1 Introduction

Over the past several decades, particle accelerators have played a fundamental role in understanding the building blocks of matter and have today a broad range of applications in medical, security and industrial sectors. Typically, they are large in size as a result of the limited accelerating electric field gradients they can sustain to accelerate charged particles for a given length. However, progress in technology is enabling a massive reduction in the size of particle accelerators. Examples of novel compact accelerator research concepts are di-electric laser accelerators being applied, for instance, on microchip-scale devices (Sapra et al. 2020) and plasma wakefield accelerators (Esarey et al. 1996) that are capable of drastically reducing the size of conventional accelerators. An accurate understanding of the low-density, non-equilibrium gaseous flow conditions under which particle accelerators typically operate is critical for their design and optimal operation. The focus of the present study is on numerical methodologies to predict transmission probability characteristics under non-equilibrium flow conditions in a laser wakefield accelerator.

A laser-driven plasma wakefield accelerator is a promising technology that has the potential to generate high-quality electron beams with energy up to several GeV over very short distances, making possible more compact and cheaper accelerator devices in the future (Tajima and Dawson 1979; Leemans et al. 2006). This work is a part of a larger project examining individual stages and overall performance aspects of a laser-driven wakefield accelerator, in support of the design phase of the upcoming Extreme Photonics Applications Centre (EPAC) in the United Kingdom. In wakefield accelerators, the targets are typically gas cells, designed for the controlled injection of plasma electrons, upon an impingement with a high-intensity laser beam. Optimal gas cell target properties are critical for the controlled generation of high-quality electron beams. Cells are typically centimetres in length with entrance and exit apertures of order millimetre diameter. They are located within a large, metre-scale vacuum chamber to allow focusing of the drive beam and diagnosis of the plasma with optical probe beams. The gas in the cell is characterised by relatively high operating pressure conditions (as high as 100 mbar), which then expands into a low-density vacuum region before the next component in the beamline. Knowledge of the gas cell density profile and gas expansion characteristics into the low-density drift regions are important for not only the controlled generation of high-quality electrons, but also in the overall vacuum design. The specific focus of the study reported here is on the transmission efficiency and gas expansion from the gas cell source region to the first low-density drift region.

One of the key parameters to be considered in the design of a vacuum device is its transmission probability characteristics. For simple geometries like tubes, analytical expressions can be derived for the transmission probability in the free molecular regime. For non-trivial shapes, these are typically computed using the test particle Monte Carlo (TPMC) method (Saksaganskii 1988). TPMC is, however, not accurate in the transition regime, where the mean free path is comparable to the system dimensions, but not large enough for the flow to be collisionless. This is particularly relevant in the context of a laser wakefield accelerator, where the flow regime typically varies from the slip to transition regime (due to the relatively high inlet gas densities) in the first low-density drift region. In such instances, the direct simulation Monte Carlo (DSMC) method (Bird 1994) is the method of choice for molecular-level gas flow simulations for any arbitrary Knudsen number. However, DSMC formulations to compute the transmission probability have not been reported in the literature to the best of our knowledge. In this work, a methodology has been therefore devised to compute the transmission efficiency using DSMC that is applicable for all Knudsen numbers. This makes it possible to characterise the transmission probability of a vacuum device for any arbitrary operating pressure condition. The validation and applicability of our methodology is demonstrated in the present study by evaluating the transmission probability and gas expansion features in the first low-density drift region of a laser wakefield accelerator device for a wide range of inlet gas density conditions.

2 Methods and numerical formulation

DSMC is the main numerical method used in the present work for studying the gas dynamics. All DSMC simulations have been carried out using SPARTA (Plimpton et al. 2019), which is a highly scalable parallel open-source code. Previous studies John et al. (2010, 2016, 2021) have demonstrated the method’s capabilities on a range of problems. Helium gas is considered with the variable hard sphere (VHS) collision model, following representative gas properties from Bird’s monograph (Bird 1994). Additionally, the cell size, time step and particle numbers that are needed for accurate DSMC calculations have been set following the well-known guidelines in (Bird 1994). Specifically, the cell sizes considered for the DSMC simulations are at least three times smaller than the mean free path. The time step is taken to be about six times smaller than \({{\Delta x_{\min } } \mathord{\left/ {\vphantom {{\Delta x_{\min } } {V_{{{\text{mp}}}} }}} \right. \kern-0pt} {V_{{{\text{mp}}}} }}\), where \(\Delta x_{\min }\) is the smallest cell dimension and \(V_{{{\text{mp}}}}\) is the most probable molecular velocity given by \(V_{{{\text{mp}}}} = \sqrt {2RT}\), where \(R\) is the specific gas constant and \(T\) is the inlet gas temperature. To minimise statistical noise, an average of at least two hundred particles per cell has been considered and the sampling phase has been carried out over a period of 200,000 time steps.

To study the gas dynamics and formulate an accurate methodology for computing the transmission probability using DSMC, a simplistic cylindrical geometry has been considered to represent the first low-density drift region of a laser wakefield accelerator. The configuration essentially consists of two cylindrical sections: a small inlet section of 2-mm diameter and 5-mm length connected to a bigger cylindrical section of 200-mm diameter and 550-mm length. A schematic of the geometry is illustrated in Fig. 1. DSMC simulations consider a two-dimensional, axisymmetric computational domain (i.e. region above the symmetry line in Fig. 1). For the axisymmetric model, particle weighting is employed to maintain uniform distributions of particles at each cell in the whole computational domain. Here particle weight is based on the distance the cell midpoint is from the axis of symmetry, multiplied by the length of the cell in the axial direction. At the inlet (denoted by I), a uniform density (corresponding to the exit of the gas cell region) is considered by applying a Maxwellian distribution. A wide range of inlet densities ranging from n = 1018 cm−3 to n = 1010 cm−3 have been considered with a constant inlet gas temperature of 273 K. The inlet Knudsen number (based on the inlet section diameter of 2 mm) for these operating conditions, ranges from 0.002 for n = 1018 cm−3 to about 24,000 for n = 1010 cm−3. At the outlet (denoted by OO′), vacuum boundary condition is applied. Planes AB and CD are selected circular sections of 26-mm diameter at which transmission probability is computed. The cylindrical walls (NO and N′O′) are considered fully transparent, while at all other walls, fully diffuse boundary conditions with a wall temperature of 273 K have been imposed.

Fig. 1
figure 1

Schematic of the computational domain (not to scale)

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The TPMC method is also used in this work for transmission efficiency calculations, but specifically in the free molecular regime for validating the DSMC methodology. MOLFLOW + (Kersevan and Pons 2009) is the TPMC code used, which is a Monte Carlo simulation tool for ultra-high vacuum systems. Unlike TPMC, which is valid only in the free molecular regime, DSMC is valid for all the flow regimes from continuum through to the free molecular regime. Therefore, it makes sense to formulate an accurate methodology to compute the transmission probability using the DSMC method for all regimes. This is particularly important in the context of the first low-density drift region of a laser wakefield accelerator, where the flow regime typically varies from the slip to transition regime, depending on the operating conditions.

The TPMC method computes the transmission probability, η, at any outlet region of interest, as the ratio of the number of molecules crossing or being pumped through this region, \({\text{Mpump}}_{{{\text{outlet}}}}\) to the total number of generated molecules at the inlet section, N. Its value in percentage can be calculated as

$$\eta = \frac{{{\text{Mpump}}_{{{\text{outlet}}}} }}{N} \times 100.$$
(1)

However, the same methodology cannot be followed for the DSMC method, as unlike DSMC that considers the motion and collision of molecules in the entire simulation domain simultaneously, the TPMC method considers only one gas molecule at a time, which is emitted from the inlet section until it is captured or pumped. In this work, we have therefore formulated the transmission efficiency for the DSMC method as the ratio of mass flow rate at the outlet section of interest to that at the inlet section as shown in Eq. (2):

$$\eta = \frac{{M_{{{\text{out}}}} }}{{M_{{{\text{in}}}} }} \times 100,$$
(2)

where \(M_{{{\text{out}}}}\) is the average inflow mass flow rate at the outlet section and \(M_{{{\text{in}}}}\) is the average mass flow rate at the inlet region of interest. The mass flow rate itself, at the inlet or outlet section, can be computed in two different ways, both of which are equivalent. The first method is to compute this during the DSMC computation as the average net number of molecules crossing the relevant section of interest per unit time as given in Eq. (3).

$$M = \frac{mN\zeta }{{{\text{d}}t}}.$$
(3)

Here \(m\) is the mass of a gas molecule, \(N\) is the net total number of molecules exiting through the relevant interface, \(\zeta\) is the ratio of physical particles to DSMC simulation particles and \({\text{d}}t\) is the time step. It can be also calculated from the macroscopic quantities at the end of the simulation (i.e. during post-processing stage) as given in Eq. (4).

$$M = \rho AV,$$
(4)

where \(\rho\) and \(V\) are the macroscopic flow density and velocity, and \(A\) is the area of the relevant section of interest. In this study, we have computed the mass flow rate during the post-processing stage following Eq. (4) for the sake of convenience.

3 Results and discussions

3.1 Gas expansion characteristics

The gas expansion characteristics from an inlet source region to the first low-density drift region are shown in Fig. 2. The inlet (denoted by I in Fig. 1) coincides with the outlet of the gas cell target region, characterised by a relatively high number density (typically of the order of 1017 to 1018 cm−3). Figure 2a and b shows the computed Mach number flow field overlaid with velocity streamlines and the number density field for an inlet number density, n = 1018 cm−3, respectively. Clearly, the gas expansion process in the drift region is characterised by a drastic drop in density accompanied by a rapid flow acceleration. The corresponding density and temperature profile along the middle section of the geometry (i.e. the symmetry plane) are shown in Fig. 3a and b, respectively. It can be noted that the rapid gas expansion process results in a drastic drop in the temperature, which increases the flow Mach number substantially (as noted from Fig. 2a).

Fig. 2
figure 2

Gas expansion characteristics for an inlet number density, n = 1018 cm−3: a Mach number field overlaid with velocity streamlines; b number density field

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Fig. 3
figure 3

Computed a number density and b temperature profile along the middle section (i.e. the symmetry plane) of the geometry for an inlet number density of 1018 cm−3

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3.2 Transmission probability characteristics

Transmission probability characteristics are examined here, starting with validation of the DSMC formulation with the TPMC method in the free molecular regime. For TPMC, the accurate design of a vacuum system is based on a three-dimensional (3D) model, realised in MOLFLOW+. To ensure that the TPMC results would provide sufficiently accurate results they must be compared with the DSMC results obtained for the same geometry. Thus, a new 3D DSMC model, equivalent to the 2D axisymmetric configuration shown in Fig. 1, was also created. A very low inlet number density of 1010 cm−3 was chosen here to ensure free molecular conditions. To rigorously test and validate the DSMC methodology for computing transmission probability (Eq. 2), we compare the computed transmission efficiency results obtained from the DSMC method with the TPMC method, at three selected planes near the outlet (i.e. AB, CD and OO′ in Fig. 1) as shown in Table 1. A sufficiently good agreement within ~ 10% can be noted between the two methods in the free molecular regime at all three sections considered. Validation of the DSMC methodology to compute transmission probability makes possible the computation of η for higher inlet gas densities (i.e. in the transition and slip regimes where TPMC is not valid). This also enables a means to provide guidelines and safety margins for using TPMC in these regimes, which is beneficial in the overall vacuum system design and analysis. Table 1 also shows good agreement between the 3D and equivalent 2D axisymmetric DSMC results, as expected. Therefore, from here on, we will consider only the 2D axisymmetric DSMC simulations for studying the transmission probability and gas dynamics.

Table 1 Transmission efficiency computed by DSMC and TPMC in the free molecular regime
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The transmission probability has been characterised for a wide range of inlet gas number densities, ranging from 1018 cm−3 (corresponding to the high-density, near-continuum regime) to 1010 cm−3 (corresponding to low-density, free molecular conditions) to study the gas dynamics for different operating conditions. This is shown in Fig. 4. It can be noted that, as expected, η is a constant in the free molecular regime due to the similarity in the nature of flow for all the cases considered in the collisionless regime. Moreover, among all the regimes, η is a maximum in the free molecular regime. This can be attributed to the fact that due to the absence of intermolecular collisions, molecules tend to travel unidirectional (in the flow direction) with minimal losses in the transverse directions. As the inlet density increases and the flow is in the transition flow regime, the computed results indicate that the transmission efficiency first decreases and reaches a minimum value in the transition flow regime. This is due to the relative increase in the importance of intermolecular collisions as the inlet density increases, which results in molecular motion being comparatively more diffuse and less unidirectional in comparison to free molecular flow. As the inlet density increases further, it can be noted that η increases. This could be due to the competing effects between the rate in the density drop and rise in the flow velocity as the flow enters the slip regime.

Fig. 4
figure 4

Transmission probability versus inlet number density computed by DSMC

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Finally, transmission probability characteristics noted from Fig. 4 indicate that TPMC can accurately calculate η in the free molecular flow regime and elsewhere overestimate it up to a factor of two when the flow regime at the source is either transitional or slip. For the vacuum system design, this implies that TPMC model provides a safety margin up to a factor of two in the transitional and slip flow regimes.

3.3 Effect of a pipe section on transmission probability

The DSMC methodology to compute transmission probability makes it amenable to assess η for a wide range of operating conditions and geometric configurations to optimise vacuum design. Here we demonstrate this by investigating the effect of having a cylindrical pipe section within the low-density chamber on η. The motivation for including a pipe section here is to capture some of the gas molecules that would be otherwise lost in the transverse direction, thereby improving η. For this study, we consider two configurations with exactly the same baseline geometrical model as considered previously in Fig. 1, but with pipe sections included, i.e. a short pipe A and a longer pipe B extending closer to the inlet section as shown in Fig. 5. The pipe sections have a diameter of 26 mm and thickness of 1 mm. The transmission probability for these two configurations is studied for a wide range of inlet gas densities, ranging from 1018 to 1010 cm−3 and compared with the baseline configuration with no pipe. Results are summarised in Table 2. The simulations reveal substantial improvement in the transmission for the configuration with the shorter pipe section, regardless of the inlet density condition considered. For the configuration with the longer pipe, enhancement of transmission efficiency depends on the inlet density conditions. At very low inlet densities, the enhancement obtained is significantly higher than that obtained with pipe A. However, at high inlet densities (e.g. 1018 cm−3), the design of the pipe (particularly its length and its proximity to the inlet section) needs to be carefully considered as it is noted that the transmission efficiency actually decreases with respect to the baseline configuration. This can be attributed to the fact that at high inlet densities, a very long pipe with an entrance closer towards the inlet section of the drift region could obstruct the flow resulting in a reduction in η. Figure 6 shows the computed density flow field for an inlet density of 1018 cm−3 obtained with the two configurations, illustrating the flow expansion near the pipe entrance location. The flow within the pipe is characterised by relatively higher densities and Knudsen numbers. It is worthwhile to also note here that there are regions immediately around the pipe where there is a near absence of gas molecules (the white regions in Fig. 6) due to the rapid supersonic expansion, which results in gas molecules predominantly following the direction of flow.

Fig. 5
figure 5

Schematic of the computational domain considering two different pipe configurations

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Table 2 Transmission efficiency versus number density for different configurations
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Fig. 6
figure 6

Computed number density fields for an inlet number density of 1018 cm−3: a short pipe and b long pipe configurations

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4 Conclusions

A methodology to compute transmission efficiency using the direct simulation Monte Carlo method has been proposed in this work, which makes possible the characterisation and design analysis of vacuum devices for a wide range of operating conditions. This was used to study the gas expansion process in the first low-density drift region of a laser wakefield accelerator for the entire Knudsen number regime. This work has not only revealed interesting insights in understanding the gas dynamics but also provides lots of scope for optimising vacuum systems. It has been demonstrated that DSMC can act as a powerful tool in complementing the TPMC method for the overall design and analysis of vacuum systems. In particular, it was shown that the TPMC model provides a safety margin up to a factor of two in the transitional and slip flow regimes, for the overall vacuum system design of a laser wakefield accelerator considered in this study.