2.1. Laser excitation of QF₆

The laser excitation rate k_A for ⁱQF₆ molecules to absorb resonant laser photons equals:

where σ_A is the laser photon absorption cross-section (cm²). QF₆ absorption bands are hot-band-broadened, their peaks becoming sharper and higher towards lower temperatures. The laser flux φ_L is given by:

Here I_L is the laser beam intensity in Watts/cm² and ε_L = hν_L the laser photon energy in cm⁻¹. Typical vibrational absorption cross-sections have (temperature-dependent) values on the order of σ_A ≈ 10⁻¹⁸ cm² at T < −150 K near a fundamental band peak (e.g., the ν₃ vibration of QF₆), σ_A ≈ 10⁻²⁰ cm² for binary combination bands (e.g., the ν₂+ν₃ vibration of QF₆), and σ_A ≈ 10⁻²² cm² for tertiary QF₆(3ν₃) absorptions. The laser photon energy ε_L must of course be equal to the vibrational excitation quantum ε_a (or ε_a + ε_b, 3ε_a) to effect resonant absorptions. For the ν₃ vibration of ³³SF₆ one has ε_L ≈ ε₃ ≈ 939 cm⁻¹ and σ_A(ν₃) ≈ 2×10⁻¹⁸ cm², while for ²³⁵UF₆ it is ε_L ≈ ε₃ ≈ 628.3 cm⁻¹ and σ_A(ν₃) ≈ 10⁻¹⁸ cm². Then Equation (2) yields:

With I_L = 1000 W/cm² for example, k_A(SF₆(ν₃)) = 1.06 × 10⁵ s⁻¹ and k_A(UF₆(ν₃)) = 8.02 × 10⁴ s⁻¹.

2.2. Spontaneous emission by QF₆ (process 1)

The spontaneous photon emission or “Einstein” rate

for a vibrational v₃ = 1 → 0 transition of the photon-active ν₃ vibration of QF₆ can be expressed by (Eerkens, 1973):

Here

is the dipole-derivative transition matrix for v₃ = 0 [harr ] 1 transitions, given by:

The derivative dipole charge z_b ≡ (z_b)₀₁ or derivative dipole moment (μ_b′)₀₁ = ez_b, and the vibrational mass M_b of vibration mode b have been calculated for some polyatomic molecules (Kraus, 1967), while emission wavelengths λ_b are usually known from spectral measurements (λ₃ ≈ 10.5 μm for SF₆ and λ₃ ≈ 15.9 μm for UF₆). More often

is measured from which values for z_b²/M_b can be derived. Then with M_b calculated or estimated, z_b can be deduced.

QF₆ molecules have two photon-active vibrations, ν₃ and ν₄. For SF₆ and UF₆, values of

, or τ_se[SF₆(ν₃)] = 0.042 s and τ_se[UF₆(ν₃)] = 0.083 s have been measured (Burak et al., 1970; Kim & Person, 1981). This yields z₃²/M₃ = 0.215 for SF₆ and z₃²/M₃ = 0.249 for UF₆. Using estimated values of M₃(SF₆) = 18.77 amu and M₃(UF₆) = 119.88 amu, one deduces that z₃(SF₆) = 2.01 and z₃(UF₆) = 5.46. Lifetimes of other QF₆ molecules for v₃ = 1 → v₃ = 0 transitions are of the same magnitude. That is 0.03 < τ_se[QF₆(ν₃)] < 0.3 s generally. Thus with transit times t_tr < 2 × 10⁻³ s, there is little chance for laser-excited QF₆(ν₃)* monomers to loose vibrational energy by photon emission during its journey in the free jet. If the ν₃ vibration is excited in QF₆:G or QF₆:QF₆ dimers, much faster predissociations prevail over relaxation by spontaneous emission; again few photons are released.

3.1. Collision rates, mean free paths; overview of reactive processes

The contact collision rate k_c of QF₆ molecules with molecules M (M = G or QF₆) is:

where n_M is molecular density (cm⁻³) of M, σ_cQ/M = π(r_oQ + r_oM)² is the collision cross-section (Ų) for QF₆ + M collisions, and u_Q/M the relative molecular velocity (cm/s) given by:

In Eq (8), n_M(cm⁻³) = 0.97 × 10¹⁹p_M(Torr)/T(K) is assumed, and M_Q/M(amu) = M_QM_M/(M_Q + M_M), where QF₆ is shortened to Q in subscripts. r_oQ, r_oG are molecular contact radii of QF₆ and G (Eerkens, 2001a). For SF₆ or UF₆ diluted in 0.01 torr Ar at T = 50 K for example, k_c(SF₆/Ar) = 7.94 × 10⁵ s⁻¹ and k_c(UF₆/Ar) = 9.88 × 10⁵ s⁻¹. Table 1 lists additional rates k_c for other gases.

Molecular parameters for SF6(ν3) and UF6(ν3) gas-phase collisional VT relaxations

Molecular parameters for SF6(ν3) and UF6(ν3) gas-phase collisional VT relaxations

The mean collision-free path [ell ]_c for a QF₆ molecule diluted in carrier gas G at temperature T is:

where σ_cQ/G is the elastic collision cross-section for QF₆ + G encounters and p_G = (1 − y_Q)p_tot is carrier gas pressure. QF₆+QF₆ collisions are neglected since mole fraction y_Q << 1 is assumed.

Most QF₆ + M encounters result in elastic collisions, but some result in the formation of transient QF₆:M dimers. For vibrationally excited QF₆* molecules, aside from elastic collisions, encounters with M can result in direct VT conversions or in VV transfers if M = QF₆ with probabilities

. Also a dimer QF₆*:M can form with probability

, which (pre-)dissociates within microseconds (Smalley et al., 1976; Beswick & Jortner, 1978; Bernstein & Kolb, 1979; Geraedts et al., 1981, 1982; Snells & Reuss, 1987; Liedenbaum et al., 1989; VanBladel & VanderAvoird, 1990; McCafferey & Marsh, 2002). The outcome of both

processes is that epithermal QF₆ and M molecules recoil off each other after conversion of V to T energy, as discussed below. Direct resonant VV transfer between an excited QF₆* and unexcited QF₆ (i.e., M = QF₆) has a high probability but does not effect overall V storage. However while the net energy distribution is not changed by VV transfers, for MLIS processes, near-resonant exchanges ⁱQF₆* + ^jQF₆ → ⁱQF₆ + ^jQF₆* generate undesirable isotopic losses (isotope scrambling). Such losses are minimized by a high dilution of QF₆ in carrier G.

3.2. Direct specular VT relaxations (process 2)

In a gas mixture of QF₆ diluted in carrier gas G, VT deexcitations of QF₆* through contact collisions QF₆* + G → QF₆ + G + K.E. occur at rates:

Here k_c is given by (8) and VT transition probability

is (Eerkens, 2001a):

with ε₃ = hν₃ and we assume ζ₃²F_A ∼ 0.25. The collision function F_C is given by:

with:

In (14), L_R is the repulsive range parameter in Å, Δε = ε₃ equals the energy quantum (in cm⁻¹) that is VT-converted, M_Q/G is the reduced mass of the collision partners in amu, and T is the gas temperature in K. For χ > ∼20, Eq. (13) reduces to the simpler SSH relation (Eerkens, 2001a; Schwartz et al., 1952; Schwartz & Herzfeld, 1954; Yardley, 1980):

VT deexcitations can also take place in collisions of QF₆* with a like QF₆ molecule, with M_G replaced by M_Q in the above expressions. However in QF₆* + QF₆ encounters, resonant VV transfers (process 3) have a much higher probability of occurring than the VT process 2.

Using molecular parameters listed in Table 1, calculated VT probabilities at different T for SF₆(ν₃)/G and UF₆(ν₃)/G with G = H₂, He, N₂, Ar, Xe, SF₆, and UF₆ are listed in Table 1. They show the enormous effect of mass M_G and T on direct VT probabilities and rates.

3.3. Direct specular resonant VV transfers (process 3)

For a direct VV transfer of v₃ in a QF₆(ν₃)/QF₆ contact collision, the probability is (Eerkens, 2001a):

where:

and:

Here ν_b, x_b, M_b, ζ_b², are the fundamental frequency, anharmonic constant, mass constant, and steric collision factor for internal vibration b in molecule QF₆ (in the present case b = 3), while in (17), we assumed ζ₃²F_A ∼ 0.25 again, and we set F_C = 1 since Δε = 0 for a VV transfer. For a two-quantum change in a QF₆QF₆ quantum box with v₃ = 1 → 0 → 1, one can as an approximation set m − n = Σ|Δv₃| = 2, with effective values m = 2 and n = 0 in Ω_anh used in Eq. (18). Note also that M_Q/Q = 0.5 M_Q. The VV rate in the gas mix is then:

with:

Here y_Q is the mole fraction of QF₆ in the QF₆/G gas mix, and k_c is the QF₆/G collision rate given by (8). The factor c_Q/Q corrects for different collision cross-sections and masses of QF₆/QF₆ interactions. Table 4 of (Eerkens, 2001a) lists some values of collision radii r_oQ and r_oG needed in (20) for molecules QF₆ and G, while the reduced mass M_Q/M = 2M_QM_M/(M_Q+M_M).

Using values for M₃, ε₃, x₃, σ_Q/Q, M_Q/Q from Table 1, one finds from (16) for example at T = 50 K, that

. For 2% SF₆ or UF₆ diluted in 10⁻² torr of Ar, one has from (19) that τ_VV(SF₆(ν₃)/SF₆) = 8.2 × 10⁻³ s, and τ_VV(UF₆(ν₃)/UF₆) = 2.36 × 10⁻³ s. These times are longer or comparable to t_tr and τ_df (< 10⁻³ s) at T < 50 K.

3.4. Dimer formation rates (process 4)

It was shown in (Eerkens, 2001a) that the dimer formation probability

can be expressed by:

where:

Here v_d is the highest (dissociation) vibrational quantum level in the dimer potential well, while M_Q, M_M, y_Q, y_M are molecular masses and mole fractions of QF₆ and M (M = G or Q ≡ QF₆). Energy ε_α = hν_α is the fundamental energy (frequency) quantum and D_α is the well-depth of the dimer bond. We use V′ to designate dimer-bond vibration as opposed to V (without a prime) for internal high-energy molecular vibrations. The dimer formation rate k_df is:

with k_c given by (8) and

by (21).

D_α values for various collision partners can be obtained from Table 4 in (Eerkens, 2001a), assuming (D_α)_Q/M = (D_αQD_αM)^1/2, while ν_α can be calculated from Eq. (19) in (Eerkens, 2001a), with values of r_o = ½(r_oQ + r_oM) taken from the same table. For 2% SF₆ or 2% UF₆ diluted in 0.01 torr of H₂ for example, one finds

at T = 50 K. With Ar instead of H₂, one obtains

at T = 50 K. In 0.01 torr of carrier gas, the dimer formation rates are then k_df(SF₆/H₂) = 1.67 × 10⁴ s⁻¹ and k_df(SF₆/Ar) = 501 s⁻¹, while k_df(UF₆/H₂) = 1.59 × 10⁴ s⁻¹ and k_df(UF₆/Ar) = 579 s⁻¹. Other calculated values of

for selected gas mixtures at various temperatures are listed in Table 2. The table illustrates the diversity of dimerization rates due to the strong effect of mass M_G and temperature T in the dimer formation process.

Dimerization parameters for SF6/G and UF6/G encounters

Figures 2 and 3 show plots of

versus gas temperature T for different SF₆/G and UF₆/G gas mixtures. The figures clearly show that towards lower temperatures, dimer formation/predissociation VT conversions are more probable than direct collisional VT relaxations. In some early (1970s) measurements of VT rates, researchers observed that vibrational relaxation rates, after initially dropping as expected, increased again towards lower temperatures (Yardley 1980). Except for light elements, this was in conflict with the collisional VT theory of Schwartz et al. (1952) and Schwartz & Herzfeld (1954), and often ascribed to possible VR (Vibration → Rotation) transfers, even though collisional VR conversions are rather improbable quantum-mechanically. By including low-temperature two-body dimer formation and pre-dissociation of V-excited species, agreement between theory and experiment is restored.

VT conversion probabilities for SF6*(ν3)/G gas mixtures.

VT conversion probabilities for UF6*(ν3)/G gas mixtures.

3.5. Collisional dissociation rates of dimers (process 5)

At monomer partial pressures p_Q = y_Qp_tot with p_Q less than the homogeneous cluster nucleation pressure p_d (see Section 7.2), the dissociation rate of dimers is (Eerkens, 2001a):

with:

Here

is the probability that a dimer QF₆:G dissociates to QF₆ + G in a collision with molecule G. The parameter a_d corrects the collision rate k_c for QF₆/G monomer collisions to the rate k_cd = a_dk_c for QF₆:G/G dimer collisions which has a larger collision cross-section and higher reduced mass. Contact radii r_oQ, r_oG, and reduced masses M_Q/M in (33) were discussed in Sections 3.3 and 3.4. Calculated values of

are listed in Table 2. At T ∼ 50 K and p_G ∼ 0.01 torr for example, Tables 1 and 2 show that 10⁴ < k_dd < 10⁵ s⁻¹, or 10⁻⁵ < τ_dd < 10⁻⁴ s. This compares with transit times of 10⁻⁵ < t_tr < 2 × 10⁻³ s.

4.1. Isotope harvesting from laser-irradiated free jets

In isotope separation techniques that utilize laser irradiated free jets of supersonically cooled gas mixtures, advantage is taken of the fact that QF₆:G dimers, QF₆^(*) monomers, and QF₆^! epithermals, migrate at different rates due to different collision cross-sections and average molecular speeds. After isotope-selective excitation of ⁱQF₆, the ⁱQF₆* will dimerize briefly as ⁱQF₆*:G which rapidly dissociates with VT conversion, creating above-thermal or “epi-thermal” ⁱQF₆^!. We label epithermal ⁱQF₆^! with superscript ^! to distinguish them from internal vibrationally excited ⁱQF₆* molecules marked by superscript * which move at thermal speeds. At low temperatures, under continuous laser pumping, the ⁱQF₆ are distributed over ⁱQF₆*, ⁱQF₆^!, ⁱQF₆:G, and non-excited ⁱQF₆ populations, while ^jQF₆ populate ^jQF₆:G dimer and ^jQF₆ monomer groups. Because of the different population fractions and different migration rates, isotopomers ⁱQF₆ flee the jet core at higher rates than the ^jQF₆, resulting in isotope separation. To summarize, the jet contains ⁱQF₆ of four types, ⁱQF₆*, ⁱQF₆^!, ⁱQF₆, and ⁱQF₆:G, which have three different average transport constants, while the ^jQF₆ are divided over two classes, thermal monomers ^jQF₆ and slower moving dimers ^jQF₆:G, each with different transport parameters (see below).

The peripheral or “rim” gases are pumped out separately from the supersonic jet core gases which exit through a skimmer at the end of the jet chamber (see Fig. 1). As they migrate through the jet core while in transit to the skimmer, laser-excited ⁱQF₆ cycle repeatedly through excitation → dimerization → dissociation → VT-conversion/epithermalization → re-thermalization. If ⁱQF₆:G dimers are laser-excited instead of ⁱQF₆, the sequence is the same except it starts with dimerization → excitation → dissociation…. Non-excited ^jQF₆ isotopomers experience only collisional dimerizations and dissociations, but at low temperatures most travel as ^jQF₆:G dimers which migrate more slowly than ⁱQF₆ monomers. Thus ⁱQF₆ escape at higher rates from the jet core than ^jQF₆, thereby enriching the rim gases with ⁱQF₆.

The mean free path between collisions of a QF₆ molecule diluted in gas G was given by Eq. (10). Table 3 lists p_G[ell ]_c values for several gases G at different temperatures. For example, at p_G = 0.01 torr, T = 50 K, and G = Ar, a SF₆ molecule experiences one collision in every 0.22 mm of travel. Thus although “thin,” the gas in a jet with 10 mm radius, still follows Maxwell-Boltzmann kinetics and obeys the diffusion laws, as assumed in what follows.

Thermalization parameters for epithermal SF6! and UF6! molecules

4.2. Radial molecular migrations and jet core escapes

To estimate the radial outward diffusion of QF₆ molecules and QF₆:G dimers from a supersonic free jet, we approximate the actual jet contour by an equivalent cylinder with radius R equal to the skimmer entrance radius (see Section 7). All molecules in the cylinder striking the “wall” at R are then assumed to leave the cylinder, as if the wall were a total molecular absorber. The non-excited and vibrationally excited monomers will be shown to escape the jet at thermal rates k_W, while epithermals escape at rates k_W1 and dimers leave the jet with a rate constant k_Wd, such that k_W1 > k_W > k_Wd s⁻¹ per molecule.

In (Eerkens, 2001b) it was shown that a batch of N QF₆ molecules diffusing through carrier gas G in a long cylinder with volume πR²L and surface 2πRL, strike the wall at rates:

if the wall is 100% absorbing. Here [ell ]_c is the mean-free-path and R is cylinder radius in cm, σ_cQ/G is the Q/G collision cross-section in Ų, p_G is in torr, T is in K, and M_Q/G = M_QM_G/(M_Q + M_G) is reduced mass in amu. These units will be used in all final expressions that follow. Table 1 lists typical values of k_Wp_G. The inverse rate k_W⁻¹ equals the average time τ_W for a QF₆ molecule in the cylinder to reach the wall, provided it is not terminated in the gas phase.

In our mathematical model, the “wall absorption” rate equals the jet core escape rate of QF₆ molecules. Because the jet moves at supersonic speed, molecules who cross the “wall” at radius R, can not return to the jet core and can be considered “absorbed” by the “peripheral” or “background” gases in the jet chamber. For a batch of QF₆ molecules entering the jet chamber, which looses QF₆ to the “wall” by radial diffusion as it moves towards the skimmer, the number left in the jet core as a function of time is:

Here N_Qo is the tagged QF₆ population entering the jet chamber at t = 0. After travel time t = t_tr, the number of QF₆ molecules remaining in the jet core is:

and the number N_Qesc or fraction Θ that escaped equals:

Here the transit time t = t_tr = L/U, where L is the travel distance and U the average bulk velocity during the jet's travel through the irradiation chamber.

For a batch of outward diffusing epithermal molecules QF₆^!, the migration rate is similar to thermals, except transport parameters must be adjusted and thermalization must be considered. Using subscript 1 to label epithermal transport parameters, the epithermal migration rate k_W1 is:

Here we used (34) and replaced T^3/2 by T₁^1/2T since the epithermal velocity u₁ is proportional to T₁^1/2. The mean free path [ell ]_c due to scattering of QF₆^! monomers by molecules G is virtually the same as for thermals and is proportional to T/p_G, if QF₆ is highly diluted in the QF₆/G gas mix.

The average epithermal temperature 〈T₁〉 during slow-down can be approximated by the logarithmic mean between the initial kinetic energy ε₀ and the final one ε_T, that is:

with:

Here a is defined as:

in which ε_Q is the original kinetic recoil energy given to QF₆ in the VT conversion by dimer predissociation, ε_a is the released quantum of vibrational energy, and ε_T = kT is the thermal kinetic energy. From momentum and energy conservation of dissociating dimer partners one deduces that:

The original kinetic energy ε₀ of a VT-energized QF₆^! molecule then equals:

In the case of laser excitation of ν₃ in QF₆ molecules, the quantum energy ε_a = ε₃ = hν₃.

Note from (42) that the heavier the dimer partner G is, the larger the fraction of the vibrational energy ε_a imparted to a recoiling QF₆ molecule. Most useful carrier gases have masses M_G that are less than those of SF₆ (M_Q = 146 amu) and UF₆ (M_Q = 352 amu). Thus for QF₆ molecules one usually has η_Q/G ≤ 0.5. Heavy particles or solid surfaces might look attractive for maximizing ε_Q, but one finds that other problems are introduced. Laser isotope separations utilizing cold surfaces are discussed in a separate paper (Eerkens, 2005).

Returning to (38), the “wall” escape rate for epithermals is:

and the epithermal escape fraction Θ₁ is:

For dimers which move thermally, the collision cross-section and reduced mass is higher than for monomers. That is in (34), one must replace σ_cQ/G and M_Q/G by σ_cQG/G and M_QG/G. Using subscripts d for dimers, the escape rate k_Wd for dimers then equals:

Here ψ_d, which is less than unity, is the migration rate correction factor for dimers compared to monomers. The QF₆/G collision rate k_c and mean-free-path [ell ]_c were given by (8) and (10). The dimer escape fraction is finally:

4.3. Slowing-down parameters for epithermal molecules

The average number of collisions that an epithermal molecule experiences till it is thermalized, and the average distance it moves, can be obtained from “Fermi Age Theory” originally developed for epithermal neutrons instead of molecules (Glasstone & Edlund, 1952). This theory shows that the average number of collisions N_T to thermalize an epithermal molecule equals:

Here ε_I, ε_II are kinetic energies of an epithermal molecule before and after a momentum-transfer collision, and the average logarithmic energy decrement ξ equals:

with:

Kinetic energy ε_Q was given by (42), and the ratio a = ε_Q/ε_T by Eq. (41). In Fermi age theory, an epithermal neutron or molecule is considered thermalized when its kinetic energy has dropped to ε_T = kT, rather than (3/2)kT (Glasstone & Edlund, 1952). The average fractional energy loss for each epithermal QF₆^! + G momentum transfer collision is given by:

If energy losses of QF₆^! during thermalization are purely due to scattering and momentum transfers, the slowing-down length or migration distance “as the crow flies,” is (Glasstone & Edlund, 1952):

where [ell ]_c and N_T were given by (10) and (49), and the thermalization “age” ϑ_T equals:

Here we assumed D[ell ]_c/ξ to be independent of energy and the “flux diffusion coefficient” is approximated by D ≈ 1/3[ell ]_c. Table 3 lists ξ, ε_Q, N_T, [ell ]_c, and s_th of SF₆ and UF₆, with various carrier gases at several temperatures T and kinetic energies ε₀.

The thermalization rate of epithermals in the gas (= a feed term for the thermal group), is:

where units are in torr, Ų, K, and amu, as before. This rate is high and forces epithermal molecules to re-enter the thermal monomer group repeatedly and be re-excited, etc, several times before reaching the wall. The same is true for dimers which are continuously formed and dissociated in collisions. That is, in a unit volume of gas, rapid exchanges take place between populations of the four migrating groups, but under steady-state conditions, the fractional populations are constant (see Section 5). Individual molecules are continuously cycled through laser-excitation, dimerization/predissociation/epithermalization, and re-thermalization in tens of microseconds as they migrate to the wall. However as long as the population fractions for each migrant class are constant, the effective escape rate per molecule per second is k_W for the thermal group, k_W1 for the epithermal fraction, and k_Wd for those in the dimer state, even though individual molecules are continuously exchanged between these different migrant groups. That is, under continuous laser pumping, all thermalizing ⁱQF₆ leaving the epithermal group are replaced by newly excited ⁱQF₆* that dimerize, and epithermalize. Thus as long as molecules are not permanently removed in the gas and populations are steady, the migration of ⁱQF₆ can be considered taking place by three independently moving species: a fraction f_it at rate k_W for excited or unexcited thermal monomers; a fraction f_i1 for epithermals at rate k_W1; and a fraction f_id at rate k_Wd for dimers. Steady-state fractional populations f_it = f_i* + f_im, f_i*, f_i!, f_im, f_id of excited, epithermal, and non-excited ⁱQF₆ monomers and dimers are considered next.

5.1. Monomer laser excitation

As mentioned, under steady monomer laser-irradiation, the ⁱQF₆ population is distributed over four distinct groups: excited species (*), epithermals (!), non-excited monomer molecules (m), and non-excited dimers (d). The fraction f_i* of laser-excited ⁱQF₆* is defined as f_i* = n_i*/n_i = [ⁱQF₆*]/[ⁱQF₆], where the brackets (by convention) indicate concentrations. Here n_i ≡ n_Qi is the density (molecules/cm³) of a selected isotopomer ⁱQF₆, n_Q = Σ_jn_Qj is the total density of all QF₆, while the isotopic abundance of ⁱQF₆ equals x_i = n_i/n_Q. Other populations of interest are the fraction f_i! of epithermal ⁱQF₆^! molecules, the fraction f_im of non-excited ⁱQF₆ monomers, and fraction f_id of non-excited dimers. We shall assume that most dimers formed in the gas mixture will be heterodimers QF₆:G if mole fraction y_Q < ∼0.05. That is, we neglect homo-dimers QF₆:QF₆ and trimers, tetramers, etc. Neglect of oligomers other than dimers is reasonable because of the short residence time of the jet in the chamber (Sec. 7.2). A material balance then gives:

Neglecting excitations of in-flight epithermals and second-step excitations of once-excited species, the excited species production rate under laser irradiation is given by the product φ_Lσ_Af_imn_i = k_Af_imn_i, where φ_L is the laser flux, σ_A is the photon absorption cross-section, and f_imn_i is the non-excited monomer population of ⁱQF₆. Then, since f_im = 1 − f_i! − f_i* − f_id, one has:

The first term on the right-hand-side of the kinetics balance (57) is the laser excitation rate per unit volume of ⁱQF₆ monomers which produces ⁱQF₆*, while the second term represents losses of vibrational excited ⁱQF₆* due to dimerizations/predissociations and direct VT and VV transfers, spontaneous emissions, and “wall” escapes of ⁱQF₆*. To-the-wall survival probabilities e_* for QF₆* in the gas phase are included in (57) to balance all sources and sinks:

Here the average travel time to the wall τ_W = k_W⁻¹. Unless the gas is very thin, e_* ∼ 0 usually. Parameters

were already reviewed, while k_A was discussed in Section 2.1.

The loss rate k_V = k_VV + k_VT from direct VT and VV transfers of the ν₃ vibration in SF₆* and UF₆* equals:

Here the collisional encounter rate k_c was given by Eq. (8),

by (12),

by (16), and c_Q/Q by (20). At enrichment-favoring temperatures T ∼ 30 K, and with p_G ∼ 0.01 torr, y_Q = 0.02, and masses M_G > ∼10 amu, Table 1 shows that

, while Table 2 gives

. Both

decrease with decreasing T, but

increases with decreasing T. Under these conditions, with k_c ∼ 10⁶ s⁻¹, one obtains k_df ∼ 10³ s⁻¹, k_VT3 ∼ 10⁻⁴ s⁻¹, k_VV3 ∼ 10² s⁻¹, while

. Thus (57) and (58) can be simplified to:

and:

The epithermal population f_i!n_i is balanced by the relation:

Here k_df was given by (30), k_th by (55), and k_W1 by (44). Since for T < 50 K, k_VT << k_df, and usually e_* << 1, e₁ << 1, the approximation in (62) applies under enrichment-favoring conditions. The to-the-wall survival probability e₁ for gas-phase epithermals in (62) equals:

Here μ₁k_c cancels out in the ratio k_th/k_W1 in the exponential of (63). Factors e₁ and e_* apply to “leaks” from internal microscopic processes per unit volume in the jet, insuring correct values for f_i* when R < [ell ]_c. Survival factors e_* and e₁ must not be confused with bulk losses Θ or Θ₁.

The population of non-excited dimers f_id is determined finally by the kinetic balance:

Under steady-state laser-irradiation and gas flow, d(f_i*n_i)/dt = d(f_i!n_i)/dt = d(f_idn_i)/dt = 0. Then ignoring k_VT since k_VT << k_df, Eqs (62) and (64) yield:

with:

and:

with:

Finally, setting (57) equal to 0 and inserting (65) and (67) yields:

The dimer formation and dissociation rates k_df and k_dd were given by Eqs (30) and (31), together with (21) through (27) and (32), while k_th was expressed by (55) and k_A by Eq (2).

Concentration fractions f_i! and f_id given by (70) and (71) are key factors in isotope separation. The higher f_i!, the higher the enrichment factor β is. Note that at low temperatures with high levels of dimerization, one has k_dd << k_df and k_dd → 0. Then the third terms in the denominators of (69) and (70) approach infinity and both f_i! → 0 and f_i* → 0, while f_id → 1 according to (71). Finally note that unless k_dd → 0, in the limit that k_A → ∞, the population f_i! becomes:

The ratio

is clearly important, and the higher

is, the higher f_i! and β.

A second source term +f_j*(1 − x_i)(n_i/x_i)k_VV could have been added to the right-hand-side of (57) to represent VV re-transfers from other (non-i) excited ^jQF₆* isotopomers which were produced by collisional VV transfers ⁱQF₆* + ^jQF₆ → ⁱQF₆ + ^jQF₆*. Here (1 − x_i) = x_j is the fraction of other isotopomers ^jQF₆ (j ≠ i), n_i/x_i = n_Q, k_VV is the VV transfer rate, and f_j* is the steady-state fraction of isotopomers ^jQF₆ that are excited due to VV transfers from ⁱQF₆*. A second balance equation would have to be written for f_j*, but if y_Q = [QF₆]/[G+QF₆] < ∼0.05, the “VV recycle” term gives only a small correction and we shall (conservatively) neglect it.

We tacitly neglected laser excitation of epithermals or higher excitations of ⁱQF₆* molecules. If the kinetic energy ε_Q is high enough, it might be possible for an excited epithermal ⁱQF₆^*! to relax its ν₃ vibrational energy by direct VT conversion collisions, since

increases with T. However for ν₃ vibrations of QF₆ one finds ε_Q < 1000 K, and the probability

is still negligible unless M_Q/G < ∼4 amu (i.e., for G = H₂ or He). “In-flight” excited epithermals ⁱQF₆^*! must first slow down before they can dimerize and experience predissociation with VT conversion again. After thermalization, still-excited ⁱQF₆* become part of the thermal population group f_i*, whose kinetics balance was already considered. Double and higher step excitations ⁱQF₆* → ⁱQF₆** → ⁱQF₆*** can also occur, since absorption frequency differences for higher vibrational steps are usually small. However subsequent dimerization/VT conversion proceeds preponderantly in one-quantum steps, so we can lump f_i** and f_i*** populations with the f_i* crowd.

5.2. Dimer laser excitation

If ⁱQF₆:G dimers are laser-excited instead of ⁱQF₆ monomers, one can write:

Here f_id is the fraction of ⁱQF₆ that are dimerized as ⁱQF₆:G and k_dd is the dimer dissociation rate given by (31). The laser rate k_A applies here to resonant absorptions by ⁱQF₆:G dimers whose internal ν₃ vibrations are slightly shifted compared to the monomer ν₃ vibration of ⁱQF₆.

Under steady-state conditions, setting time derivatives equal to 0, one obtains:

with:

Usually k_A >> k_dd, k_th >> e₁k_W1, and e₁ << 1 if p_G ∼ 10⁻³ torr, in which case the approximations in (76) and (78) apply. Combining (76) and (77) yields:

and:

Here we added subscripts d to f_i! and f_id to distinguish dimer laser-excitation fractions f_i!d and f_idd from expressions for monomer laser-excitation fractions f_i! and f_id given in Section 5.1.

Comparing (79) with (70), we see that the expressions for f_i!d and f_i! are almost the same except for the last term in the bracket. For dimer-excitation the last term k_dd/k_df → 0 if k_dd → 0, and:

On the other hand, for monomer pumping we found f_i! → 0 when k_dd → 0 since the last term k_df/k_dd → ∞. Thus at very low temperatures where k_dd → 0, only laser excitation of ⁱQF₆ dimers can promote isotope separation. However because monomers have higher and sharper peak absorption cross-sections (σ_A) than dimers, it is more advantageous to pump monomers instead of dimers, as long as k_dd is not entirely 0. That is, with the same laser intensity or flux φ_L, one usually finds (k_A)_monomer ∼ 10(k_A)_dimer. Temperatures and pressures for optimum enrichments using either monomer or dimer laser pumping will be analyzed in the next section.

6.1. Enrichment factor β_m for monomer laser excitations

Assuming continuous laser irradiation of the jet and steady jet flow, the mole fractions of thermals (ⁱQF₆ + ⁱQF₆*), epithermals (ⁱQF₆^!), and thermal dimers ⁱQF₆:G throughout the jet will be f_itx_iy_Q = (1 − f_i! − f_id)x_iy_Q, f_i!x_iy_Q, and f_idx_iy_Q, where y_Q is the mole fraction of QF₆ in the gas mix and x_i = [ⁱQF₆]/[Σ_j(^jQF₆)] is the isotopic abundance of ⁱQF₆. Here the fraction of excited and non-excited thermal monomers is f_it = f_i* + f_im = 1 − f_i! − f_id. Fractions f_it, f_i!, f_id, f_jt, and f_jd are constant but the number of ⁱQF₆ and ^jQF₆ in the jet drops as the jet travels through the chamber to the skimmer due to escape fractions Θ, Θ₁, and Θ_d.

Using a prime to label the isotopic abundance of ⁱQF₆ in core-escaped gas and no prime for incoming un-irradiated feed gas, the isotopic composition x_i′ of QF₆ in the escaped gas is:

or:

Here β_m = x_i′/x_i is the enrichment factor for ⁱQF₆ in the jet-escaped rim gases, and φ₁, φ_d equal:

Parameters μ₁, ψ_d, κ₁, ω_d, and k_W were given by (40), (47), (66), (68), and (34), while mole fractions f_i*, f_i!, and f_id were given by (69), (70), and (71). Mole fractions f_jm and f_jd of other isotopic monomers ^jQF₆ and dimers ^jQF₆:G are:

with escape fractions Θ and Θ_d. Also of course x_j = 1 − x_i. Note that f_jd given by (87) agrees with f_id expressed by (71) for k_A = 0. Also note if x_i << 1, the enrichment factor becomes:

with:

The enrichment relation (83) or (88) has three important components, f_i*, φ₁, and c_d. The laser-pumped excited fraction f_i* increases with laser radiation intensity and the higher f_i* is, the higher β_m. However f_i* also depends on the availability of monomers and, as mentioned, if k_dd → 0 or ω_d → 1, one finds f_i* → 0. This happens at low temperatures when all monomers become dimerized during the jet flight and there are no monomers left for laser excitation. Since the factor f_i* can never exceed 1, only the epithermal factor φ₁ and dimer factor c_d can make β_m larger than 2. The larger φ₁ and c_d are, the higher β_m is.

Figures 4 and 5 show plots of β_m at different pressures and temperatures for several SF₆/G and UF₆/G gas mixtures, assuming a fixed k_A = 10⁵ s⁻¹ and jet core with radius R = 1 cm and length L = 20 cm. The curves show maximum β_m's around T = 25 K for SF₆/G and T = 35 K for UF₆/G at pressures between p_G = 0.005 and 0.05 torr. High values of β are driven by high values for c_d and φ₁. The latter has high values when ε_Q/ε_T = η_Q/Gε_a/(kT) is high. Since ε_a is fixed for a given QF₆, one expects low values of T and a high value for η_Q/G = M_G/(M_G+M_Q),(i.e., high M_G), to yield a high φ₁ and β. Indeed Figures 4 and 5 show heavy SiBr₄ (M_G = 348) as carrier to give the highest enrichment, but its poor aerodynamics gives problems (see Sections 7 and 8). Low temperatures for good enrichments can only be attained temporarily in supersonic jets. Feasibility of attaining such low temperatures will be investigated in Section 7.

Enrichment factors βm(T) for SF6(ν3) monomer excitation at fixed pG.

Enrichment factors βm(pG) for SF6(ν3) monomer excitation at fixed T.

Enrichment factors βm(T) for UF6(ν3) monomer excitation at fixed pG.

Enrichment factors βm(pG) for UF6(ν3) monomer excitation at fixed T.

Inspection of (83) and (88) shows that the observed peaks of β_m around T = T_max ∼ 25K for SF₆/G and T = T_max ∼ 35 K for UF₆/G, occur just before ω_d → 1 or ω_d ≈ 0.995, where:

For T much less than T_max, almost total dimerization occurs as

. Under these strong dimerization conditions, there are no monomers left for laser excitation during the supersonic jet flight in the chamber, and thus β → 1. One finds that epithermals contribute very little to β_max, and the major isotopic effect is due to dimers.

6.2. Enrichment factor β_d with dimer laser excitation

Isotope enrichment factors under direct selective laser excitation of dimers ⁱQF₆:G can be deduced in the same manner using the fractions f_idd and f_i!d obtained in Section 5.2. One has:

where λ₁ (≈ k_A/k_th), φ₁, φ_d, and c_d = ω_d(1 − φ_d) were given, respectively, by (78), (84), (85), and (89) with ω_d = k_df/(k_df + k_dd) defined by (68), and where f_idd = [1 + k_A/k_th + k_A/k_df + k_dd/k_df]⁻¹ was given by (80). For x_i << 1, the expression for β_d reduces to:

For low temperatures where k_dd << k_df and ω_d → 1, one has that c_d → (1 − φ_d) and (92) becomes:

with:

Figures 6 and 7 show enrichment factors β_d(T) for ³³SF₆/G and ²³⁵UF₆/G for carriers Ar and Xe at various temperatures and pressures, assuming R = 0.4 cm and a laser excitation rate of k_A = 10⁴ s⁻¹ (simulating (σ_A)_dimer ∼ 0.1 (σ_A)_monomer). The plots show peaks of β_d in the same general region (20–40 K) where β_m peaks, but in addition there are some high β_d values at very low pressures (p_G ∼ 0.001 torr) and temperatures (T < ∼ 0.5 K), where complete dimerization has set in. This region is unattractive however since such low jet expansion temperatures and pressures are difficult (costly) to achieve, and product cuts Θ_o are very low there (see below).

Enrichment factors βd(pG,T) for SF6:G dimer laser excitations.

Enrichment factors βd(pG,T) for UF6:G dimer laser excitations.

6.3. Overall product cuts Θ_o for monomer and dimer excitations and depletion factor γ

The enrichment factor β alone is insufficient to characterize the performance of an isotope separation process and a minimum of two performance parameters are needed. Besides β, the product “cut” Θ or tails isotope depletion factor γ must be specified to define a separation process (Benedict et al., 1981). Usually the product output cut Θ_o is specified which is defined by:

From a mass balance, one can show that the isotope depletion factor γ in the “Tails” stream (this γ not to be confused with γ = c_p/c_v) is related to β and Θ_o by (Benedict et al., 1981):

Here a single prime refers to product, a double prime to tails, and no prime to the feed stream.

In the case of monomer excitations, the product cut Θ_om according to (95) equals:

which reduces to:

where Θ = 1 − exp(−k_Wt_tr), Θ₁ = 1 − exp(−μ₁k_Wt_tr), Θ_d = 1 − exp(−ψ_dk_Wt_tr), φ₁ = Θ₁/Θ, φ_d = Θ_d/Θ (see Eqs (37), (45), (48), (84), (85)). The parameter c_d = ω_d(1 − φ_d) was previously defined by (89), ω_d = k_df/(k_df + k_dd) by (68), while fractions f_it = f_im + f_i* = 1 − f_i! − f_id, f_i! = κ₁f_i*, and f_id = ω_d{1 − (κ₁ + 1)f_i*}, with f_i* given by (69) and κ₁ by (66). We also used the relations x_j = 1 − x_i, f_jd = ω_d, and f_jm = 1 − f_jd = 1 − ω_d.

For the dimer excitation case, one deduces similarly:

where λ₁ ≈ k_A/k_th was given by (80) and f_idd by (82).

If x_i << 1, both Θ_om and Θ_od reduce to:

with Θ given by (37) and ω_d by (89). Figures 8 and 9 give calculated curves of Θ_o for SF₆/G and UF₆/G in the pressure and temperature ranges where β_m and β_d are high. For Θ_o > ∼0.3, the cylindrical jet approximation may become less realistic, and the equations for calculating β_m, β_d, and Θ_o are less reliable. Note that at temperatures below T = 1 K, where β_d has some secondary peaks, the product cuts Θ_o are very small (Θ_o < ∼10⁻³), and economically unattractive.

Isotope product cuts Θ(T) for 33SF6 at fixed pG.

Isotope product cuts Θ(pG) for 33SF6 at fixed T.

Isotope product cuts Θ(T) for 235UF6 at fixed pG.

Isotope product cuts Θ(pG) for 235UF6 at fixed T.

For a usable laser isotope separation process, one would like Θ_o > ∼0.08 and β > ∼1.8 or γ < ∼0.93 according to (96). From Figs. 4 through 9, it is clear that cuts Θ_o are higher the higher T and p are, contrary to the enrichment factor β which is higher the lower T and p are. Thus compromise temperatures and pressures must be chosen to optimize separations. For example for SF₆/Xe, one might select T = 25 K and p_Xe = 0.007 torr, which gives β = 2.00 and Θ_o = 0.10. For UF₆/SF₆, a possible selection might be T = 35 K, p_SF6 = 0.02 torr, yielding β = 1.95, Θ_o = 0.23. These are reasonable process parameters, and allow ³³SF₆ to be enriched from 0.75% to 96% in seven stages for example, or ²³⁵UF₆ from 0.73% to 2.8% in two, or from 0.73% to 5.4% in three stages. This is much better than in Diffusion (β ∼ 1.002) or Ultracentrifuge (β ∼ 1.1) separation plants that enrich ²³⁵UF₆ employing hundreds of stages.

For the MLIS/CRISLA scheme, staging is simple since Feed, Product, and Tails streams are all gaseous and chemically the same except for isotope concentrations. In MLIS schemes other than CRISLA, the product is chemically changed and must be reformed between stages.

7.1. Free jet flow characteristics

To ascertain that the free-jet low temperatures and pressures desirable for useful isotope separations are attainable without excessive cluster growth, we briefly examine some supersonic expansion relations obtained from the adiabatic expansion relations (Liepmann & Puckett, 1953):

Here subscripts o and t on T, p, A refer to upstream reservoir (stagnation) and throat conditions of the gas mixture which is expanded from a large reservoir through a small constricting circular opening or a slit (the “throat”) with cross-sectional area A_t. No subscripts indicate downstream conditions in the supersonic jet stream. A is the cross-sectional area through which the jet core flows and U is the supersonic bulk-flow velocity of the jet at a downstream station. U_so is the sound velocity of the gas in the reservoir, given by (Liepmann & Puckett, 1953):

with M being the atomic mass of the gas atoms or molecules in amu. Gas coefficients γ = c_p/c_v equal γ = 1.67 for monatomics He, Ar, Xe, γ = 1.4 for diatomic gases (H₂ and N₂), γ ≈ 1.3 for SF₆, CH₄, and Br₂, and γ ≈ 1.07 for UF₆ and SiBr₄.

Low-pressure background gases in the jet chamber create a diffuse boundary layer around an expanding supersonic free-jet core, as the latter experiences oscillating expansions and contractions that form a series of Mach cells with normal and oblique shocks. Figures 10 and 11 show free-jet structures measured by (Eerkens, 1957, Eerkens et al., 1958) and (Chow, 1959) for reservoir pressures of p_o = 0.8 and 3.9 torr, and throat diameters of D_t = 2.22 and 1.27 cm, with jets exhausting into vacuum chambers pumped down to less than 0.01 torr. The latter conditions are close to the expansions needed for successful CRISLA operations. The jet contours of Figures 10 and 11 also show that it is reasonable to approximate the jet core by a gas-filled cylinder with a “vacuum wall” as assumed in Section 4.2 to simplify mathematics. As long as radial diffusion losses from the jet are not excessive (Θ_o ≤ ∼0.3), jet expansion parameters using (101)–(104) provide useful first-order estimates. Note that because the flow is supersonic, the downstream skimmer (see Fig. 1), can not perturb the upstream flow pattern.

Measured free-jet contour and pressure distribution for G = Air; po = 3.86; p∞ = 0.21 torr (after Eerkens et al., 1958; Chow, 1959).

Measured free-jet contour and pressure distribution for G = Air; po = 0.80; p∞ = 0.15 torr. (after Eerkens et al., 1958; Chow, 1959).

Theoretical calculations of two-dimensional jets, though available (Adamson & Nichols, 1959), are complex and involve graphical constructions. While the one-dimensional flow relations (101)–(104) can not predict the observed two-dimensional “Mach diamonds or cells” of supersonic free jets, they do provide average order-of-magnitude downstream values of the supersonic flow expansion cross-sections (A/A_t) and accompanying pressures and temperatures. To attain gas expansions to T ∼ 25 K and p ∼ 0.01 torr with a noble gas carrier for example, one calculates reservoir pressures of 5 torr and throat diameters of 0.532 cm, which are quite manageable. Table 4 lists calculated values of the required reservoir tank pressures p_o and required circular throat radii R_t or rectangular slit widths W_t, for various carrier gases containing a few percent QF₆, which are expanded in one step to pressure p = 0.01 torr and various temperatures in a vacuum chamber. The reservoir temperature is assumed to be T_o = 300 K in these one-step-expansion calculations, while a final jet radius of R_j = 1 cm is assumed in all cases. Calculations with SiBr₄ as carrier gas G shows it to be impractical due to a very low γ. Low-energy vibrations (hν_a ∼ kT) in this molecule retain too much energy at low temperatures.

Calculated one-step supersonic jet expansion parameters for QF6/G with G = He, Ar, Xe (γ = 1.67); G = H2, N2 (γ = 1.4); G = SF6 (γ = 1.33); G = SiBr4 (γ = 1.07)a

To improve laser beam overlap and flow control, it can be advantageous to pass the gas first through a contoured supersonic nozzle, before allowing it to expand as a free jet into the jet chamber, as illustrated in Figure 1. The gas can be pre-expanded smoothly in a nozzle (with less flaring) for example from ∼5 torr at T = 300 K in the feed chamber to about 0.1 torr and T ∼ 63 K at the nozzle exit, using an axi-symmetric Mach-3.4 nozzle with exit diameter of 0.6 cm (including boundary layer) and throat diameter of 0.26 cm. The gas mix subsequently enters the jet chamber where it expands further to p ≈ 0.005 torr and T ∼ 25 K with some flaring, forming a quasi-cylindrical jet. Actually any available Mach-3 or Mach-4 supersonic nozzle with throat radius 0.2–0.5 cm and exit radius 5–7 cm can be adapted for this function.

For increased flow rates slit nozzles with two-dimensional (instead of axisymmetric) jets can be used. In this case, the laser beam is passed cross-wise through the free jet (Eerkens, 1998). Pre-expansion of the gas mix through a two-dimensional supersonic slit nozzle is again desirable to tailor the free jet for better laser beam overlap.

7.2. Nucleation and particle growth

To determine whether significant concentrations of (QF₆)_N clusters could develop during the transit time t_tr as the jet expands to the desired final temperature and partial pressure, we examine conditions that cause nucleation. Typically the final partial pressure of QF₆ is p_Q ∼ 2 × 10⁻⁴ torr if the final gas jet pressure is p ≈ p_G ∼ 10⁻² torr and y_Q = 0.02. At adiabatic expansion temperatures of T ≤ 100 K, this pressure p_Q is well above the equilibrium vapor pressure p_eQ of both SF₆ and UF₆ (and other QF₆ species) listed in Table 5. Under normal equilibrium this would cause QF₆ to condense, but in the gas phase the process is retarded because of the reduced surface tension (Kelvin effect) and reduced binding energy per monomer of small clusters. The actual pressure p_dQ at which QF₆ vapor would start to nucleate and grow (QF₆)_N clusters in the gas is much higher than p_eQ. As reviewed in Eerkens (2003), depending on pressure p_Q and temperature T, a “critical” particle size r_* or monomer loading N_* can be calculated, requiring time t_c to grow. N-mers with N < N_* do not grow since they can not keep additional monomers on their surface during repeated collisions. That is, all sub-critical N-mers (dimers, trimers, …) exist only transiently in extremely small concentrations (Eerkens, 2001a).

Calculated nucleation and particle growth parameters for (QF6)N

At T > ∼150 K, the concentration of dimers at any instant is much lower than that of monomers, while trimer populations are orders-of magnitude smaller than those of dimers, etc. Thus nucleation is of no concern. However for a condensable vapor which is suddenly cooled in an expanding free jet to T < 100 K, significant particle (“snow”) formation might occur since N_*(T) drops rapidly as T drops and consequently the concentration of critical N_*-mers can increase steeply. Even then, this can only happen if the time constant t_c for growth is less than t_tr (∼10⁻³s). Although heterodimers QF₆:Ar and hetero-N-mers (QF₆)_N:Ar_K also form, we briefly examine growth of homo-N-mers (QF₆)_N, and assume the results for hetero-NK-mers are similar.

To determine N_* as a function of T and p_Q or φ_Q = [ell ]n(p_Q/p_eQ), a quartic equation must be solved, while an estimate of t_c = t_c(N_*) requires analysis of a kinetics rate equation (Eerkens, 2003). Computer-calculated values of N_*(T) for some selected temperatures considered earlier, are listed in Table 5. For a given N_*, t_c can be calculated via the relations (Eerkens, 2003):

Here D_α and ε_α are the dimer-bond well-depth and fundamental frequency discussed in Section 3.4, while t_c is the time it takes for 20% of the QF₆ to have condensed into growing particles. Units in Eq. (110) are: T(K), M_Q(amu), n_Q(molecules/cm³), and Ω_Q is the volume of one QF₆ molecule in the condensate in ų. The latter is readily calculated from the relation:

where ρ_Q is the density of QF₆ (liquid or solid) condensate, tabulated in Physics Handbooks. Values of D_α, ε_α, and Ω_Q for SF₆ and UF₆ are listed in Table 5. Some t_c times calculated by (105)–(111) are tabulated in Table 5, showing the shortest time t_c ∼ 0.1 s. Thus for jet transit times of t_tr < 10⁻³ s and p_Q < 2 × 10⁻⁴ torr, there is not enough time for cluster development. However note from (110) that τ₄ ∝ n_Q⁻¹ so clustering can occur if p_Q > 10⁻² torr.

Note from Table 5 that t_c exceed 100 seconds both at high and low temperatures. At high T this is caused by the high value of N_*(T). For example oligomers with N = 20, needed to start SF₆ nucleation at T = 74 K, are extremely rare and have vanishingly small populations (< 10⁻⁸⁰ cm⁻³). At temperatures of T = 10 K, the high t_c values are due to low collision rates.