Separation and Conversion Dynamics of Four Nuclear Spin Isomers of Ethylene

Molecules with three or more nuclei of nonzero spin exist as discrete spin isomers whose interconversion in the gas phase is generally considered improbable. We have studied the interconversion process in ethylene by creating a sample depleted in the B_2u nuclear spin isomer. The separation was achieved through spatial drift of this isomer induced by resonant absorption of narrow-band infrared light. Evolution of the depleted sample revealed conversion between B_2u and B_3u isomers at a rate linearly proportional to pressure, with a rate constant of 5.5 (±0.8) x 10^–4 s^–1 torr^–1. However, almost no change was observed in the A_g isomer populations. The results suggest a spin conversion mechanism in C₂H₄ via quantum relaxation within the same inversion symmetry.

Department of Physics, University of Toyama, Toyama 930-8555, Japan.

^* To whom correspondence should be addressed. E-mail: zdsun{at}unbsj.ca

Nuclear spin isomers and their stability are fundamental concepts in quantum mechanics (1). In accordance with Pauli's principle, all molecules possessing identical nuclei with nonzero spin have distinct nuclear spin isomers (1). However, despite continuous study following the first separation and conversion of ortho- and para-H₂ in 1929 (2), the interconversion dynamics of three or more isomers in larger polyatomic molecules remain poorly understood. In astronomy and astrophysics, the abundance ratios of nuclear spin isomers in the interstellar medium (ISM) are key parameters in probing the formation conditions in the past and anticipating subsequent processes in the future evolution of planetary materials and protostellar environments (3–5). It is widely assumed that the conversion probabilities among nuclear spin isomers for the various molecules in the ISM are zero, even over time spans of millions of years. However, this is not necessarily the case (6–10).

To date, separation and conversion of nuclear spin isomers have been successfully studied for only a small number of polyatomic molecules: CH₃F (6, 7), ¹³C¹²CH₄ (8), H₂CO (9), and H₂O (11). Among the separation methods (6, 9, 11), the light-induced drift (LID) (12) technique is one of the more powerful and sensitive tools. The principle of LID can be briefly described as follows: Let a powerful laser pass through a closed cell containing a low-pressure gas mixture of a laser-absorbing species and a nonabsorbing buffer gas. When the laser frequency is tuned to, for example, the red wing of the spectral Doppler absorption profile, a certain velocity class of absorbing molecules moving toward the laser will be excited as a result of the Doppler effect. Because the excited molecules usually have a larger cross section than the ground-state molecules, their mean free path will be smaller than that of the ground-state molecules. This produces a drift of the absorbing species moving in the direction of the laser beam with respect to the buffer gas and results in a concentration difference between the two ends of the closed cell. So far, however, insights from LID studies have been limited to molecules with only ortho and para isomers (gaseous CH₃F and ¹³C¹²CH₄). For a molecule with more than two nuclear spin isomers, such as the four isomers (A_g, B_1g, B_2u, and B_3u) of ¹²C₂H₄ ethylene, the possibility of interconversion remains experimentally unexplored. To address this question, we have assembled a spectrometer, following the design of Nagels et al. (7), to separate and monitor potential interconversion among the ¹²C₂H₄ nuclear spin isomers.

Ethylene has a simple structure and a point group (D_2h) of high symmetry. There are two zero-spin ¹²C nuclei and four hydrogens with active spins of . However, unlike ortho/para hydrogen, one cannot visualize the isomers by flipping the spin of individual nuclei. The symmetry characteristics of the four nuclear spin isomers of C₂H₄ are listed in Table 1 (13). Here the coordinate system and group theoretical definitions are the same as those given in the textbook by Herzberg (14) and that by Landau and Lifshitz (1); the x-y plane with the x axis parallel to the C=C double bond is the molecular plane, and the z axis is vertical to it. The four nuclear spin species correspond to different classes of J_Ka,Kc rotational levels in the ground rovibrational state, where J, K_a, and K_c refer to the quantum numbers for rotational angular momentum and its projections along the x and z axes, respectively. An energy level of C₂H₄ is of even or odd parity with regard to the inversion operation E* in D_2h(M) in the molecular symmetry group (10). As the parity is given by (–1)^Kc (15), the subscripts g or u in the symmetry representations correspond to even or odd parity of an energy level.

Table 1. Species of nuclear spin isomers (NSI) of C₂H₄ (1, 14). W is the statistical weight, I is the total spin of four equivalent hydrogen nuclei, and even and odd refer to whether K_a and K_c are even or odd integers.

NSI W I Ka Kc Ag 7 2, 0 Even Even B1g 3 1 Odd Even B3u 3 1 Even Odd B2u 3 1 Odd Odd

We now report the successful use of LID to deplete the population of the B_2u isomer in a sample of gaseous ethylene, followed by monitoring of the subsequent spin conversions for the return to equilibrium. We measured isomer concentrations by recording the absorption intensities of spectral lines with appropriate J, K_a, and K_c quantum numbers. Our experimental setup uses two CO₂ lasers (Edinburgh Instruments PL3 as the separation laser and a home-built laser as the probe) and three glass cells (for separation, test, and reference) (16). We measured the spin conversion rates for ¹³CH₃F with this setup and obtained good agreement with the published results (6, 7).

For the ethylene study, the experimental schemes are shown in Table 2, where the reported results from high-resolution infrared spectroscopy (17) were used to calculate the frequency offsets between the C₂H₄ transition frequencies and the CO₂ laser frequencies. Application of the LID technique for the separation of nuclear spin isomers requires that a molecular transition be near-coincident with a CO₂ laser line. Here, the 10P44 laser line with a power of 6 W was used. Its frequency was tuned about 20 MHz above the center frequency by adjusting the laser cavity length to set it in the red wing of the 9_0,9 10_1,9 10_1,9 line of the ₇ band of ethylene. This frequency selectively excited the B_2u isomer, with the other three isomers acting as a buffer gas. The B_2u molecules drift, by the LID effect, along the direction of the separation laser beam in the separation cell, thereby depleting the B_2u species and enriching the A_g, B_1g, and B_3u species at the entrance end of the cell; this direction of drift corresponds to an increase in the collision cross section upon excitation. The nonequilibrium population was then transferred through a valve from the near end of the separation cell to the test cell. For high sensitivity, we measured differential absorption by splitting the probe beam to acquire simultaneous data from the test cell and the reference cell with a population at thermal equilibrium. We determined normalized absorption intensity differences for appropriate probe lines to observe the initial degree of isomer depletion or enrichment. At an ethylene pressure of 1 torr, the probe was tuned through five absorption lines belonging to one of the species B_2u, B_3u, or A_g (cases 1 to 5 in the seventh column of Table 2 together with the corresponding absorption coefficients in the sixth column) (18). The depletion of the B_2u species was about 3%, with 1% or less enrichment of the other three isomers.

Table 2. Experimental schemes and determined absorption coefficient ß (cm^–1 torr^–1), the percentage of enrichment or depletion (negative values) at a pressure of 1 torr for a 3-min separation period, and pressure dependence of conversion rate = kp + y of C₂H₄ at a temperature of 300 K, where k, p, and y are in units of s^–1 torr^–1, torr, and s^–1, respectively. Rovibrational transition is from the ground state to the ₇ = 1 state. Frequency offset f denotes the C₂H₄ transition frequency minus the CO₂ laser frequency.

Case and number Rovibrational transition NSI Laser line f (MHz) ß Enrichment k (10-4) y (10-4) Separation 90,9 101,9 B2u 10P44 61 Probe 1 90,9 101,9 B2u 10P44 61 0.019 ± 0.001 -2.55 ± 0.50 5.76 ± 1.12 1.02 ± 1.98     2 50,5 41,3 B2u 10P10 -100 0.059 ± 0.002 -3.55 ± 0.50 5.79 ± 0.59 1.59 ± 1.17     3 61,5 62,5 B3u 10P26 112 0.097 ± 0.003 0.91 ± 0.10 5.05 ± 0.57 2.57 ± 0.83     4 43,1 32,1 B3u 10R22 102 0.083 ± 0.002 0.94 ± 0.10 5.41 ± 0.92 2.14 ± 1.13     5 63,4 52,4 Ag 10R28 -228 0.191 ± 0.001 0.76 ± 0.15 Average of cases 1 to 4 5.5 ± 0.8 1.8 ± 1.3

The equilibration kinetics of the B_2u-depleted sample were measured as follows: For the first 1-min period, the separation laser was blocked and the valve was kept open to record the zero baseline of the differential signal in the first period. Then, in the second period, the separation laser was unblocked and its beam was introduced into the separation cell for 3 min to generate the nonequilibrium distribution in the test cell. Then the valve was closed, and the decay curves due to isomeric conversion were monitored during the third period. Typical signals are shown for probing B_2u (Fig. 1), B_3u (Fig. 2A) and A_g (Fig. 2B) populations. Very similar signals were also observed for alternative B_2u and B_3u probe resonances (cases 2 and 3 in Table 2). We tried to monitor the B_1g population dynamics but were not successful because the line intensity of the resonant 26_10,16 27_9,18 transition was too weak. The signals in the third period show the relaxation due to the conversion among spin isomers. A model function A exp(–t)+ B (where A is the integrated intensity, is the observed conversion rate constant, and B is the baseline offset) was fitted to the decay data of Fig. 1 to give the solid smooth curve shown with a rate constant = 8.09 (±0.10) x 10^–4 s^–1.

Fig. 1. Recorded differential absorption signal at lock-in time constant of 0.3 s using the 10P44 probe CO2 laser line at a pressure of 1.44 torr. The trace in the first period is the zero-difference baseline. The trace in the second period shows depletion of 2.46% (±0.20%) for 3 min, and the trace in the third period shows the conversion after the valve is closed. [View Larger Version of this Image (27K GIF file)]

Fig. 2. Recorded differential absorption signals at lock-in time constant of 0.3 s using probe CO2 laser lines of (A)10R22 at a pressure of 0.98 torr and (B) 10R28 at a pressure of 1.02 torr. The enrichments are 0.91% (±0.05%) and 0.89% (±0.05%) at the end of the second period of (A) and (B), respectively. The spin conversion rates observed in the third periods of (A) and (B) are 7.55 (±0.04) 10–4 s–1 and 5 (±5) x 10–5 s–1, respectively. [View Larger Version of this Image (29K GIF file)]

The data clearly show that the concentration of the A_g species is almost constant in time, whereas monoexponential kinetics are observed for recovery of the depleted B_2u population and decay of the enriched B_3u population. Furthermore, the B_2u signal does not return to the original zero-difference baseline, and the B_3u signal overshoots the baseline and asymptotically approaches a new equilibrium level. These general phenomena can be qualitatively explained using Curl's theory of state mixing (19). We assume that conversion of nuclear spin isomers of C₂H₄ is allowed between the B_2u and B_3u isomers, and between the A_g and B_1g isomers, but forbidden between species of opposite inversion symmetry. Specifically, molecular "doorway" states are posited, between either B_2u and B_3u or A_g and B_1g, that are so close in energy that the weak intramolecular nuclear spin-rotation and spin-spin interactions of C₂H₄ can induce mixing between them. This mixing is interrupted by collisions, which promote interconversion between either the B_2u and B_3u or the A_g and B_1g states, through the quantum relaxation process proposed by Chapovsky for ortho- and para-CH₃F (20). Therefore, the time rate of change of the number density of one species is determined by the net number of doorway transitions within species of like inversion symmetry. The concentrations of the B_2u and B_3u species relax exponentially toward a common depleted equilibrium level, whereas those of the A_g and B_1g species retain their initial enriched level with no large relaxation. Net population is thus transferred from the B_3u to the B_2u state (reflected in the absorption signal of the B_2u population not reaching the zero-difference baseline, and the B_3u signal passing the baseline).

From the near-constancy of the signal in the third period of Fig. 2B, it appears that spin isomer conversion between states of opposite inversion symmetry is negligible, as is the impact of molecular collisions with the cell wall over the 30-min time range studied. However, over a longer time frame, it is speculated that these factors could cause eventual reequilibrium of the isomer populations to the initial thermal ratios (zero-difference baseline).

The theory of quantum relaxation in ortho-para conversion (20) predicts that, at low pressure, the spin conversion rate should vary linearly with the total gas concentration p. Thus, the observed first-order rate constant is = kp + y, and varying the pressure allows extraction of the bimolecular rate constant k. So far, this behavior has been observed for CH₃F (6, 7) and ¹³C¹²CH₄ (8). For C₂H₄, we measured more than 100 conversion tracks at different pressures and observation times, probing at each of the four B_2u and B_3u resonances (Table 2, cases 1 to 4). The mean values of are plotted in Fig. 3 as a function of pressure. The data fit reasonably well to a linear pressure dependence. Rate constants from the fits for each probe wavelength agree well within the experimental errors (Table 2) and give an average of 5.5 (±0.8) x 10^–4 s^–1 torr^–1.

Fig. 3. Observed conversion rates as a function of pressure for probing (A) the B2u species by the 10P10 line and (B) the B3u species by the 10P26 line of probe CO2 laser. [View Larger Version of this Image (15K GIF file)]

Thus, our spin conversion observations for C₂H₄ are well accounted for by the model of quantum relaxation. The results provide evidence of the weak intramolecular hyper-fine interactions in C₂H₄ and suggest that the conversion mechanism among nuclear spin isomers of polyatomic molecules in general is quantum relaxation with conserved inversion symmetry.

References and Notes

Received for publication 12 September 2005. Accepted for publication 27 October 2005.

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