Center-line intensity of a supersonic helium beam

Supersonic helium beams are used in a wide range of applications, for example, surface scattering experiments and, most recently, microscopy. The high ionization potential of neutral helium atoms makes it difficult to build efficient detectors. Therefore, it is important to develop beam sources with a high center-line intensity. Several approaches for predicting the center-line intensity exist, with the so-called quitting surface model incorporating the largest amount of physical dependences in a single analytical equation. However, until now only a limited amount of experimental data has been available. Here we present a comprehensive study where we compare the quitting surface model with an extensive set of experimental data. In the quitting surface model the source is described as a spherical surface from where the particles leave in a molecular flow determined by Maxwell-Boltzmann statistics. We use numerical solutions of the Boltzmann equation to determine the properties of the expansion. The center-line intensity is then calculated using an analytical integral. This integral can be reduced to two cases, one which assumes a continuously expanding beam until the skimmer aperture and another which assumes a quitting surface placed before the aperture. We compare the two cases to experimental data with a nozzle diameter of 10 $\mu \mathrm{m}$, skimmer diameters ranging from 4 to 390 $\mu \mathrm{m}$, a source pressure range from 2 to 190 bars, and nozzle-skimmer distances between 17.3 and 5.3 mm. To further support the two analytical approaches, we also perform equivalent ray-tracing simulations. We conclude that the quitting surface model predicts the center-line intensity of helium beams well for skimmers with a diameter larger than $120\mu \mathrm{m}$ when using a continuously expanding beam until the skimmer aperture. For the case of smaller skimmers the trend is correct, but the absolute agreement is not as good. We propose several explanations for this and test the ones that can be implemented analytically.

Figure 1

Illustration of all variables used in the ellipsoidal quitting surface model. Here $P$ is a point on the quitting surface from which a particle leaves in a straight trajectory until $P^{\prime }$, a point placed on the detector plane. The point on the quitting surface is given by the set of Cartesian coordinates $(x,y,z)$, which can be related to the polar coordinates $r$, $\alpha$, and $\rho$ for integration. Here $x_{\mathrm{S}}$ is the distance from the nozzle to the skimmer and $x_{\mathrm{D}}$ is the distance from the nozzle to the detector. Therefore, $a=x_{\mathrm{D}}-x_{\mathrm{S}}$. The angles $\beta$ and $\theta$ can also be expressed in terms of $r$, $\alpha$, and $\rho$.

Figure 2

Diagram of the section of the quitting surface considered in the ray-tracing simulation; only the angle ${\delta }_{\mathrm{m}}$ seen by the detector through the skimmer contributes to the intensity at the detector. Here $R_{\mathrm{F}}$ is the radius of the quitting surface, $y$ is the distance between the axis of symmetry and the projection of the maximum-angle ray on the quitting surface, $r_{\mathrm{S}}$ is the skimmer radius, $r_{\mathrm{D}}$ is the radius of the detector, $a$ is the distance between the skimmer and the detector, $d$ is the distance from the skimmer to the point where the maximum-angle ray crosses the symmetry axis, $x_{\mathrm{D}}$ is the distance between the nozzle and the detector, $x_{\mathrm{S}}$ is the distance between the nozzle and the skimmer, and $x$ is the distance from the point of emission of the maximum-angle ray to the nozzle plane.

Figure 3

Sketch of the experimental setup used for the center-line intensity measurements. A skimmer is used to select the supersonic beam, followed by two apertures. Vacuum pumps are placed in each chamber to reduce interactions of reflected particles with the beam. Here $R_{\mathrm{F}}$ is the radius of the quitting surface, from where the gas particles are assumed to leave following a molecular flow.

Figure 4

Example of the alignment procedure, here done for a cold source at 60 bars and a 390-$\mu \mathrm{m}$-diam skimmer. The nozzle is moved relative to the skimmer in 50-nm steps for three values of $x_{\mathrm{S}}$. The optimum alignment position of the nozzle relative to the skimmer is obtained by finding the center point of the maximum of intensity. The complete intensity plot of the beam is shown in the top left corner.

Figure 5

Drawings of the skimmers used for the center-line intensity measurements: (a) the Beam Dynamics skimmers, with diameters of 120 and 390 $\mu \mathrm{m}$, (b) the glass microskimmers mounted on copper, with diameters of 4 and 18 $\mu \mathrm{m}$, and (c) the Kurt skimmer, with inserted apertures of 5 and 100 $\mu \mathrm{m}$.

Figure 6

Plot of the ray-tracing simulation (dashed lines) compared with Eqs. (9) and (11) (respectively, circles and crosses for $S_i=S_{\parallel }$ and triangles for $S_i=S_{\perp }$, superposed). The solid (light gray) green descending line shows the effect on the center-line intensity of varying the distance between the skimmer and the detector $a$. The ascending dashed blue (upper) and red (lower) lines show the intensity change when varying the radius of the pinhole in front of the detector $r_{\mathrm{D}}$ and the radius of the skimmer $r_{\mathrm{S}}$. The center-line intensity and the variable values have been normalized to 1 in order to show all dependences in a single plot. The calculations are done at a fixed skimmer position $x_{\mathrm{S}}=11.3$ mm (the center position). Here $a$ is varied between 0.5 and 2 m, $r_{\mathrm{D}}$ is varied between 10 and 100 $\mu \mathrm{m}$, and the radius of the skimmer $r_{\mathrm{S}}$ is varied between 1 and 10 $\mu \mathrm{m}$. While a variable is varied, the others are kept fix at the maximum value of their span ($a=2$ m, $r_{\mathrm{D}}=100\mu \mathrm{m}$, and $r_{\mathrm{S}}=10\mu \mathrm{m}$). The source temperature is 115 K and the source pressure is 161 bars. Both the ray-tracing simulation and the center-line intensity model assume a quitting surface placed just before the skimmer position ($R_{\mathrm{F}}=11.2$ mm).

Figure 7

Plot of measured and predicted intensities for a warm source (300 K), $120\text{−}\mu \mathrm{m}$ (pink solid line) and $390\text{−}\mu \mathrm{m}$ (red dashed line) skimmers, and three values of $x_{\mathrm{S}}$: 5.3 mm (circles), 11.3 mm (upward arrows), and 17.3 mm (asterisks). The intensities are computed assuming that the expansion stops at the skimmer with $S_i=S_{\perp }$. Note that, for the larger skimmer, the center-line intensity becomes independent of the distance between the skimmer and the nozzle, so all the curves collapse in one simulated curve (in good agreement with what is observed experimentally). The difference in intensities between the two skimmers is due to the fact that they were obtained using different pinholes in front of the detector (see Table 1 and Fig. 3).

Figure 8

Plot of measured intensities for a warm source (300 K) and $120\text{−}\mu \mathrm{m}$ (pink solid line) and $390\text{−}\mu \mathrm{m}$ Beam Dynamics skimmers (red dashed line). The measured intensities are compared to Eq. (12), with the expansion stopped before the skimmer and $S_i=S_{\parallel }$. Note how after the quitting surface has surpassed the skimmer, the model loses its predictability (light gray for $x_{\mathrm{S}}=17.3\mathrm{mm}$; dark gray indicates the whole span for the different values of $x_{\mathrm{S}}$). The difference in intensities between the two skimmers is due to the fact that they were obtained using different pinholes in front of the detector (see Table 1).

Figure 9

Plot of measured and predicted intensities for a cold source (125 K) and the Beam Dynamics skimmers: $120\mu \mathrm{m}$ (pink solid line) and $390\mu \mathrm{m}$ (red dashed line). The intensities are computed using Eq. (12) and assuming that the expansion stops at the skimmer with $S_i=S_{\perp }$. The intensities are plotted for three values of $x_{\mathrm{S}}$: 5.3 mm (circles), 11.3 mm (upward arrows), and 17.3 mm (asterisks). Note how for $P_0>40$ bars and the $390\text{−}\mu \mathrm{m}$ skimmer (red), in the case of $x_{\mathrm{S}}=5.3\mathrm{mm}$, skimmer effects are clearly present and the center-line intensity is significantly lower than for the other two $x_{\mathrm{S}}$ positions. All measurements were taken with $r_{\mathrm{D}}=25\mu \mathrm{m}$ (see Table 1 and Fig. 3).

Figure 10

Plot of measured and predicted intensities for a warm source and the glass skimmers: $18\mu \mathrm{m}$ (black dashed line) and $4\mu \mathrm{m}$ [green (light gray) solid line]. The intensities are computed using Eq. (12) and assuming that the expansion stops at the skimmer with $S_i=S_{\perp }$.

Figure 11

Plot of measured and predicted intensities for a cold source and the glass skimmers: $18\mu \mathrm{m}$ (black dashed line) and $4\mu \mathrm{m}$ [green (light gray) solid line]. The intensities are computed assuming that the expansion stops at the skimmer with $S_i=S_{\perp }$.

Figure 12

Plot of measured and predicted intensities for the $100\text{−}\mu \mathrm{m}$ Kurt skimmer (black dashed line) and a 120-$\mu \mathrm{m}$ (pink solid line) Beam Dynamics skimmer for a warm source. The intensities are computed assuming that the expansion stops at the skimmer with $S_i=S_{\perp }$. Note how strong discrepancies are not observed except for the case of the $100\text{−}\mu \mathrm{m}$ Kurt skimmer. Two discrepancy modes can be observed, a very significant one for $x_{\mathrm{S}}=5.3\mathrm{mm}$ and a less significant one for the rest of the nozzle-skimmer distances.

Figure 13

Measured center-line intensities per area in $\mathrm{counts}/(\mathrm{s}*{\mathrm{m}}^2)$ for a warm source and for the following skimmer apertures: 120-$\mu \mathrm{m}$ Beam Dynamics [yellow (light gray) (top)], 390-$\mu \mathrm{m}$ Beam Dynamics [red (dark gray) (top)], 18-$\mu \mathrm{m}$ glass skimmer [black (bottom)], and $4\text{−}\mu \mathrm{m}$ glass skimmer [green (light gray) (bottom)]. The circle, triangle, and asterisk markers correspond to nozzle-skimmer distances $x_{\mathrm{S}}$ of 5.3, 11.3, and 17.3 mm, respectively.

Figure 14

Measured center-line intensities per area in $\mathrm{counts}/(\mathrm{s}*{\mathrm{m}}^2)$ for a cold source and the following skimmer apertures: 120-$\mu \mathrm{m}$ Beam Dynamics [yellow (light gray) (top)], 390-$\mu \mathrm{m}$ Beam Dynamics [red (dark gray) (top)], 18-$\mu \mathrm{m}$ glass skimmer [black (bottom)], and $4\text{−}\mu \mathrm{m}$ glass skimmer [green (light gray) (bottom)]. The circle, triangle, and asterisk markers correspond to nozzle-skimmer distances $x_{\mathrm{S}}$ of 5.3, 11.3, and 17.3 mm, respectively.

Figure 15

Measured differences between cold source and warm source beam intensities per area in $\mathrm{counts}/(\mathrm{s}*{\mathrm{m}}^2)$ for the following skimmer apertures: 120-$\mu \mathrm{m}$ Beam Dynamics (blue solid line) and 390-$\mu \mathrm{m}$ Beam Dynamics (red dashed line). The circle, triangle, and asterisk markers correspond to nozzle-skimmer distances $x_{\mathrm{S}}$ of 5.3, 11.3, and 17.3 mm, respectively. The solid line indicates that where experimental data were missing, data were extrapolated from the closest experimental points.

Figure 16

Measured differences between cold source and warm source beam intensities per area in $\mathrm{counts}/(\mathrm{s}*{\mathrm{m}}^2)$ for the following skimmer apertures: 18-$\mu \mathrm{m}$ glass skimmer (black dashed line) and and $4\text{−}\mu \mathrm{m}$ glass skimmer [green (light gray) solid line]. The circle, triangle, and asterisk markers correspond to nozzle-skimmer distances $x_{\mathrm{S}}$ of 5.3, 11.3, and 17.3 mm, respectively. The solid line indicates that where experimental data were missing, data were extrapolated from the closest experimental points.

Figure 17

Predicted value of $S_{\perp }$ for a cold source (125 K) according to the numerical calculation of the supersonic expansion presented in Sec. 2a. The circle, triangle, and asterisk markers correspond to nozzle-skimmer distances $x_{\mathrm{S}}$ of 5.3, 11.3, and 17.3 mm, respectively.

Figure 18

Diagram of the supersonic expansion for the case of a radius of the quitting surface radius higher that the distance between the nozzle and the skimmer. The quitting surface is assumed to expand unaffected by the skimmer aperture, except by collimation. Here $R_{\mathrm{F}}$ is the radius of the quitting surface, $y$ is the distance between the axis of symmetry and the projection of the maximum-angle ray on the quitting surface, $r_{\mathrm{S}}$ is the skimmer radius, $r_{\mathrm{D}}$ is the radius of the detector, $a$ is the distance between the skimmer and the detector, $d$ is the distance from the skimmer to the point where the maximum-angle ray crosses the symmetry axis, $x_{\mathrm{D}}$ is the distance between the nozzle and the detector, and $x_{\mathrm{S}}$ is the distance between the nozzle and the skimmer.

Figure 19

Plot of measured intensities for a warm source (300 K) and the following Beam Dynamics skimmers: $120\text{−}\mu \mathrm{m}$ (pink solid line) and $390\text{−}\mu \mathrm{m}$ (red dashed line) Beam Dynamics skimmers. The measured intensities are compared to Eq. (12), with the expansion stopped after the skimmer and $S_i=S_{\parallel }$.