In low field, a perturbation calculation at the second order is straightforward:
We consider a diatomic molecule in_its I F (Y = 0)Sound state, placed in a strong electric field E. If the dipole d is not too small and the field is not too large, it is a good approximation to neglect the coupling terms to other vibrational or electronic levels (i.e., to neglect the effect of polarizability) and to write the Hamiltonian
.E=B j 2 - dE cos 8
1~1’7
J(J
+
J ( J + 1) - 3M2 ’) -k w 2 U ( J l)(U- 1 ) ( U
+
+ 3) (4a)
case J = 0:
The extension to fourth order is not too difficult, but the expression thus obtained is voluminous. 0 1995 American Chemical Society
13592 J. Phys. Chem., Vol. 99, No. 37, 1995
Loison et al.
E (MV/m) 0
2
6
4
8
E (MV/m)
C " ' I " ' i " ' l " ' l " ' l " ' ~ 1
m 0
-20
4
6
8
1 0 1 2
40 41 42 43 30 44 31
20
10
2
0
1 0 1 2
11
22 10
32
-
0
5
10
15
20 33
21
21
20
20
5
0
10
15
20
w
0
Figure 1. Plct of the rcduced energies &'/B as a function of w = dE/B
Figure 2. Ploj of the ofientation ( j , M ( cos O l ] , M ) as a function of w
for the levels J,M with J = 0-4. The upper horizontal scale gives the corresponding electric field for IC1 molecules in their ground states.
for the levels J,M with J = 0, 1, 2 . Note the existence of levels with negative orientations when w is small. Upper horizontal scale is as in Figure 1.
In high field, the Schrodinger equation which describes the motion of a point on a sphere can be written in spherical coordinates 8, 9: sin e$
$1
+
(ZMICOS elS,M)
L: - w cos e VI = -VI B
(5)
By analogy to the pendulum in classical mechanics, we assume that 6 is small and approximate sin 6 by 6 and cos f3 by 1 (02/2). Then eq 5 becomes the Schrodinger equation in polar coordinates for a 2-dimensional harmonic oscillator. The energy levels are given by Messiah'* N,M)
B
= --w
TABLE 2: Calculated Orientations 0,Mlcos @l],M)in the Low-Field and High-Field Regimes
+ &(25-
[MI
+ 1)
(6)
ems
=
2 - 2 6 - (3'"
(7)
A more refined treatment of this type has been made and is quite efficient, but more c o m p l e ~ . ' ~ -We ~ ' will use only this simple approximate form as it gives an efficient prediction of the M-averaged orientation (see below). From these two approximations, we can understand most of the features of the energies as a function of the field. As remarked by Friedrich and H e r s ~ h b a c h , ~the , ~average .~ orientation (3,Mlcos f-l~,M)is related to the energy derivative with respect to the field by the Hellman-Feynman theorem 1a 8 (I,M~COS 813,M) = -- B a-w
Using this formula and the approximate eqs 4 and 6, we get the orientations reported in Table 2. The orientation can also be calculated from the eigenstates produced by the numerical diagonalization of the Hamiltonian. The resulting values are plotted as a function of o in Figure 2. In low field, the J = 0 level excepted, the orientation sign depends on M,and the orientation is negative for the M = 0 level. This is explained by the associated classical trajectory: the pendulum spins in the vertical plane, with a high velocity
high field
0
J=O
f-
J S O
0 ( 3 M 2 - J ( J + 1)) J(J 1)(u - 1)(U 3)
1--
3
+
1
%G
1 - u-IMI
+
+1
z/z;;
TABLE 3: M-Averaged Orientation (cos @ ) j (~,+.@,MIcos @lJ,M))/(U1) Calculated in the Low- and I-hgh-Field Regimes. In Low Field, only the J = 0 Level Presents a Nonvanishing M-Averaged Orientation Term Linear in o
+
(cos 6)j
The vibrational quantum in energy units is equal to B ( ~ U ) ' / ~ = (2B dE)'". This formula predicts the energies of the lowest levels when they are localized in 8 space near 8 = 0. This is verified if the second term in eq 6 is very small with respect to the first. The localization occurs very slowly; for instance the rms 8 value in the lowest state is
low field
low field
high field
l--
1
3J2+3J+ 1 2J+l
6
near 8 = 0 and a lower velocity near 8 = n. In high field, the levels described by our approximate treatment are localized near 8 = 0 and have strong orientation. The average cos 8 value converges like w - " ~ toward 1; this slow convergence is a quantum effect, directly related for the I = 0 level to the zeropoint motion. For reasons which will be clear in section 3, we will need the M-averaged orientation as a function of J ; these values are given in Table 3. In Figure 3, we compare the results of these approximate formulae with those of the numerical calculations. The unexpected result is that the high-field formulae predictions are in very good agreement with the numerical values, even when the average cos 6 value is far from 1. Another interesting result of practical importance is that for modest w values (w 5 5 ) only the J = 0, 1, and 2 levels present a noticeable orientation.
3. Orientation of a Molecular Beam The rotational distribution will be described by a Boltzmann distribution, with a rotational temperature Tmt. It is well known that this is not a fully realistic description of a beam, but if deviations have been observed in many cases, there is no general formulation which explains and predicts these deviations.
Molecular Axis Orientation by the Brute Force Method
J. Phys. Chem., Vol. 99,No. 37, I995 13593
E W/m)
(cos
e),,= 1cos e exp 1exp(
+-)
dQ =
kTrot 1 -(12) tanh(dE/kTrot) dE
In the limit dE > B. As the (21 1) sublevels of j,M are equally populated, this explains why the M-averaged orientation (cos e)] is useful. The total orientation can be written
+
using the traditional notation. Here the energy levels depend on the position of the molecule in space when entering the fringing field, and v is the molecular beam velocity. The energy levels are plotted in Figure 1 as a function of the field, and they do not present any localized avoided crossing (one should consider only levels with the same M value). The LandauZener parameter can be re-expressed as Vlz/(AFv)= z, where z is the time over which the wave function changes its character. For the level J,M, this is done in the time necessary to go from zero field to a field E such that dE BJ Vl2. Therefore the time z is given by
(3
d
VZM
BJ
where aElax is the gradient of electric field, roughly given by E,,,=# where 1 is the condenser plate spacing. We then obtain 2n A = -V,,t
where
h
2- t
trot(J)
+
(1 1 4
where zrot(J) = 2dd2B(J 1) is the rotational period for the level J. With typical distances 1 (-low3 m) and velocities (-lo3 d s ) , the time z is of the order of lo-’ s, while the rotational period is typically less than s even for a heavy molecule like IC1 with J = 1. The Landau-Zener parameter is usually larger than lo3, and the behavior is fully adiabatic.
(1 1b)
5. Effect of the Hyperfine Structure on the Orientation Process: Discussion in the Case of IC1
In the low-field regime, one obtains (cos 6 ) = dE 3kTrOt
dE
(cos 6 ) = 3B
if B > kTrot
The comparison with the result of classical thermodynamics is interesting:
The hyperfine structure of the X state of IC1 is well known.23 It is dominated by the electric quadrupolar term of the iodine nucleus (I = V2), but the similar term due to the chlorine nucleus
13594 J. Phys. Chem., Vol. 99, No. 37, 1995
Loison et al.
E W/m) 0
2
4
6
8 1 0 1 2
C ' " l " ' I ' " I " ' l " ' l " ' ~ 1 40 41 42 43
20
as 31
10
m
32 20
33
21
3
0
-10
22 10
11
00
-20
0
5
15
10
20
w
Figure 4. Same plot as in Figure 1 , in the case of IC1 molecule, with its hyperfine structure taken into account. At the present scale, the effect of the hyperfine structure is just to broaden the lines. The highlighted region near the 2,0/3,3 crossing expanded in Figure 7.
(I = 3/2 for both isotopes) is not negligible. A small magnetic spin rotation term has been measured for the iodine nucleus. We limit the discussion to the case of one nuclear spin I, but the calculations taking into account both nuclei. We have already done the calculation of the Stark hyperfine energies of IC1 X (and A) states and obtained excellent agreement with experiment.* Figure 4 presents a plot of the Stark hyperfine energy levels as a function of the field strength. The pattern is not strongly modified by the presence of hyperfine structure, but this structure has two main effects: (i) a complex decoupling occurs in the low-field region and (ii) avoided crossings appear at high field. For instance, near w = 19, the 2,M levels 2,O and 3,3 cross in the absence of hyperfine structure, but this crossing is replaced by a series of crossings and avoided crossings when the hyperfine structure is taken into account. First we consider the low-field decoupling region. The zerofield eigenstates IJ,I,F,MF)evolve and give rise to the high-field eigenstates J~,M,MI). This high-field coupling scheme is valid when the Stark splitting between two j,M levels (differing only by this M value) is larger than their hyperfine coupling term. Equation 4a can be used to evaluate the Stark splitting and the field necessary to decouple the hyperfine structure. This field depends strongly on M , as the energy shifts vary in @. Moreover, as opposite M values are degenerate, the decoupling of the hyperfine structure is never complete. Figures 5 and 6 present an enlarged view of the energy levels issued of the lowfield levels J = 0 and J = 1, as well as their cos 8 value. We have also calculated an average over the hyperfine component of the orientation, and we have compared it to the result of the calculation ignoring hyperfine structure. The first important point is that these two orientations are almost indistinguishable from one another. The second point is that the decoupling is reasonably slow (as the Stark splitting is proportional to I?) and probably adiabatic. The nature, diabatic or adiabatic, of this transfer has little importance in fact, since the hyperfine sublevels of a given J level have, in zero field, the same population. The high-field avoided crossings are easily understood. The energy level j,M is split by the hyperfine coupling into (21 1) levels, with almost parallel curves. Then, where the two curves of the ~ I N andI &,M2 levels were crossing, this simple picture would predict (21 1)2 crossings. However some of them are replaced by avoided crossings because of the noncrossing rule which applies to levels having the same MF value. We have illustrated this case by the pattern of hyperfine levels
+
+
:-0.02 1 Figure 5. Effect of the hyperfine structure on the = 0 level of ICl. Upper panel: energy curves C/B as a function of w or of the electric field E (MV/m). Middle panel: orientation value (cos 0) as a function of the hyperfine level. Lower panel: comparison of the orientation value of the various hyperfine levels with the orientation calculated for the same rotational level without hyperfine structure. The decoupling region appears clearly (w < 5 ) , and decoupling is very slow in the present case J = 0.
in the region where the 1,M levels 2,O and 3,3 cross (see Figure 7). The Landau-Zener parameter A can be evaluated from the hyperfine structure matrix element PFS, the difference Ad of local dipole moment, the field gradient, aElax, and the velocity Y
With VHFSIh= 10 MHz, Ad = 0.3 D, aElax = lo9 V/m2, v = 500 m/s, one gets A = 5 . This is just an order of magnitude, and depending on the crossing (which fixes FFS) and on the field gradient, the behavior can be made diabatic (A > 1). In this latter case, the orientation will be affected since a large fraction of the population of the two lowfield levels is diverted from what would be an adiabatic transfer in the absence of hyperfine structure. Fortunately, this problem occurs only with large w values (w > 19) which are not easily reached at the present time. For ICl, this w value is obtained with an electric field E = 10.6 MVIm, which has not been applied during the experiments.'~~ However, in the case of more polar molecules with comparable B values, this value of w value can be easily reached. Finally, we have studied only the lowest crossing of an infinite series, and we expect similar behavior for all these crossings.
J. Phys. Chem., Vol. 99, No. 37, 1995 13595
Molecular Axis Orientation by the Brute Force Method
2.2
We may recall some well-known considerations concerning oriented targets. Strictly speaking, these considerations apply to spin '12 for which orientation is the only anisotropic character. The question is usually to optimize the measurement of the orientation effect on some property. If 3% the intensity of the beam and @ is its orientation, the signal is given by
-
S=/3fll fa&) pll>-ll
where a and ,L? are proportionality factors, a measuring the sensitivity to orientation, and +I- conespond to the two possible orientations. The orientation effect is therefore
A J = S, - J-= 2/3~~97@
0.2
(19)
and the noise on this difference is proportional to &(if the noise is not dominated by background). Therefore, the signal to noise ratio (SNR)is proportional to
0.1 YI
2
%
(18)
0.0
0 0
SNRa I / N Y
-0.1 IN>-10
-0.2 0.5
0.0
1.5
1.0
'
2.0
0
Figure 6. Effect of the hyperfine structure on the = 1 level of ICI. Same information as in the upper and middle panels o^f Figure 5. Decoupling is more complex but also more rapid than for J = 0 as the M = 0 and IMI = 1 levels separate rapidly when w increases. Near w = 0.7 a series of avoided crossings occurs and _explainsthe irregular behavior of the orientation curves. In the cases J,M = O,O/l,O/l,l,we
have averaged the calculated orientation over the hyperfine structure, and its value agrees wiLh the orientation in the absence of hyperfine structure for the same J,M level within a 0.1% relative deviation as soon as o > 1.
E (MV/m) 10.5
10.8
10.7
10.8
10.9
\
C " l " " i " " l ' " ' l " " l 1
7.9
7.5
F IYI,I-I.s lW,I-O.S
19.2
19.4
19.6
19.8 20.0
w
Figure 7. The high-field crossing j,M = 2,0/3,3 becomes a complex series of avoided crossings when the hyperfine structure is taken into account. Here only MF = 1 levels are represented, for sake of clarity. The plot has been prepared with only 20 different w values (Le., a step Am = 0.0477),and this explains the imperfect definition of the sharpest avoided crossings.
6. Orientation of an IC1 Molecular Beam
(a) Optimizingthe Measurement of the Orientation Effect. The classical formulae (eqs 12 and 13) indicate that the orientation is an increasing function of the ratio EIT,,. The discussion of the quantum behavior proves that this is oversimplified, but the trends are clearly correct: one should minimize the rotational temperature and maximize the electric field. As these two parameters appear to be completely independent, there is no trade-off between these two optimizations.
(20)
which proves that one should optimize the product 9@. This discussion neglects the fact that the molecular axis distribution is more complex than the one of a spin I12 and must be described by a complete set of average values of the Legendre polynomials Pl(cos 0 ) as discussed in ref 20. (b) Practical Values of the Electric Field E and of the Rotational Temperature Trot. The electric field in dense matter can be very large (about lo3 MVIm), but in vacuum it is more severely limited by breakdown.24 Values of the order of 20 MVIm have been used in molecular beam experiments (e.g., in ref 25) and higher values up to 50-80 MV/m seem possible for small distances I between the electrodes ( I 1-2 These very high values would open many new possibilities, but they are surely difficult to realize. In particular, the field which initiates the breakdown is the field at the electrode, not the one applied to the molecular beam; the holes in the electrodes which are necessary for crossed-beam experiment^',^ have the effect of increasing the field on the electrodes with respect to the useful field. Several experiments have already characterized the rotational distribution of IC1 seeded in various gases.26-28 The most extensive study is presented in ref 26 which has used laserinduced fluorescence to measure this distribution. There is some excess of population of high-J levels but the l o w 4 population (up to J = 5-6) is very well represented by a temperature. The data of refs 27 and 28 concem only J = 0, 1, and 2, selected by electric focusing, and the measurements give slightly smaller rotational temperatures than the LIF measurement of ref 26. The rotational temperatures measured by ref 26 for IC1 seeded in the following rare gases are 5 K for He, 2.6 K for Ne, 1.6 K for Ar, and 3.2 K for Kr under the following expansion conditions: source pressure po x nozzle diameter d = p& = 5.7 Torr cm, IC1 molar fraction x 0.5%. The lowest value is obtained for IC1 in argon (a similar result was obtained for z229). (c) Expected Orientation Effect. We have calculated the orientation as a function of the rotational temperature Trotand the field E. All the effects of hyperfine structure have been neglected in this calculation. The curves (cos 6 ) =flu) are plotted in Figure 8 for a series of rotational temperatures down to 1 K. Clearly, in favorable cases, a very large orientation can be produced. To describe this orientation in greater detail, we have calculated the distribution of the molecular axis
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13596 J. Phys. Chem., Vol. 99, No. 37, 1995
Loison et al.
Acknowledgment. We thank R6gion Midi-PyrinBes for financial support and the European Community for giving a postdoctoral position to one of us (R.W.) through the H.C.M. program. B. Pioline had taken part in this work at an early stage and we thank him for his help. References and Notes
In this paper, we have discussed the orientation of the axis of diatomic molecules by the “brute force” method of Loesch and Remscheid. After recalling a series of well-established results conceming the case of a molecule without nuclear spins, we have then discussed the effect of nuclear spins on the orientation and found that it is weak, as long as we can neglect the effect of high-field avoided crossings. The optimization of an oriented IC1 beam is under way in our laboratory, and we have discussed this optimization in light
(1) Loesch, H. J.; Remscheid, A. J . Chem. Phys. 1990, 93, 4779. (2) Friedrich, B.; Herschbach, D. R. Nature 1991, 353, 412. (3) Rost, J. M.; Griffin, J. C.; Friedrich, B.; Herschbach, D. R. Phys. Rev. Lett. 1992, 68, 1299. (4) Loesch, H. J.; Remscheid, A. J . Phys. Chem. 1991, 95, 8194. Loesch, H. J.; Moller, J. J . Chem. Phys. 1992, 97, 9016. Loesch, H. J.; Moller, J. J . Phys. Chem. 1993, 97, 2158. ( 5 ) Friedrich, B.; Rubahn, H. G.; Sathyamurthy, N. Phys. Rev. Lett. 1992, 69, 2487. (6) Block, P. A.; Bohac, E. J.; Miller, R. E. Phys. Rev. Lett. 1992, 68, 1303. (7) Friedrich, B.; Herschbach, D. R.; Rost, J. M.; Rubahn, H. G.; Renger, M.; Verbeek, M. J . Chem. SOC.,Faraday Trans. 1993, 89, 1539. (8) Durand, A.; Loison, J. C.; ViguC, J. J . Chem. Phys. 1994, 101, 3514, C. R. Acad. Sci. Paris 1994, 11319, 739. (9) Slenczka, A.; Friedrich, B.; Herschbach, D. R. Chem. Phys. Lett. 1994, 224, 238. (10) Wang, S. X.; Booth, J. L.; Dalby, F. W.; Ozier, I. J . Chem. Phys. 1994, 101, 5464. (11) Friedrich, B.; Herschbach, D. R. Z . Phys. D 1992,24,25. Slenczka, A.; Friedrich, B.; Herschbach, D. R. Phys. Rev. Leu. 1994, 72, 1806. Friedrich, B.; Slenczka, A,; Herschbach, D. R. Chem. Phys. Lett. 1994, 221, 333. (12) Wu, M.; Bemish, R. J.; Miller, R. E. J . Chem. Phys. 1994, 101, 9447. Bemish, R. J.; Wu, M.; Miller, R. E. J . Chem. Phys. 1994, 101, 9457. (13) Brouwer, F. Ph.D. Thesis, University of Utrecht; Paris, H. J., Ed.; Amsterdam, 1930. (14) Hughes, H. K. Phys. Rev. 1947, 72, 614; 1949, 76, 1675. (15) Ramsey, N. F. Molecular Beams; Oxford University Press: London, 1956. (16) Stolte, S. Nature 1991, 353, 391. (17) Huber, K. P.; Henberg, G. Constants ofDiatomics Molecules; Van Nostrand: New York, 1979. (18) Messiah, A. Mkcanique Quantique; Dunod: Paris, 1967. (19) Peter, M.; Strandberg, M. W. P. J . Chem. Phys. 1957, 26, 1657. (20) Bulthuis, J.; Van Leuken, J.; Van Amerom, F.; Stolte, S. Chem. Phys. Lett. 1994, 222, 378. (21) Buthuis, J.; Van Leuken, J. J.; Stolte, S. J . Chem. SOC.,Faraday Trans. 1995, 91, 205. (22) Zener, C. Proc. R . SOC.1932, A137, 696. (23) Herbst, E.; Steinmetz, W. J . Chem. Phys. 1972, 56, 5342. (24) Latham, R. V. High voltage vacuum insulation; Academic Press: London, 1981. (25) Knight, W. D.; Clemenger, K.; de Heer, W. A,; Saunders, W. A. Phys. Rev. B 1985, 31, 2539. (26) Hansen, S. G.; Thompson, J. D.; Kennedy, R. A,; Howard, B. J. J . Chem. SOC., Faraday Trans. 2 1982, 78, 1293. (27) Liibbert, A.; Rotzoll, G.; Giinther, F. J. Chem. Phys. 1978,69,5174. (28) Keil, D.; Liibbert, A.; Schiigerl, K. J . Chem. Phys. 1983, 79,3845. (29) McClelland, G. M.; Saenger, K. L.; Valentini, J. J.; Herschbach, D. R. J . Phys. Chem. 1979, 83, 947.
of t h e existing literature.
JP950665 1
w
Figure 8. Calculated orientation of an IC1 beam as a function of the field E (or w ) for different rotational temperatures Trot. The curve numbers are the same in Figure 9: number 2, Trot= 10 K; number 3, Trot= 5 K; number 4, Trot= 3 K; number 4, Trot= 1 K.
Figure 9. Polar plot of the distribution of the molecular axis W(0) given by eq 21. The calculation is made for w = 24 and a series of rotational temperatures. The circle numbered 1 represents W(0) for an isotropic medium and the curves numbered 2 to 5 represent (with the same normalization) the function W ( e ) for the rotational temperatures Trot= 10 K (number 2 ) , 5 K (number 3), 3 K (number 4), and 1 K (number 5 ) .
Figure 9 presents a polar plot of W(0) for various rotational temperatures in a high field ( E = 13.1 MV/m or o = 24). This distribution is extremely perturbed for temperatures Trot= 1-3 K, but even for Trot= 5 or 10 K, the orientation effect remains quite substantial.
7. Conclusion
