Method for traveling-wave deceleration of buffer-gas beams of CH

Cryogenic buffer-gas beams are a promising method for producing bright sources of cold molecular radicals for cold-collision and chemical-reaction experiments. In order to use these beams in studies of reactions with controlled collision energies or in trapping experiments, one needs a method of controlling the forward velocity of the beam. A Stark decelerator can be an effective tool for controlling the mean speed of molecules produced by supersonic jets, but efficient deceleration of buffer-gas beams presents new challenges due to longer pulse lengths. Traveling-wave decelerators are uniquely suited to meet these challenges because of their ability to confine molecules in three dimensions during deceleration and their versatility afforded by the analog control of the electrodes. We have created ground-state $\mathrm{CH}\left(X{}^2\Pi \right)$ radicals in a cryogenic buffer-gas cell with the potential to produce a cold molecular beam of ${10}^{11}$ molecules/pulse. We present a general protocol for Stark deceleration of beams with a large position and velocity spread for use with a traveling-wave decelerator. Our method involves confining molecules transversely with a hexapole for an optimized distance before deceleration. This rotates the phase-space distribution of the molecular packet so that the packet is matched to the time-varying phase-space acceptance of the decelerator. We demonstrate with simulations and an analytic one-dimensional model that this method can decelerate a significant fraction of the molecules in successive wells of a traveling-wave decelerator to produce energy-tuned beams for cold and controlled-molecule experiments.

Figure 1

Spectra of $X{}^2\Pi (v^{\prime \prime }=0,N^{\prime \prime }=1,J^{\prime \prime }=1/2)$ cryogenic CH molecules. The main figure shows the $Q$-branch transitions; the frequency offset is 25 723.4 ${\mathrm{cm}}^{-1}$. The inset shows the $P$-branch transition plotted on the same scale; its frequency offset is 25 698.2 ${\mathrm{cm}}^{-1}$. The spectra were obtained from 1 to 2 ms after the ablation pulse. The experimental measurements are shown as points, and the fit to a Gaussian (two Gaussians in the case of the $Q$-branch transition) is shown as a solid line. The data were taken at a cell temperature of 5 K, ablation energy of 0.1 J, and helium buffer gas density of $1\times {10}^{16}$ ${\mathrm{cm}}^{-3}$.

Figure 2

Measured optical density on the $Q$-branch transition as a function of time after the laser ablation pulse. The experimental measurements are shown as solid squares, and the fit to an exponential is shown as a solid line; the exponential time constant is 2 ms. The data were taken at a cell temperature of 5 K, ablation energy of 0.1 J, and helium buffer gas density of $1\times {10}^{16}$ ${\mathrm{cm}}^{-3}$.

Figure 3

Stark shift of the $\lambda$ doublet states of the ${}^2{\Pi }_{\frac{1}{2}}$ energy level for $|J,M\rangle =|\frac{1}{2},\pm \frac{1}{2}\rangle$. This calculation does not take into account coupling to higher-lying rotational levels.

Figure 4

Time series schematic of the molecule acceptance scheme. As the molecular packet (ellipse) approaches the decelerator entrance (represented by the dotted line), its longitudinal PSD rotates and stretches. The velocity of molecules arriving at the decelerator entrance decreases over time. If the decelerator well velocity changes in a manner that matches the changing molecule arrival velocity, molecules can be captured and decelerated in successive potential wells (rectangles).

Figure 5

Linear and $1/t$ acceptance functions. The linear (black) curve uses a constant acceleration designed to bring an index molecule with 210 m/s velocity to a standstill with 600 decelerator rings. The nonlinear [red (gray)] curve tracks the center velocity of molecules entering the decelerator after traveling through a 1.5-m hexapole guide. The inset shows how the depth of the potential well experienced by the molecules changes with acceleration.

Figure 6

Simulated final longitudinal phase-space distribution for linear deceleration using a 1.5-m hexapole and 210 m/s index molecule. Molecules are frozen at their phase-space coordinates when they exit the simulation by moving beyond the inner diameter of the hexapole or slower or the simulation ends. The shaded region represents the location of the decelerator.

Figure 7

Simulated time-of-flight trace for deceleration of a buffer-gas beam. The parameters are the same as those for Fig. 6. The molecules in the first broad peak have a velocity of 180 m/s. The numerous narrow peaks correspond to decelerated molecules with an average velocity of 25 m/s.

Figure 8

The velocity acceptance width $\Delta v$ used in the analytic model as a function of acceleration. For a decelerator length of 1.22 m, an acceleration of $-{10}^5\mathrm{m}/{\mathrm{s}}^2$ is enough to decelerate an index molecule from 495 to 25 m/s.

Figure 9

The fraction of molecules decelerated using a linear chirp for hexapole guide lengths of (left) 0.5 m, (middle) 1.5 m, and (right) 4 m. The results of 3D simulations are shown as black squares, and the solid lines are 1D model predictions. All simulations used 600 decelerator rings. Each point in these plots represents separate linear chirp simulations run with a unique choice of index molecule velocity.

Figure 10

A visualization of the overlap of the PSD of the molecular packet and the molecules that are accepted into the de-celerator using the 1D model. The gray regions show the PSD of the molecular beam as it enters the decelerator after traveling through a 1.5-m hexapole guide. These gray regions are identical in each frame. The black regions show which molecules will be accepted after the acceptance process is complete. The top row shows the overlap for linear acceptance schemes, while the bottom row shows the overlap for $H/t$ acceptance schemes.

Figure 11

A comparison of the 3D simulation results, 3D separatrices, and 1D model separatrices. One 3D separatrix is superimposed on one filled well from the beginning, middle, and end of the decelerated molecule PSD. The 1D model approximates the PSA as a continuous region in the position coordinate, so we show only the velocity boundaries (horizontal lines). The difference can only be discerned when the scale is increased to show individual wells. We see that the model separatrix does accept molecules between individual wells, which, although unphysical, results only in a consistent overestimate of how many molecules can be accepted. This accounts for the excellent agreement between the model and simulations in Fig. 9 after a constant scaling.

Figure 12

One-dimensional model predictions of the fraction of molecules decelerated using the $1/t$ chirp for hexapole guide lengths of 0.5, 1.5, and 4 m. The number of rings required to decelerate the molecular pulse using a particular hexapole guide length is shown for an index molecule with a velocity of 231 m/s (dashed line), which is 180 m/s plus three times the longitudinal velocity width.