Molecular Beam Nozzle Optimization

Transcription

1 Molecular Beam Nozzle Optimization Anton Mazurenko 30 June, 2010 This document describes the analysis and optimal design of the oven nozzle used in the neutral atoms experiment. The capillary geometry and the interaction with the cooling laser beams are considered. Capillary Geometry and Model λ 2R The problem consists of an array of capillaries attached to an oven, followed by a zeeman slower and 4 transverse columnating beams. The mean free path inside the chamber is about 200 µm. Therefore, the Knudsen number K n = 1. According to a variety of sources, the number density profile is linearly decreasing to zero at the end of the channel 1. At the source, the pressure is near 1 torr and the temperature is o C. The source number density can be approximated as being in the interval [10 21, ] molecules/m 3. This approximation comes from two methods. The first was from the mean free path, which is known, and where the scattering cross section is derived from the atomic radius (which is not necessarily accurate). The other method was from the ideal gas law, because the strontium is near the dilute limit. Both approximations are very worrying, but they gave relatively consistent results, so it is not egregious to approximate the density to one order of magnitude. In this regime, it is necessary to consider both the intermolecular collisions and the wall collisions. The interaction with the wall is modeled by the particle forgetting its previous velocity and reflecting off the wall with a cosine distribution around the normal vector 1 S. Adamson and J F. McGlip. Measurement of gas flux distributions from single capillaries using a modified, uhvcompatible ion gauge, and comparison with theory. Vacuum, 36(4): , July 1985 at that spot 2. The interactions between molecules were poisson- 2 S. Adamson, C. O Carroll, and J F. distributed over short regions (the mean free path varies over long McGlip. The angular distribution beams formed by single molecular flow intervals) and scattered the test-particle in a random direction. Because the number density at the end was so small, the interactions regime. Vacuum, 38(6): , 1988 between separate beams in the array were neglected, and the analysis was done for a single beam. Simulation Methodology The simulation I wrote largely used the methods of Adamson 3. The 3 S. Adamson, C. O Carroll, and J F. simulation started at a random place on a plane perpendicular to McGlip. The angular distribution beams formed by single molecular flow the tube, and at a random, cosine-distributed velocity (to the polar regime. Vacuum, 38(6): , 1988 angle). There was a major computational difficulty in this simulation % of the particles that started at the end of the tube were

2 molecular beam nozzle optimization 2 backscattered for nontrivial tube lengths (L > 500µm). In nature, this is not a major problem because the particle can reenter the capillary easily. In computation, however, this was not realistic, as the particle can bounce around the oven for a long time. Thus, any particle that passed the base of the tube was counted as backscattered. The backscattering led to small samples and high computation times. To resolve this problem, I started the simulation 2/3 of the way through the tube, where K n = 4. The justification was that the straightening of the beam was mostly due to the wall collisions 4, and that the intermolecular collisions dominate and destroy these effects until the longer mean free path makes intermolecular collisions negligible. Trial and error showed K n = 4 to be an acceptable value. A major limitation of the algorithm was the velocity approximation, in that the problem was treated purely geometrically - the particles velocity consisted of only a direction, but no time dependence came into play. To overcome this, a velocity distribution was added after the angular distribution was established. The velocity disribution used was 5, and seen in Fig 1 f (z) = f M (z)/ ψ(z) 0 dz f M(z)/ ψ(z) where f M = (4/ π)z 2 exp( z 2 ), z = v/ 2kT/m, and ψ(z) = z exp( z2 ) + (π 1/2 /2)(1 + 2z 2 )erf(z) 2π 1/2 z 2 Every exiting particle was assigned an exit velocity, which was to be used later in the cooling stage. There is some ambiguity in the velocity distribution to use because this is a zero limit of a distribution derived for K n >.1 because another source 6 gives a diffirent distribution, based on experimental results. However, K n <<.1 in this case, and it is likely that the present situtation is far closer to equation (1) than to the experimetnal result. The velocity distribution was not sufficient to establish the flux, because the number density was approximated to be zero. The conductance of a capillary 7 in the interemediate flow regime is C = π 128ηL (2R)4 P T/M(2R) 3 L Where η is the viscosity, P is the average pressure, and L, R are the length and radius, and everything is in CGS. The net flux is related to this by (1) (2) 4 S. Adamson and J F. McGlip. Measurement of gas flux distributions from single capillaries using a modified, uhvcompatible ion gauge, and comparison with theory. Vacuum, 36(4): , July D.R. Olander, R.H. Jones, and W.J. Siekhaus. Molecular beam sources fabricated from multichannel arrays. iv. speed distribution in the centerline beam. Journal of Applied Physics, 41(11): , October 1970 Figure 1: The velocity distribution from capillary (Blue), Maxwell-Boltzmann (Purple) 6 G.C. Angel and R.A. Giles. The velocity distribution of atoms issuing from a multi-channel glass capillary array and its implication on the measurement of atomic beam scattering cross sections. J. Phys. B: Atom. Molec. Phys, 5(1):80 88, May A. Roth. Vacuum Technology. North- Holland, 3 edition, 1989 Ṅ = Cn

3 molecular beam nozzle optimization 3 Results The simulation consiseted of 400 test particles. Of these, only about 55% gave useful results, as the rest were backscattered into the oven. Though these results agree with rough predictions of other papers, they are in the worst case an upper bound on the deviation. The results are presented in figures 2 and 3. The y-scale on the figures if only relative, and has no bearing on the real flux. 0.3 cm 0.6 cm 1.0 cm The results make qualitative sense and clearly show the proper behavior in the extreme cases. Due to the relatively small sample size, it was not possible to achieve a better resolution. Figure 2: Broad view of the Monte- Carlo simulation of the angular spread of molecular beam, row 1: 550 µm, row 2: 400 µm, row 3: 300 µm

4 molecular beam nozzle optimization cm 0.6 cm 1.0 cm Cooling After the atoms leave the nozzle, they are columated by 4 transverse beams, and are slowed by the zeeman slower, and these effects must be considered. The force on a particle is given by fig 4, and the timedependent transverse velocity is given by fig 5. The assumed laser parameters were a power of 10mW, detuning δ = 32(10 6 ) spontaneous decay rate γ = 2.1(10 8 ) Hz. The derived quantities are defined as per F = F + + F (3) Figure 3: Narrow view of the Monte- Carlo simulation of the angular spread of molecular beam, row 1: 550 µm, row 2: 400 µm, row 3: 300 µm. y: number of trials, x: radians. The numerical algorithm neglected 2/3 of the tube in all cases, 6.6 mm, 4 mm, 2 mm for 10 cm, 6mm, 3mm lengths. F ± = ± hkγ 2 s s 0 + [2(δ ω D )/γ] 2 (4) The zeeman slower increases the beam s deviation by about 2mm, meaning that to hit the aperture, a beam must have a spread of no more than 20mrad. Fig7 shows the angular distribution of the beam after the transverse slowing. The deviation was computed by the formula from 8, and the maximum acceptable deviation was computed from the figure 6. Figure 4: The force felt by atoms, as a function of transverse velocity. x = t f 0 dt hk γ p t/m (5) Figure 5: The time dependent angle, during the cooling, for a v = 450m/s. 8 Harold Metcalf and Peter van der Straten. Laser Cooling and Trapping. Springer

5 molecular beam nozzle optimization 5 where the velocity is approximated to fall linearly as a funtion of x, to about 31 m/s 9. 9 Marcin Bober, Jerzy Zachorowski, and Wojciech Gawlik. Designing zeeman slower for strontium atoms towards optical atomic clock Useful Flux Using the flux data, columation data and the target deviation, it is clear that the problem is that high flux means higher deviation. The useful flux was computed by determining the fraction of the atoms that had a deviation of less than 20 mrad after the laser columation stage, and multiplying that fraction by the net flux. Essentially, I normalized the angular data to the derived flux. Figure 8 (a), shows the net flux from various capillaries. Figure 8 (b) shows the flux that goes into the aperture to the MOT chamber. From these figures, it is clear that only tubes will be needed to sustain an acceptable flux, and we will be able to increase it well beyond that. Recommendations Approximating the nozzle as an infinite plane (the nozzle area» capillary area) for the purposes of packing, and assuming optimal (honeycomb) packing density, I rescaled the absolute and useful fluxes and estimated the absolute lifetime of the source, shown in figures 10 (a),(b), 9. These results are show that the best nozzle is probably a 300 µm, 1cm capillary array. The tight packing and columated beam means that while its usful flux is only 57% of the other extreme, its lifetime is 2.5 times longer. The flux it will provide is likely sufficient for the experiment, being higher than the flux used by other experiments. The data also show that there is not much gain from going to a 400 µm capillary, as the greater flux is cancelled by the greater diver- Figure 6: Scale schematic of the zeeman slower, nozzle output. The target aperture is shown at right, the nozzle at left.

6 molecular beam nozzle optimization cm 0.6 cm 1.0 cm gence, resulting in worse lifetime. That means that if we would make two nozzles, the second should be a 550 µm diameter, 1cm length. The 1 cm length is optimal, because the lifetime would otherwise be unreasonably short. Figure 7: Narrow view of the Monte- Carlo simulation of the angular spread of molecular beam after transverse slowing, row 1: 550 µm, row 2: 400 µm, row 3: 300 µm, y: number of trials, x: radians

7 molecular beam nozzle optimization 7 (a) (b) Figure 8: (a) Total flux from capillary, (b) Useful flux from capillary

8 molecular beam nozzle optimization 8 References S. Adamson and J F. McGlip. Measurement of gas flux distributions from single capillaries using a modified, uhv- compatible ion gauge, and comparison with theory. Vacuum, 36(4): , July S. Adamson, C. O Carroll, and J F. McGlip. The angular distribution beams formed by single molecular flow regime. Vacuum, 38(6): , G.C. Angel and R.A. Giles. The velocity distribution of atoms issuing from a multi-channel glass capillary array and its implication on the measurement of atomic beam scattering cross sections. J. Phys. B: Atom. Molec. Phys, 5(1):80 88, May Figure 9: y: days minimum lifetime of source, total operating time. Weekends, 8 hour days, and not running the machine continously means that the actual lifetimes will be 10 times this. The lifetime of the previous nozzle was much longer. Marcin Bober, Jerzy Zachorowski, and Wojciech Gawlik. Designing zeeman slower for strontium atoms towards optical atomic clock. Harold Metcalf and Peter van der Straten. Laser Cooling and Trapping. Springer. D.R. Olander, R.H. Jones, and W.J. Siekhaus. Molecular beam sources fabricated from multichannel arrays. iv. speed distribution in the centerline beam. Journal of Applied Physics, 41(11): , October A. Roth. Vacuum Technology. North-Holland, 3 edition, (a) (b) Figure 10: Maximum possible flux from capillary array, accounting for packing. Also compared to the previous nozzle (prev.). (a) total, (b) useful