### Summary

- Top of page
- Summary
- Introduction
- Materials and methods
- DSMC results
- Electron beam transfer
- Analysis of results
- Discussion
- Conclusion
- References

The gas density of argon along the axis of a pressure-limiting aperture in an environmental scanning electron microscope is found by the direct simulation Monte Carlo method. The aperture is made on a thin material plate, producing the sharpest possible transition region between the specimen chamber and the differentially pumped region downstream of the gas flow. The entire regime from free molecule to continuum flow has been studied, which covers any size of aperture diameter and any pressure from vacuum to one atmosphere. The amount of electron beam transmitted without scattering at any point along the aperture axis is found in the range of accelerating voltage between 1 and 30 kV for argon. The electron beam transmission is further computed for helium, neon, hydrogen, oxygen, nitrogen and water vapour. This study constitutes the basis for the design and construction of an environmental scanning electron microscope having an optimum electron beam transfer, which is the primary requirement for an optimum performance instrument.

### Introduction

- Top of page
- Summary
- Introduction
- Materials and methods
- DSMC results
- Electron beam transfer
- Analysis of results
- Discussion
- Conclusion
- References

The environmental scanning electron microscope (ESEM) has been researched and developed since 1970s, although the first commercial instrument was launched in 1988 by ElectroScan Corporation, Boston, USA. Throughout this period, a large number of publications have appeared both on the development of this technology and on its applications, as is shown by an early bibliography (Danilatos, 1993), and this trend continues unabated to date. The principal development of this technology has appeared in two major surveys (Danilatos, 1988, 1990a) and other key works, as will be shown later.

A successful ESEM involves the synergy of several techniques operating in unison. Among others, these include differential pumping that separates the high vacuum of the electron gun chamber from the high-pressure specimen chamber, novel electron detection techniques in a gaseous medium, gas control and specimen handling, together with specimen transfer from ambient conditions to the specimen chamber of the microscope. The differential pumping technique allows the electron beam to be formed and transferred from the vacuum of the electron optics column to the high-pressure gaseous environment of the specimen chamber. As such, this technique must be designed in an optimum way to allow maximum electron beam transfer, that is, with minimum electron scattering and loss. The optimum conditions have been confirmed and established by work done with a laboratory experimental prototype ESEM and much of this work has been published at various stages of development.

However, the commercial instruments have not fully implemented these findings for an optimum operation yet. As a result, a large number of users of commercial ESEM are severely constrained with regard to maximum pressure attainable, minimum accelerating voltage possible, amount of freedom to specimen movement, detector positioning and design, field of view limits, column contamination and image astigmatism, electron gun life and instrument maintenance. These, in turn, result in frequent specimen damage or difficulty in specimen handling, loss of practical resolution and a general limitation on the number of applications feasible.

One study on two commercial models of ESEM has already been published, in which it is shown that a significant fraction of primary electron beam is lost before the beam enters the specimen chamber (Danilatos, 2001). In particular, it was found that an older model performs better than a later one. Furthermore, it was found that both of those models performed worse than a laboratory ESEM employing a thin-plate pressure-limiting aperture (PLA). The thin aperture case was presented for comparison purposes under the same conditions of chamber pressure and aperture size as the commercial instruments. It can be shown that the thin aperture represents the minimum gaseous mass thickness that the electron beam has to overcome during its transfer. For this reason, an extension of that study in the entire pressure range and aperture sizes that may be encountered in the design and construction of any ESEM is now presented herein.

To cover the required range of conditions as completely as possible, it is necessary to include a minimum amount of results necessary for use by the designer or user of an ESEM. The scope of this paper is to provide a comprehensive study that immediately determines the optimum design specifications for an ESEM in regard to electron beam transfer.

### Materials and methods

- Top of page
- Summary
- Introduction
- Materials and methods
- DSMC results
- Electron beam transfer
- Analysis of results
- Discussion
- Conclusion
- References

There have been extensive studies of rarefied gas flows through nozzles of various shapes in the field of gas dynamics research, but only limited information is available on flows through an orifice made on a thin plate, especially as such information relates to ESEM. In particular, we are interested in establishing the gas density both in the far and in the near field of a PLA as is used in an ESEM. We often have to work very close to this aperture at high pressure and low-kV beam and we need to know the gas density along the length of beam travelled. A study has already been published for the case of a thin aperture, that is, an orifice having 0.5-mm diameter on a 0.1-mm-thick plate with a downstream diverging lip at 45° angle, as is shown in Fig. 1 (Danilatos, 2000a). The same aperture is used in the present study, which constitutes a continuation in part of the previous work.

The complexity of the study of gas flow arises from the fact that the gas densities and orifice sizes of interest correspond to the entire regime from free molecule to continuum flow, as the specimen chamber pressure may vary from vacuum to near atmospheric, and the orifice diameter is of the order of 1 mm and preferably much less. The available theoretical solutions are inadequate or impractical to use, especially in the rarefied or transition gas flow regime. Theoretical and numerical methods (Beylich, 1979, 1984) as well as experimental ones (Liepmann, 1961; Anderson *et al.*, 1966; Ashkenas & Sherman, 1966) can be used up to a certain point, but they are all insufficient for our purposes.

The situation has been addressed by the advent of the direct simulation Monte Carlo (DSMC) method (Bird, 1995) and its adaptation to the present work. This has opened a relatively easy and practical way to study systems in which the variation of geometry and gas state are critical in the determination of an optimum condition of operation. This method allows for the study of flows of complex geometry as, for example, in the study of pumping properties of an annular supersonic jet, which has led to a novel differential pumping system (Danilatos, 2000b). It has also been applied to study the effect of varying the lip angle and thickness of an orifice wall, the effect of specimen presence upstream of the orifice entrance and the effect on two combined orifices in a two-stage differential pumping system. According to these studies, provided the angle of the edge (measured from the axis of the aperture) is greater than 45°, the flow is, for all practical purposes, close to that for an infinitely thin wall aperture.

The DSMC method uses some thousands or millions of simulated molecules for the computer modelling of a real gas flow. The position and velocity of the molecules are stored and modified in the computer over time as the gas flows within given boundary conditions. The number of simulated molecules needed depends on the gas density and spatial extent of the flow field, which determine the computation time required with a given computer. The program starts with initial conditions of pressure on either side of the aperture, and after a transition time period, a steady state flow is achieved, in which we are interested. In reality, the transition to the steady-state flow is achieved within a fraction of a second, but the computation time with the earlier PC computers required some days to simulate the steady state. The results included in this paper were obtained concurrently and are part of the work published earlier on single orifice (Danilatos, 2000a).

Argon is used as the test gas at given input (stagnation) particle number densities *n*_{0} between 4.94 × 10^{20} m^{–3} and 1.235 × 10^{24} m^{–3}, which correspond to input pressures *p*_{0} between 2 Pa and 5000 Pa at 293°K temperature, and in all cases, the gas exhausts in vacuum downstream of the aperture. The similarity of flow principle has also been tested and verified by varying the orifice diameter *D* up to five orders of magnitude inversely to the input pressure. By keeping the product *p*_{0}D constant, it was found that the same flow property reproduced when we normalized the field over the input flow property and over the aperture diameter. The flow properties (density, temperature, velocity, etc.) were studied for different values of the parameter *p*_{0}D, which has direct engineering use in ESEM technology (e.g. in detection efficiency, gain, etc.) and was preferred over the Reynolds number frequently employed in the gas dynamics field. This is an important situation, because for every value of *p*_{0}D, there corresponds a family of pressure–aperture values that cover a wide range of instrument design and operation. For example, with *p*_{0}D= 2.5 Pa-m, there corresponds the pair 50 000 Pa–50 μm and also the pair 5000 Pa–500 μm of pressure–diameter values and so on. Argon has been used in the past, as it is used now, as a standard gas against which comparisons can be made at various stages of development and design. Argon is the preferred gas because it is inert (i.e. ‘clean’ for the microscope) and, being monatomic, the electron scattering cross-section is better established for subsequent computation of electron beam scattering. Other gases have also been used in DSMC studies and presented in this paper.

The computer program yields, among other parameters, the molecule (particle) number density, velocity, the Mach number and temperature in the flow field, together with the transfer rate of molecules across interfaces between zones or through the aperture. Here, we provide only the variation of particle number density along the axis of the aperture, as this directly determines the electron beam scattering and transmission.

### DSMC results

- Top of page
- Summary
- Introduction
- Materials and methods
- DSMC results
- Electron beam transfer
- Analysis of results
- Discussion
- Conclusion
- References

The value of particle density computed by the DSMC method is normalized by dividing by the input value of density and then an average is found along the axis of the aperture from the values at three radii of *r*= 0.05, 0.1 and 0.2 mm from the axis. The distance along the axis is also normalized over the diameter of the aperture. The computation has been done with fixed values of *p*_{0}D= 0.001, 0.005, 0.025, 0.05, 0.1, 0.25, 0.5, 1.0 and 2.5 Pa-m. Only three typical cases, at the shown values of *p*_{0}D, are given in Fig. 2 for illustrative purposes, corresponding to gas flow in free-molecule, rarefied flow (transition) and continuum flow regimes. However, the corresponding beam transmissions are given complete in the following section. The computations have covered the distance along the aperture axis from four diameters ‘below’ (i.e. upstream) the aperture up to 20 diameters ‘above’ (i.e. downstream) the aperture. However, for clarity, only the distance up to four diameters above the aperture is shown in the figures, because the values of density are too small to be drawn visibly above this distance. Nevertheless, the complete distance is used in the computation of beam transmission later.

We note in Fig. 2 that the density curve is almost antisymmetrical around the ordinate axis for *p*_{0}D= 0.001 Pa-m corresponding to free-molecule flow. However, as we move in the transition flow region (e.g. *p*_{0}D= 0.025 pa-m) and beyond in the continuum flow (*p*_{0}D >1 Pa-m), the density curve is skewed (or pushed) to the right, that is, downstream.

The reason for stopping the computations at four diameters below the aperture is that it is found that the particle density is constant and equal to the stagnation density in the chamber below that point. The reason for stopping the computations at 20 diameters above the aperture is that it is found that the gas density above this point is negligible in the contribution to electron beam scattering by the remaining gas above it. The latter applies well to the aperture system of the present study, but this distance (above which the scattering is negligible) may be different for other design systems (Danilatos, 2001).

### Electron beam transfer

- Top of page
- Summary
- Introduction
- Materials and methods
- DSMC results
- Electron beam transfer
- Analysis of results
- Discussion
- Conclusion
- References

As given in previous works, the electron beam scattering process is governed by the Poisson distribution probability *P*(*x*):

- (1)

which gives the probability for an electron to undergo *x* number of collisions, when the average number of collisions per electron is *m*. Knowledge of the parameter *m* allows us to immediately find the electron beam current *I* that is transmitted completely without any scattering by the gas molecules, when the initial incident beam current in vacuum is *I*_{0}. The fraction of transmitted beam *f* (or electron beam transmission) is given by the exponential equation:

- (2)

The parameter *m* is found from the function of particle number density *n*(*z*) along the axis *z* and the total scattering cross-section σ_{T} of the gas:

- (3)

The integration above is performed between any two limits defining the gas layer of interest. In the present case, we are first interested in taking the definite integral from 20 diameters above the aperture progressively to a lower distance until four diameters below the aperture in the same direction in which the electron beam travels along the axis, and then we plot the result. The integral factor in Eq. (3) has been referred to as the molecular (particle) thickness, which is the total number of gas particles in a column of gas with unit base area. This quantity normalized over the input number density and the orifice diameter yields a dimensionless constant ζ, the normalized molecular (particle) number thickness, and is given by the integral:

- (4)

The latter form is directly applicable to the normalized functions of particle density presented in this work. The stagnation number density *n*_{0} relates to the chamber pressure *p*_{0} by

- (6)

where *k* is Boltzmann's constant and *T*_{0} is the absolute chamber temperature. Thus, the electron beam transmission is given by the simple formula:

- (7)

By this method, the beam transmission is plotted along the beam axis as shown in Figs. 3–11, each corresponding to a fixed value of *p*_{0}D for all the cases for which the gas density was computed previously, and each case with a set of fixed values of beam accelerating voltage in kV, as they are more likely to be used in ESEM. Thus, from each figure, we can immediately obtain the electron beam intensity at any point along the axis for electron beams of 1, 2, 3, 4, 5, 10, 15, 20 and 30 kV. The scattering cross-sections σ_{T} depend strongly on both the nature of the gas and the accelerating voltage, and the values used for argon are given in Table 1. The numerical values of gas density used in the distance from four diameters below the aperture to the 20 diameters above the aperture are closely spaced along the axis and are very large in number to be tabulated in this paper, but the set of graphs may provide sufficient information for further analysis by other workers.

Table 1. Total electron scattering cross-sections of various gases, σ_{T}× 10^{20} m^{2}, for different accelerating voltages. Accelerating voltage, kV | Argon | Neon | Helium | Nitrogen | Oxygen | Hydrogen | Water |
---|

1 | 6.77 | 1.72 | 0.287 | 4.41 | 4.24 | 0.723 | 2.87 |

2 | 3.62 | 0.946 | 0.165 | 2.41 | 2.32 | 0.406 | 1.58 |

3 | 2.50 | 0.662 | 0.118 | 1.68 | 1.61 | 0.287 | 1.10 |

4 | 1.92 | 0.513 | 0.0923 | 1.3 | 1.25 | 0.224 | 0.855 |

5 | 1.56 | 0.421 | 0.0763 | 1.06 | 1.02 | 0.184 | 0.701 |

10 | 0.827 | 0.226 | 0.0420 | 0.571 | 0.547 | 0.101 | 0.377 |

15 | 0.572 | 0.158 | 0.0296 | 0.398 | 0.381 | 0.0708 | 0.263 |

20 | 0.442 | 0.122 | 0.0232 | 0.309 | 0.296 | 0.0554 | 0.205 |

25 | 0.363 | 0.101 | 0.0192 | 0.254 | 0.244 | 0.0458 | 0.169 |

30 | 0.309 | 0.0862 | 0.165 | 0.218 | 0.208 | 0.0393 | 0.144 |

The parameter ζ is calculated by numerical means by first storing the values of the number density function *n*(*z*) produced from the DSMC method in a spreadsheet. In total, 1200 values are used in the distance from −4 to +20 diameters along the aperture axis. The spreadsheet produces the parameter ζ on a separate column from which the electron beam transmission is calculated by Eq. (7) in another column by the use of constants taken from Table 1 for each case given in the figures. Similar processing is done for the other gases given as well as for the results in the analysis and discussion that follow.

### Analysis of results

- Top of page
- Summary
- Introduction
- Materials and methods
- DSMC results
- Electron beam transfer
- Analysis of results
- Discussion
- Conclusion
- References

The results above constitute the basis for extracting information in the design of specifications for an optimum ESEM. However, some salient corollaries may not be immediately obvious without a minimal analysis attempted below.

The five parameters used, namely, aperture diameter, chamber pressure, specimen distance from aperture, beam accelerating voltage and beam transmission along the axis, are interrelated so that, given any four of them, the fifth can be uniquely determined. Alternatively, given any three of them, the remaining two parameters may be plotted against each other. A minimum number of *p*_{0}D values are provided, which are sufficient to allow reliable interpolation between these values. The parameter of normalized particle density, such as in Fig. 2, is incorporated in the beam transmission graphs and the value of it for any specimen distance greater than four diameters below the aperture can be reliably taken as equal to unity. The database can be used by a computer program to calculate these parameters in any desired fashion, and a complete presentation may be obtained via the Internet from the present author in the future.

We note in Fig. 3 that even with the lowest 1-kV beam, more than 93% of the beam is transmitted down to four diameters below the aperture. For example, with a 50-μm aperture, the maximum pressure can be 20 Pa (i.e. low vacuum) for this case of gas density distribution along the axis. As we move up with values of *p*_{0}D, the transmitted fraction of beam diminishes: for an intermediate value of *p*_{0}D= 0.025 Pa-m (Fig. 5), we still have 18% of transmitted beam with even the lowest 1-kV beam. Thus, with a 50-μm aperture, the maximum pressure can be 500 Pa, which ushers ESEM conditions of operation. With *p*_{0}D= 0.25 Pa-m (Fig. 8), we observe a better than 10% transmission at four diameters below the aperture for all beams above 10 kV. For example, with a 50-μm aperture, we can have 5000 Pa in the chamber, or with a 500-μm aperture, we can have 500 Pa with a specimen placed at 2-mm distance from the aperture. In the extreme case, where *p*_{0}D= 2.5 Pa-m (Fig. 11), no useful amount of electron beam survives at four diameters even at 30 kV. However, there is still a useful electron beam transmitted in the specimen chamber at shorter specimen distance from the aperture.

Therefore, it is important to determine the closest distance of the specimen from the aperture where the pressure is practically the same as the recorded stagnation chamber pressure, because the pressure decreases significantly as we approach the aperture. From the graphs of Fig. 2, we have calculated the normalized number thickness figures for each layer of gas one diameter thick and the results are given in Table 2. Integration of Eq. (4) inside a unity layer (i.e. one diameter thick) with a unity number density yields unity. Indeed, we note that these figures are very close to unity for all the gas layers below one diameter from the aperture, but large deviations from unity are observed in the layers immediately below and above the aperture. These observations apply first in the case when the region is free from any specimen present. The situation has already been further examined also in the case of a flat specimen placed below the aperture at different distances, because the specimen is expected to modify the flow field properties (Danilatos 1991, 1992). However, those studies have shown that even in the presence of a typical specimen, the pressure drop is insignificant or small, provided the specimen is placed no closer than one diameter from the aperture. Hence, we may assume that the flow is the same or close to the specimen-free case, as is presented in the present paper. Therefore, we can set one diameter as the limiting working distance at which it is possible to operate an ESEM and still assume that the recorded chamber pressure is practically the same as the pressure experienced over the observed specimen area. For this criterion of operation, we can extract from the previous plots the transmission factors at one diameter below the aperture versus beam accelerating voltage at fixed values of *p*_{0}D as shown in Fig. 12.

Table 2. Normalized particle thickness for unity layers below (negative) and above (positive) the aperture at three values of *p*_{0}D for argon. Layer | *p*_{0}D= 0.001 Pa-m | *p*_{0}D= 0.025 Pa-m | *p*_{0}D= 1 Pa-m |
---|

−4 | 0.99 | 1.003 | 0.988 |

−3 | 0.986 | 1.002 | 0.985 |

−2 | 0.969 | 0.995 | 0.984 |

−1 | 0.832 | 0.880 | 0.944 |

1 | 0.192 | 0.225 | 0.331 |

2 | 0.03 | 0.039 | 0.068 |

3 | 0.011 | 0.014 | 0.025 |

4 | 0.006 | 0.007 | 0.013 |

Further, we can define another criterion for those applications that allow the placement of a specimen at even shorter distance, that is, within one diameter from the aperture. For example, small specimens placed on top of a thin needle, while they can be moved freely in the vicinity close to the aperture, experience both a variable pressure and a cooling effect on account of the gas flow properties. Another example is work at or near atmospheric pressure, where the specimen is placed very close to the aperture, and the lowering of pressure over the viewed area may have no detrimental effect on the application, given that the absolute value of pressure remains very high. In fact, imaging has been undertaken successfully for this condition and beyond, in some special cases, by inserting the needle above the aperture while still inside the gas flow, and images of the upper side of the aperture have been shown by Danilatos (1985). Detection was successful both with a system of scintillating back-scattered electron (BSE) detectors placed above and below the aperture and with the gaseous detection device via a system of wires placed both below and above the aperture (Danilatos, 1990b). Gaseous scintillation avalanche detection has also been used in all those cases of operation (Danilatos, 1986, 1993). In other words, there are specimens that are not affected by a lower pressure than that recorded in the chamber, nor are they affected by gas streaming over their surface as the gas flows through the aperture. For all those applications, the ultimate criterion is the amount of electron beam transmitted down to the limiting position where imaging is still usable. The subject of establishing a lowest fraction of useful beam is relative to the application and to the amount of contrast produced by way of specimen nature and detector design as well as to the initial current intensity *I*_{0} available (or usable) in the beam depending on the electron gun type of microscope. Therefore, it is justified to extract information from the previous plots as to what happens, for example, at zero distance from the aperture, a condition to which we may refer as the ‘physical performance limit’ of operation in the design of an optimum ESEM.

As a result of the latter criterion as an ultimate or limiting operational case, the corresponding gas particle thickness, that is, the definite integral ζ_{c},

- (8)

has been defined as a characteristic normalized number thickness constant for any thin aperture, to which we may refer as ‘critical particle thickness’, for short, in this work. From this, we get a corresponding critical electron beam transmission *f*_{c}:

- (9)

The values of ζ_{c} have been tabulated in a previous work (Danilatos, 2000a) at or very close to the aperture axis, whereas herein, we provide the averages of gas density among three radii along the aperture axis. These averages, being slightly less than the peak values, are used now, because a converging (cone of) electron beam scanned around the axis of the aperture experiences an average gas density rather than the peak value. In either case, the difference is small for any practical purposes, and the average values 〈ζ_{c}〉 are now provided as data points for argon in Fig. 13.

Following the same procedure as in previous work, the points have been fitted again with the curve:

- (10)

where the new constants *a*, *b*, *c*, *d* and *e* are given in Table 3, with *y*≡〈ζ_{c}〉 and *x*≡*p*_{0}D. The overall estimated error on the graphs is ±5% towards the continuum flow end (right) and less than ±1% towards the molecular flow end (left).

Table 3. Constants of Eq. (10) for various gases. Gas | *a* | *b* | *c* | *d* | *e* |
---|

Argon | 4.0915 | 23.5058 | 45.9232 | 1.3150 | 2.9067 |

Helium | 4.1206 | 11.6470 | 29.8438 | 24.2696 | 47.5001 |

Hydrogen | 4.0961 | 73.2387 | 281.1271 | 1212.3227 | 2752.7335 |

Neon | 4.0226 | 9.7785 | 20.2649 | 11.7383 | 23.6059 |

Nitrogen | 4.0554 | 23.3237 | 54.0130 | 147.3236 | 350.4308 |

Oxygen | 4.0628 | 24.0250 | 55.6308 | 90.8792 | 215.0981 |

Water vapour | 4.0394 | 26.1171 | 60.6801 | 28.1944 | 72.3453 |

Following the same procedures as with argon, other gases have been studied with the DSMC method and the corresponding constants of curve fits are given also in Table 3 for helium, neon, nitrogen, oxygen, hydrogen and water vapour (data points are omitted for clarity). These new results partly confirm some theoretical considerations in the previous paper, especially towards the free molecule flow and for monatomic gases, but significant deviation occurs towards the continuum flow of molecular gases. The expectation previously expressed that the specific-heat ratio could affect the critical particle thickness seems to be born out of the new studies, as the gases appear to be grouped according to the number of degrees of freedom with reference to the equipartition of energy theorem. A modified theory of another possible transformation formula among gases is beyond the purposes of this paper.

Now, we can use the fitted curve of Eq. (10) for argon to show a family of critical transmission versus *p*_{0}D curves for a set of fixed accelerating voltages, as in Fig. 14. We can do the same for any other gas if we are given reliable experimental electron scattering cross-sections. No such data have been found yet for the purposes of Table 1, except the equations and computations provided for several gases first by Danilatos (1988), based on existing theoretical derivations of scattering cross-sections of gases, which may or may not be reliable. Those computed values are reproduced in Table 1. On the basis of these values, we make a comparative presentation of the critical transmission *f*_{c} for various gases for two typical (low and high) accelerating voltages of 5 and 30 kV in Figs. 15 and 16, correspondingly.

### Discussion

- Top of page
- Summary
- Introduction
- Materials and methods
- DSMC results
- Electron beam transfer
- Analysis of results
- Discussion
- Conclusion
- References

For high accelerating voltage and low *p*_{0}D, the beam is transmitted with negligible loss. However, the highest demand of ESEM usage is at low kV and preferably high *p*_{0}D conditions, which necessitate working as close to the aperture as possible. This makes the present investigation imperative, as it is clear that the beam may undergo critical or catastrophic loss even before the beam enters the specimen chamber. It is important to note that argon gas has the worst transmission rate of all gases presented here, which rank in the following order transmission-wise:

Nitrogen is very close to oxygen and hence very close to air. Then follows water vapour as an even better performing gas, which is fortunate, as this is the most used gas in ESEM. This also explains why many users ‘prefer’ water vapour gas as a general imaging gas, apart from its ‘wetting’ properties. Should an application require an increased pressure *per se*, then helium is on top of the transmission rate list. This classification of gases is important to bear in mind, if we use argon as a reference gas for grading the electron transmission performance of an instrument. At present, argon renders the best reference to calibrate or determine the performance specifications of any given ESEM instrument. In this sense, the main purpose of this work is accomplished, and the results for other gases are included herein for immediate comparative and practical purposes, but they should be subjected to further investigations in the future, in continuation and in conjunction with the theoretical and practical groundwork presented herein.

Notably, with reference to Fig. 15, for 30% transmission, we have *p*_{0}D= 0.663 Pa-m for argon and *p*_{0}D= 1.781 Pa-m for water vapour, a difference that has a significant practical outcome. For example, with a 500-μm aperture, the corresponding pressure is 1326 Pa for argon but the corresponding pressure increases to 3562 Pa for water vapour. Furthermore, by the use of a 50-μm aperture with argon again, the corresponding pressure increases to 13 260 Pa. The latter finding for a small aperture is not a mere theoretical exercise, because it has a real practical consequence. To start with, the much smaller PLA does not necessarily imply a smaller field of view if it is employed in conjunction with a new configuration of scanning coils and a pair of apertures according to the patent US 6809322 B2 (Danilatos, 2004). At low magnification, fields of view greater than 1 and 2 mm have been achieved with apertures as small as 50 and 100 μm. By this approach, the advantages are immense. Relatively low-kV electron beams are feasible at relatively high pressures (i.e. in comparison with existing commercial instrument performance). Additionally, the gas leak rate through the small PLA is greatly reduced in comparison to the same field of view achieved by the means hitherto employed by ESEM manufacturers. In fact, this study shows that it is the wrong way to use large apertures in order to increase the field of view, as is the current practice in commercial instruments. The leak rate *N* through the PLA is given by:

- (11)

where *s* is the characteristic aperture speed as given by Danilatos (2000a). The speed *s* is a function of *p*_{0}D. Thus, for any fixed value of *p*_{0}D (similarity condition), the leak rate is proportional to the diameter of the aperture. Therefore, in the above numerical example, by going from a 500-μm to a 50-μm PLA, we reduce the leak rate 10-fold, and at the same time, we increase the pressure 10-fold and still maintain a 30% electron beam transmission rate at 5 kV! Increasing the diameter of the PLA as a means to achieve large field of view has catastrophic results on the performance of ESEM, which, in practice, is reduced to a low-vacuum instrument only.

The results of this work are consistent with practice. For example, work has been performed routinely at room temperature and saturation water vapour pressure with low-kV beam without the need to use a cooling stage to lower the water pressure. Furthermore, very good imaging has been achieved up to a pressure of one atmosphere with a 100- to 150-μm aperture. In the light of the present analysis, this experience may be interpreted in either of the two ways: (a) very low transmissions can still produce satisfactory imaging or (b) the scattering cross-sections used in the calculations are overestimated. Therefore, further work is required (a) to experimentally quantify simultaneously the electron beam transmission, together with the absolute incident beam current required to produce satisfactory imaging on a standard specimen, and (b) to experimentally confirm or correct the electron scattering cross-section of various gases.

In relation to the proposal of experimentally establishing the scattering cross-sections of gases, we may consider the following approach: for a homogeneous gas layer with constant pressure and temperature, Eq. (7) reduces to an exponential function of distance only:

- (12)

where *I* is the measured beam current at the exit plane and *I*_{i} is the ‘initial’ current at the entry plane of a gas layer of thickness *z*. By plotting the logarithm of *I/I*_{i} current ratio versus distance *z*, we can immediately deduce the scattering cross-section from the slope of the straight-line graph. Furthermore, by taking additional measurements as close as possible to the aperture, we can establish the point at which the measurements deviate from the straight line. This deviation signifies that we have gone beyond the area of gas homogeneity and entered the region of gas flow with variable properties. The latter region may be a combined effect of the interaction of the measuring probe device with the aperture geometry. If the probe is sufficiently small so as not to affect the gas flow in any significant manner, then the deviation from the straight line would be entirely due to the depletion zone of the gas flowing below the aperture. The latter situation can be readily seen by replotting the same curves, for example, of Fig. 7 but on log-linear axes, as in Fig. 17, where the straight lines deviate within the first one-diameter gas layer below the aperture.

For the measurement of cross-sections, care should be taken to initially use a dilute gas with long working distance and low electron beam current. This will ensure a weak ionization of the gas in order to avoid any speculated effects from a strongly ionized medium that might result in self-focussing of the electron beam or other unexpected effects bearing on the accuracy of measurements (Danilatos, 1988). After we obtain definitive values for the cross-sections, we may proceed to repeat the same measurements progressively at elevated gas pressure and high beam current (with strongly ionized gas) in order to establish the existence or not of any deviation from the expected trend of beam transmission; this would indicate the presence of new phenomena worth studying in the ESEM. Therefore, the entire task is reduced to making and using an appropriate electron current-measuring device. The latter probe requires care on account of the presence of ionization currents and contamination effects. This problem has been surveyed in detail by Danilatos (1988, pp. 171–178), and a shielded Faraday cage, together with a small-diameter, heated platinum wire, was designed to overcome these difficulties. The electron beam is scanned across the edge of the heated wire and the transmitted beam current is recorded either by an electrometer or by the profile of the beam on an image. This device was initially used to demonstrate the constancy of beam diameter with increase of pressure, but it can also be used at constant pressure to measure the transmitted beam current versus distance. From such measurements, the scattering cross-sections can be determined readily and reliably, once and for all. By such means, an experimentally obtained cross-section σ_{Te} would correspond to an experimental electron transmission *f*_{e}. The latter enters into a ratio relationship over the transmission provided in Eq. (7) as follows:

- (13)

By the simple derivation above, we can see when the precision in cross-section values can become important. That is, when the factor *n*_{0}Dζ becomes large enough, it can produce a significant effect on the ratio of transmissions for a given difference in cross-sections; this varies greatly from the lowest values of *p*_{0}D with practically no effect to the highest ones with significant effect, all cases of which are applicable to ESEM design and operation.

The design, construction and use of the above electron current probe device have additional benefits as it can be used for the experimental determination of the particle number thickness of gas at any point along the aperture axis, provided the device does not affect the flow properties of the gas. Having established the correct experimental cross-section for a given gas, we can then measure the beam transmission at any given point and given pressure, from which we can deduce the particle number thickness from Eq. (7). In an optimum design instrument, the experimentally measured transmissions for argon should be the same as those obtained for argon on the basis of the DSMC work. Conversely, with such an optimum instrument, we can then proceed to establish the same optimum design characteristics by the use of any other gas also as a calibrating medium. As a spinoff, the probe-measuring device may also be used as a pressure gauge of the instrument following an appropriate calibration. Therefore, it is proposed that any commercial instrument should be accompanied with an electron current probe device in order to demonstrate the claimed performance of the instrument, or check its performance at regular intervals on account of contamination, gas leakage or other malfunction, and conduct other experiments and measurements. It is also a future task to miniaturize the probe device by nanotechnology microfabrication so that measurements can be made at a close range to the aperture as well.

Finally, we should clarify that although we have established the absolute physical limits for electron beam transfer in an ESEM using the differential pumping method between specimen chamber and electron optics column, those limits can still be overcome in an ESEM employing the method of reverse-flow PLA (Danilatos, 2000b). By such a method, an annular gas jet flows around the PLA in the direction of the electron beam, thus creating a pumping action in its core and no gas jet forms above the aperture. This significantly reduces the electron beam loss above the aperture that would otherwise occur by conventional differential pumping.

Later publications will present work on (a) flat apertures with varying thickness wall, which corroborates the claim that the thin aperture is an optimum design, and (b) conical apertures with a thin rim at various cone angles, which shows that the characteristic particle number thickness is close to that of a thin aperture, provided the cone angle is greater than 45°.

### Conclusion

- Top of page
- Summary
- Introduction
- Materials and methods
- DSMC results
- Electron beam transfer
- Analysis of results
- Discussion
- Conclusion
- References

This study has dealt first with the computation of the density gradients of argon as it flows through the PLA of an ESEM. With the density known as a function of position, the particle number thickness has been found by integration along any gas layer that is penetrated by the electron beam. Based on this, the intensity of beam transmitted without any scattering has been established along the aperture axis, which is necessary for a differential pumping system as is used in ESEM. The geometry of the PLA determines the gas gradient and the resulting beam transmission. A sharp edge and small-diameter aperture represent the sharpest gradient possible in the transition from high to low pressure, and it is for this geometry that this work has been presented. The results cover the complete pressure range from near vacuum to one atmosphere, together with any aperture diameter that may be used as a PLA in an ESEM. These parameters cover the entire gas flow regime from free molecule to continuum flow. Furthermore, the electron beam transmission has been presented for accelerating voltages between 1 and 30 kV.

Argon has been used as the reference gas to compare past, present and future measurements, and the results presented for a thin aperture can serve as a benchmark in the design of specifications of an optimum ESEM instrument. The critical particle thickness has been used to define an ultimate physical limit of operation. The limits of beam transmission and operation have also been presented for other gases, including water vapour, pending verification of their electron scattering cross-sections derived from basic theoretical formulas. The ESEM can be used in the future for expanding the present work according to certain proposals made.

Most important, in order to fully utilize the present findings, a redesign of the configuration of the scanning coils and PLAs is necessary and due if we wish to eliminate the ‘tunnel’ vision at low magnifications. The practice of increasing the PLA diameter, as a means to increase the field of view, quickly reduces the working pressure range on account of increased gas leak and increased gas density gradients in the aperture holder of commercial instruments. This problem has been overcome with a latest development (Danilatos, 2004) that allows the use of a much smaller and thinner PLA with simultaneous much greater field of view than that hitherto used. The practical consequences of such means are enormous:

Clearly, an ESEM optimally designed to operate at high pressure is automatically superior in performance at low pressure or low vacuum without further ado. A proper ESEM is the one that allows the user to operate at room temperature, under fully wet conditions, at low kV (less than 5 kV), with sufficient space to move the specimen, or place detectors, and allows maximum number of applications and ancillary devices (like a microinjector) to be used. We get better resolution on delicate specimens with much less beam damage at higher magnifications and resolutions. A smaller PLA means less probability to contaminate the upper column and allows long electron gun life, less servicing requirements and less instrument downtimes. Contrary to this, the consequence of using a cylindrical instead of a thin-wall PLA has far more destructive effects in addition to the beam loss. The cylindrical geometry (used in commercial instruments) has a large inside surface, which is prone to faster contamination. Any debris in contact with the surface gets irradiated and firmly stuck on the surface. At low pressure, or if the user wishes to revert to vacuum operation, any contamination inside the PLAs will create serious astigmatism on account of charging. The charging can be so great by the incident beam that even at increased chamber pressure, the gaseous ionization may not suffice to balance off the amount of charging and hence imaging becomes problematic and a general malady for the instrument over all. Furthermore, the position and shape of the second (upper) PLA, together with the overall aperture holder assembly, are critical in preventing contamination of the upper column, which can drastically reduce the lifetime of the electron gun and affect normal or efficient electron probe formation. Manufacturers should address these problems both in new instruments and in the ones already out in the market. In this context, the present paper makes a contribution in the elimination of existing problems towards an optimum design ESEM.