Transfer Matrix of the Uniform Field
Equations (1.19) and (1.20) provide a detailed analysis of the trajectory's exit ordinate and slope after traversing a uniform field In the paraxial case (α₁ ≈ 1, as illustrated in Fig 1.13), the expressions simplify significantly, yielding sinα₁ ≈ r₁, cosα₁ ≈ 1, r₁ V₂/V₁, and r₂ ≈ 2Lr₁.
In the general case, with r 1 = 0, one has therefore r 2 =r 1 + 2L
Fig 1.13 Paraxial case of acceleration of charged particle through a uniform field or in matrix form r r
The transfer matrix coefficients of the uniform electrostatic field are thus a 11 = 1, a 12 = 2L
This is valid not only for acceleration (V 2 /V 1 >1), but also for decel- eration (V 2 /V 1 1), drift (V 2 /V 1 = 1) or deceleration (V 2 /V 1 < 1) From the above matrix coefficients those of the uniform field can be derived by performing the transitionsR 1 , R 2 →
Acceleration of Charged Particles Emitted
A practical example of acceleration of charged particles from a spherical surface are tip sources of electrons or ions (Fig 1.17) In these cases
Fig 1.17 Acceleration of charged particles between concentric spherical emitting and accelerating equipotential surfaces the acceleration energyeV a is usually large in comparison to the initial energyeV 1
The tip radius \( R_t \) is significantly smaller than the distance to the acceleration electrode, making its shape irrelevant, as the primary acceleration occurs within a few tip radii in the field defined by the tip Consequently, particles that begin at a glancing angle from the surface exhibit a slope of 2 after being accelerated by \( V_a \).
V 1 /V a against the surface normal, as compared to
In the planar case, the virtual starting point is positioned at a distance of R t /2 behind the emitting surface, resulting in the emission being projected onto a virtual surface with a radius of R t /2 Notably, all trajectories exhibit a maximum slope of 2.
V 1 /V 2 to the normal of their start- ing points, when extended backwards to the center of the semi-sphere, have a distance of R 2 t 2
The concept of a virtual source radius allows for the realization of microsources using straightforward methods For instance, thermionically emitted particles possess initial energies below 1 eV, which can be effectively accelerated with a suitable voltage.
At an acceleration voltage of 10 kV, the virtual source radius is significantly smaller, measuring 100 times less than the tip radius This relationship holds true only when the acceleration voltage is not obstructed by a Wehnelt electrode, as the presence of this electrode can substantially modify the electric field surrounding the tip.
Passage of Charged Particles Through an Electrode
an Electrode with Round Aperture
An acceleration field, depicted in Fig 1.15, is concluded by a planar electrode featuring a centrally located aperture for the passage of accelerated particles (Fig 1.18) This aperture leads to a bulging of the equipotential surfaces towards the field-free space Near the hole, the field strength reveals a radial component, E r, which serves to deflect particles away from the axis.
The round opening functions as a lens, demonstrated by inserting a fictitious cylinder through the hole, with one end in a uniform field and the other in free space By applying the conservation of field line flux in the absence of space charge, we derive the relationship: r²πE_a + 2rπ.
A charged particle passing from the left at the distancer from the axis experiences a radial momentum: mv r =− eE r dt=−e v z
Fig 1.18 Diverging action of aperture terminating an acceleration field
The lens action of an aperture in an acceleration field is derived by substituting the relationship v_z = dz/dt, leading to dt = dz/v_z Given that the trajectory is paraxial, v_z is treated as a constant By integrating the expression from (1.35), we arrive at the equation mv_r = er.
The trajectory suffers a kink ∆r (Fig 1.20) given by, with (1.37), ∆r v r /v z = erE a /2mv z 2 With mv z 2 /2 = eV, the particle energy, we then obtain
The trajectory deflection ∆r thus is proportional to the distance from the axis This is the characterization of a lens The focal length is given by (comp (1.4))
In practical scenarios, the acceleration energy \( eV_a \) is significantly greater than the initial particle energy Therefore, the initial particle energy at the aperture can be substituted with \( eV_a \), leading to the equation \( E_a = V_a / L \) (refer to Fig 1.18).
The aperture acts as a diverging lens with the focal length −4L [4].
A particle initiating its motion off-axis from point A, perpendicular to the surface, seems to originate from the axis at a distance of 3L behind the emitting surface This scenario involves a virtual surface positioned at a distance of L.
Fig 1.20 Deflection of trajectories passing through the aperture
The imaging of the emitting surface, facilitated by the accelerating field and the aperture positioned behind it, is captured at a distance of 4/3L to the left of the aperture, which corresponds to L/3 behind the emitting surface.
This can easily be derived geometrically (see Fig 1.21) or by calcu- lation with the lens formula (1.8):
The focal length f 2 is that of the aperture according to (1.39).
General Aperture
Equation (1.39) is just a special case of the general formula for the lens effect of an aperture separating two regions of different field strengths (Fig 1.22) [4]: f = 4V a
The relationship between the aperture voltage (V_a) and the field strengths (E_1 and E_2) is given by the equation E_2 - E_1 Here, V_a represents the particle energy measured against the particle source, with E_1 and E_2 indicating the electric field strengths on either side of the aperture Positive field strengths accelerate particles, while negative field strengths decelerate them When E_1 is greater than E_2, the focal length is negative, resulting in a diverging lens effect; conversely, if E_1 is less than E_2, the aperture behaves as a converging lens, indicated by a positive focal length.
The article illustrates several scenarios involving lenses, categorized into diverging and converging types Cases (a) to (e) represent diverging lenses, characterized by a concave curvature in the potential curve when viewed from the zero potential side In contrast, cases (f) to (k) depict converging lenses, which feature a convex curvature This distinction is crucial for understanding the behavior of different lens types in optics.
The transfer matrix of such an aperture is thus (comp (1.6)) r r
The properties of lenses composed of more than one aperture can be calculated by transfer matrix multiplication.
The configuration depicted in Fig 1.24 can be described using the matrix product of transfer matrices, which includes the first aperture, a uniform field, and the second aperture In this setup, particles exhibit varying energies before and after the arrangement, specifically eV 1 and eV 2, and this system is referred to as an "immersion lens." This term is analogous to light optics, where it denotes a lens with differing indices of refraction on either side Additionally, the arrangement illustrated in Fig 1.25 is defined by the matrix product of five transfer matrices.
first aperture ×first uniform field ×second aperture ×second uni- form field ×third aperture.
It represents a so-called einzel lens when the particle energy is the same before and behind the lens, eV 3 =eV 1
In this way, the lens properties, such as focal lengths and position of principal planes, can be calculated.
When the voltage V3 differs from V1, a three-aperture immersion lens is utilized Unlike a two-aperture immersion lens, where the focal lengths remain constant for a specific V2/V1 ratio, the three-aperture immersion lens allows for greater flexibility and adaptability in optical performance.
Fig 1.23 Various cases of aperture lenses: (a) to (e) diverging lenses, (f) to (k) converging lenses
Fig 1.24 Uniform field between two aperture lenses, constituting an “immersion lens”
Fig 1.25 Combination of three apertures and two fields constituting an einzel lens (V 3 = V 1 ) or an immersion lens (V 3 = V 1 ) possibility of adjusting the focusing action for any ratioV 3 /V 1 by varying
V 2 Such lenses are electrostatic “zoom lenses”.
When calculating lenses using this method, optimal results are achieved primarily when the aperture diameters are small relative to the distances between them If this condition is not met, the fields between the apertures become non-uniform, leading to significant discrepancies in the axial potentials compared to the potentials applied to the aperture electrodes This limitation can be expressed by the inequality L ∆V / E, which must be satisfied on both sides of an aperture where an electric field is present, with ∆V representing the potential difference to the adjacent aperture electrode.
When the aperture diameter exceeds the specified condition, the formula remains effective, provided that the axis point within the aperture is considered instead of the potential of the aperture itself The difference between these two measurements increases with larger aperture diameters, especially in relation to the distances between adjacent apertures This discrepancy is particularly notable when V a reaches its maximum or minimum values.
Passage of Charged Particles Through an Electrode
an Electrode with Slotted Aperture
When the aperture in an electrode separating spaces of different field strength is not circular but a slot (Fig 1.26), such that its width
Fig 1.26 Slotted aperture between different fields
(y-direction) is small in comparison to its length (x-direction), the lens action is only in the y-direction.
The formula for the focal length (comp (1.40)) is then f = 2V a
The lens action is twice as strong as that of a circular aperture, but occurs only in one azimuthal direction.
In the x-direction no focusing of the trajectories occurs but only a variation of the incremental change of the refractive index Parallel trajectories continue being parallel (Fig 1.27).
When particles emitted from a planar surface are accelerated through a slotted electrode, the defocusing effect caused by the slot must be considered, particularly when the voltage is set at V1 V a.
The virtual subject located at the distanceL behind the emitting sur- face is now imaged to the emitting surface itself with the magnification r b /r a = 1/2.
The beam radius at the aperture is of course the same as with a round aperture r B =r a + 2L
Fig 1.27 Action of slotted aperture between different fields: (a) lens action, (b) bending of trajectories
Fig 1.28 Imaging of emitting surface by acceleration field and slotted aperture but the maximum beam angle in the direction of the lens action is now α max V 1
Emission Lenses
The usual requirement in dealing with charged particles emitted from a surface is to form them into a beam with a certain energy and shape,
The emission lens, also known as a cathode lens for electrons, operates in two distinct modes to effectively form an image of a surface.
The simplest emission lens is obtained by adding a second apertured electrode to the acceleration electrode (Fig 1.29).
The two electrodes and their apertures create an immersion lens, which, in conjunction with the uniform acceleration field, forms the emission lens This lens can be analyzed by multiplying the matrix of the first aperture lens with that of the combination of the field along L2 and the second aperture lens.
⎠, where the focal lengths f 1 and f 2 of the two apertures are given by
The matrix coefficients are found to be m 11 = 1− 1 f 1
Several options are open to shape the beam for given distances L 1 and
L 2 by variation of the voltage ratioV 1 /V 2 The valueeV 2 is the energy of the beam after passing through the lens.
Telescopic imaging is a crucial application for beam sources, where particles emitted from various points on a surface travel along parallel trajectories after being accelerated by the emission lens Notably, particles that begin parallel to the axis maintain this parallelism post-lens, with any angular spread resulting solely from their initial energy.
The condition for telescopic imaging is that r 2 be independent of r 1 , or in particular that forr 1 = 0 also r 2 = 0 This is the case for m 21 = 0 (1.48)
Fig 1.30 Telescopic imaging through emission lens by converging–diverging apertures
Fig 1.31 Telescopic imaging through emission lens by diverging–converging apertures
The emission lens thus has the focal length f =−1/m 21 =∞.
Two solutions can be easily calculated using a pocket calculator, as illustrated in Figs 1.30 and 1.31 For the equation L1/L2 = L, the ratios V1/V2 are determined to be 0.34 and 2.8 The first solution, depicted in Fig 1.30, shows the first aperture functioning as a focusing lens while the second acts as a defocusing lens Conversely, the second solution in Fig 1.31 presents the first aperture as a defocusing lens and the second as a focusing lens This configuration is referred to as an accel-decel arrangement, where particles are accelerated in L1 and decelerated in L2 A comparison of both solutions reveals that, for a specified final beam energy eV2, the field strength varies accordingly.
In scenarios depicted in Fig 1.31, the electric field strength (L1) is significantly elevated, which is crucial when space charge limitations affect the emitting surface, as it enables a higher emission current density However, for a specified beam energy (eV2), the voltage (V1) required in these cases may reach excessively high levels, potentially leading to breakdown issues Additionally, it is essential to evaluate other beam parameters, including the location of the virtual emitting surface.
Table 1.1 Telescopic imaging with emission lens after acceleration field (L 1 =
The calculated values assume that the aperture diameters are significantly smaller than the distances between the apertures While this scenario is uncommon in real-world applications, adjusting V1 allows for the condition where f approaches infinity to be achieved.
The virtual object radius r a is found (Fig 1.31) from r 1 =r 0 ,r 1 = 0 forr 0 = 0: r a = (r 2 ) r
1 =0=m 11 r 1 , r a r 0 =m 11 (1.50) The maximum beam diameter and aperture angle are obtained with (1.46) and (1.47) by inserting r 1m V 0 /V 1 (comp (1.32)), r 1 r 0 + 2L 1 r 1m : r 2m =m 11 r 0 + (2L 1 m 11 +m 12 )
V 0 /V 1 , (1.52) where r 0 is the radius of the emitting area on the surface.
With the above relations the properties of the emission lens with telescopic imaging are completely described.
Table 1.1 presents the values for L1, L2, and L A significant application of an emission lens featuring two aper- tured electrodes is in emission microscopy, which creates a magnified image of the surface from emitted particles For effective imaging at a distance much larger than L1 and L2, the emitted trajectories must remain parallel to the axis after passing through the emission lens This condition is expressed mathematically as r1 = 2L1r1 and r2 r0 = 0 = m21r1 + m22r1.
Again, there are, for any ratioL 1 /L 2 , two solutionsV 1 /V 2 meeting the above condition (Figs 1.32 and 1.33) With the same example as above
L 1 =L 2 =L, these solutions are V 1 /V 2 = 0.18 and V 1 /V 2 = 4.5 [5].Again, as in telescopic imaging, the second solution represents a large accel–decel voltage ratio, and the same applies as above.
Fig 1.32 Emission microscopy: Imaging of surface to infinity by converging– diverging apertures
Fig 1.33 Emission microscopy: Imaging of surface to infinity by diverging– converging apertures
The focal length of the emission lens is found from the relation (with r 1 =r 0 ,r 1 =r 0 = 0). f = r 0 r 2 =− 1 m 21 (1.54)
The backfocal plane is where a trajectory starting with r 0 = 0,r 0 = 0 crosses the axis Its distance from the second electrode of the emission lens is found from (with r 1 =r 0 , r 1 =r 0 = 0) g r 2
Table 1.2 Imaging of a planar surface to infinity with a two-aperture emission lens after the acceleration field (L 1 = L 2 = L)
The distance of the principal plane from the second electrode of the emission lens is h=g−f.
In the backfocal plane, all beamlets originating from individual points on the surface intersect the axis, resulting in a total beam that has a waist The radius of this waist is determined by the formula r²_max = 2L₁m₁₁r₁m + m₁₂r₁m = (2L₁m₁₁ + m₁₂).
When an aperture stop is positioned in the back-focal plane, it effectively removes a peripheral portion of the beam waist, allowing for the discrimination of particles with higher initial energies, particularly those with larger initial angles (α 1) This technique is commonly utilized in emission microscopy to enhance lateral resolution, which is influenced by the parameter V 0.
To create a surface image at a specific distance (b), rather than at infinity, the focal length (f) must be adjusted slightly This adjustment can be achieved by modifying the ratios V1/V2, resulting in a shorter focal length for Fig 1.32 or a longer focal length for Fig 1.33 The magnification (M) is determined by the ratio of the distance (b) to the focal length (f), expressed as M = b/f.
Values for the exampleL 1 =L 2 =L are listed in Table 1.2.
Using a two-electrode emission lens, the ratio V1/V2 is fixed for a specific beam energy and distances L1 and L2, which also establishes the field strength at the emitting surface (E1 = V1/L1) However, introducing a third aperture electrode enhances operational flexibility This addition allows for the adjustment of the field strength at the emitting surface to any desired value within certain limits, enabling the beam to be accelerated to a final energy (eV3) with customizable focusing conditions by varying the potential (V2) of the second electrode.
The system can be analyzed using a transfer matrix approach, achieved by multiplying the matrix corresponding to the lens in Fig 1.29 with the matrix from Fig 1.21 This method follows the procedure established for the two-electrode system Typically, this results in two possible solutions.
Fig 1.34 Emission lens with three apertures, two operating modes indicated in Fig 1.34 When V 1 = V 3 , the three electrodes act as an einzel lens (see below).
The accuracy of treating an optical system with transfer matrices of uniform fields and aperture lenses is primarily valid when electrodes are planar and parallel, and aperture diameters are relatively small compared to electrode distances However, in practical applications, electrodes often need to be conical to accommodate primary radiation, leading to larger bores to minimize lens aberrations This results in a transition from a segmented axial potential distribution to a smooth curve with gradual bends Consequently, lens actions are more broadly distributed along the potential curve rather than being sharply localized at the apertures The first-order optical properties of the system are entirely determined by the axial potential distribution, as dictated by the Laplace equation, divV = 0.
, which for rotationally sym- metric systems can be written as
Fig 1.35 Axial potential distributions of two emission lenses with similar optical properties (comp Fig 1.23)
When the axial potential distributionV(z) r=0 is given, the paraxial po- tential distribution is also fixed through (1.57), and with it the paraxial (first order) focusing properties of the system.
Even with significant differences in electrode design, similar axial potential distributions lead to comparable optical properties in different systems A key benefit of particle optics over light optics is the ability to continuously adjust optical properties through straightforward variations in potential.
In a triode system with large apertures, the potential V1 can be set to low or negative values when V1 is less than V2, leading to the term "Wehnelt" electrode The primary acceleration from the surface is achieved through field penetration from the second electrode (V2), resulting in a strong focusing effect near the surface, where the beam waist, or crossover, occurs within the acceleration field This configuration allows the beam crossover to act as a source for further imaging, while the Wehnelt electrode effectively controls the beam intensity.
Immersion Lenses
The transfer matrix of an immersion lens composed of two planar aper- tured electrodes has been given already In the general case there is no
Fig 1.36 Immersion lens with acceleration (V 2 > V 1 ) or deceleration (V 2 < V 1 )
field before the first electrode (Fig 1.36) The focal length of the first aperture is given by
The matrix coefficients are the same as given earlier except that 1/f 1 is different here.
The two possible cases with acceleration (V 2 /V 1 >1) and deceler- ation (V 2 /V 1 < 1) are shown in Fig 1.36 For V 2 /V 1 = 1 the matrix coefficients become those for a drift space of lengthL.
Focal length f 2 and distance g 2 of the focal plane in the V 2 space are given by f 2 =−1/m 21 , (1.59) g 2 =−m 11 /m 21 (1.60)
The distance of the principal plane from the second electrode is h 2 =g 2 −f 2 = 1−m 11 m 21 = m 12 m 21 f 1 (1.61) When (1.59) is carried out, it simplifies to f 2
Since both V 1 and V 2 have to be positive to let particles pass through the apertures, the focal length f is always positive for any ratio V 1 /V 2
It is confirmed that, except for the trivial case where V2 equals V1, the relationship between V1 and V2 can be verified by substituting values into equation (1.62) for both V1/V2 less than 1 and V1/V2 greater than 1 Consequently, operating a lens with telescopic imaging, as depicted in Fig 1.30 (which includes an acceleration field and an immersion lens), is not feasible This indicates that any alteration in beam energy is inherently associated with the focusing of the beam by the accelerating or decelerating field.
An immersion lens exhibits distinct focal lengths on its entrance and exit sides, which is crucial in electrostatic optics where particle trajectories are reversible The optical parameters on the entrance side, denoted as f1, g1, and h1, can be determined by applying the inverse values of V1/V2 using equations (1.59)–(1.61) Specifically, the focal length f1 can be derived from equation (1.62).
V 2 /V 1 +V 1 /V 2 −2 (1.63) When we form the ratio f 2 /f 1 we find with (1.62) and (1.63) f 2 f 1 V 2
This is a general property of all immersion lenses It is analogous to immersion lenses in light optics, where f 2 /f 1 =n 2 /n 1 , the ratio of the refractive indices on both sides.
Immersion lenses are commonly designed with tubular electrodes, which allows for a smaller lens diameter relative to the beam diameter This design results in a smoother axial potential distribution, reducing spherical image aberration It is important to note that the field penetration within the tubes significantly decreases to negligible levels at a depth approximately equal to the inner tube diameter, necessitating that this area remains free of any structural components that could disrupt the field distribution.
The lens properties are determined experimentally or by computa- tion A variety of lenses with different geometries have been published in tabulated form [6–9].
In tubular lenses, the principal planes are consistently positioned on the "slower" side of the lens field, as illustrated in Fig 1.37 Additionally, it is common for these planes to be interchanged, resulting in the light trajectories crossing P2 before reaching P1.
Fig 1.37 Immersion lens with tubular electrodes
Fig 1.38 Graphic construction of imaging through immersion lens
When the focal lengths and principal planes are known for a given ratioV 1 /V 2 – their position depends onV 1 /V 2 – then the imaging prop- erties can be easily found (Fig 1.38).
Trajectory 1, going through the axis points of P 1 and P 2 , is only refracted: r 2 = r 1
V 1 /V 2 (comp (1.21)) For trajectory 2, passing through the lens at some distance r 1 from the axis, we have r 2 =r 1
V 1 −r 1 f 2 , (1.65) where the first term denotes the refraction and the second the focusing. The equation of trajectory 1 in the r–z coordinate system (V 2 -space) is thus r(V 2 ) =r 1 +zr 2 =r 1 +z r 1
, (1.66) and withr 1 =ar 1 (a= object distance) andf 1 /f 2 V 1 /V 2 r(V 2 ) =r 1 a+z
Trajectory 1 crosses the axis (r(V 2 ) = 0) at the distance z = b This inserted in (1.67) yields the imaging equation for immersion lenses
The inverse image distance is thus found to be
The magnification is found to be (see Fig 1.38)
From the figure one can extract the relations r a a−f 1 = r b f 1 ; r b b−f 2 = r a f 2
Fig 1.39 Einzel lens built of three planar apertured electrodes
The imaging equation, also known as Newton's imaging equation in light optics, is applicable to various lens configurations, including two-electrode immersion lenses and those with differing axial potentials This flexibility extends to three-electrode immersion lenses, which are commonly used due to their ability to independently adjust energy change and beam focusing within specific limits These lenses are designed similarly to einzel lenses, with the distinction that the first and last electrodes operate at different potentials.
Einzel Lenses
Einzel lenses, characterized by three electrodes with the first and third at the same potential, are always focusing lenses similar to immersion lenses An example of a symmetric einzel lens made of apertured electrodes illustrates a potential distribution that resembles a saddle surface, with equipotential lines acting as topographic level lines These lenses can operate in two modes: decel–accel and accel–decel, as depicted in the axial potential distribution The lens strength, denoted as D/f, is also a critical factor in their functionality.
The refractive power of an einzel lens varies with the voltage of the middle electrode, exhibiting two distinct branches corresponding to different operational modes The decel–accel mode is preferred in practical applications due to two main factors: it allows the center electrode to maintain a voltage similar to the particle source, which is typically floating with respect to ground, enabling easy voltage supply through a divider Additionally, this mode achieves high refractive power and short focal lengths with voltages comparable to the source In contrast, the accel–decel mode necessitates a separate voltage supply with opposite polarity, requiring significantly higher voltages to achieve the same refractive power.
The accel-decel mode offers advantages for longer focal lengths, as it minimizes both spherical and chromatic image aberrations compared to the decel-accel mode This reduction occurs because the particle trajectories remain closer to the optical axis, resulting in a smaller relative energy spread (∆V/V) within the lens field By selecting this mode, the diameter of the center electrode can be reduced relative to the outer electrodes, leading to lower voltage requirements for achieving a specific focal length.
On the decel–accel branch of the working curve, when the voltage ratioV L /V B is increased to even more positive values, the lens strength
The focal length of a lens experiences a maximum, while the lens strength leads to a minimum focal length As the lens strength increases, the focus converges towards the lens field By carefully selecting the lens geometry, it is possible to achieve a focal length minimum precisely at the ratio of V L to V B equal to one.
Fig 1.41 Einzel lens geometry advantageous for accel–decel mode
Fig 1.42 Geometry of einzel lens having maximum refractive power with V L = V B
(middle electrode at source potential) The focal length is then f ≈ 2D For V L = 0.5
In certain applications, it is beneficial for both the source and the lens to operate on the same voltage supply When the ratio of the lens voltage (V L) to the beam voltage (V B) exceeds +1, the saddle point potential matches that of the source, causing particles to decelerate to zero energy At a slightly higher potential, these particles are reflected, effectively switching the beam off Figures 1.42 to 1.45 illustrate various examples of einzel lenses.
In lens construction, achieving optimal concentricity of the lens elements is crucial, alongside ensuring adequate electrical insulation for stable voltage While distances and contours are important, they are less critical as long as the design maintains rotational symmetry.
Fig 1.43 Example of einzel lens with decel–accel mode
Fig 1.44 Example of einzel lens with accel–decel mode
Fig 1.45 Example of einzel lens where the middle electrode is centered and insu- lated by six precision ceramic balls
Summary Optical parameters and dispersive properties of electrostatic sector fields are discussed.
This article explores the behavior of charged particles in uniform electrostatic fields, focusing on two scenarios: large-angle deflection and acceleration or deceleration It then examines beam steering using deflection elements for small angles and sector fields for larger angles, emphasizing that the beam's energy remains unchanged during its passage through these devices.
Parallel Plate Condenser
Beam steering, whether for adjustment or scanning, typically utilizes a pair of parallel plates with an applied deflection voltage The deflection characteristics of these plates can be readily derived from fundamental principles.
The field can be assumed to be sharply terminated at both ends by
To achieve effective boundaries, the effective length (L) of the deflector must exceed its physical length, accounting for fringe field effects A particle enters the field along the x-axis of the x-y coordinate system with an initial energy of eV0 The central plane at y = 0 maintains a potential of V0, while the deflection voltage is symmetrically applied at V0 Consequently, the particle experiences a deflection force described by the equation eE = m¨y.
We can integrate with respect to time, whereby starting time and trans- verse starting velocity are set to be zero atx,y= 0: ¨ y= e mE, y˙= e mE t, y = eE
The axial starting velocity is v 0 2eV 0 /m We can replace t in
The virtual deflection center is located at x d =L−y 1 /y 1 =L/2 (2.5) For small deflection angles the tangent can be replaced by the angle φ= EL
The particle energy, which increases slightly inside the field to e(V 0 +Ey), drops back to eV 0 after passage of the fringe field at the exit.
The trajectories of singly charged particles have been calculated, revealing that multiply charged particles follow the same paths in a deflection field when accelerated by the same voltage, as their charge cancels out However, if these multiply charged particles enter the deflection field with the same energy as singly charged particles, they experience greater deflection due to the increased force, which is proportional to the number of elementary charges (n) multiplied by the electric field strength (E).
The deflection angle φ is influenced by the particle energy eV0, leading to energy dispersion among the particles Specifically, particles with energy eV0(1 + δ) experience slight variations in deflection Here, δ represents the relative energy deviation, calculated as δ = ∆V / V0.
By differentiating (2.6) with respect toV 0 one obtains
Thus, the value of the energy dispersion factor is identical with the deflection angleφ.
Cylindrical Condenser
The energy dispersion in a parallel plate condenser is limited due to the small deflection angles achievable To effectively utilize an electrostatic deflection field for energy dispersion, larger deflection angles are necessary While a uniform field has been previously discussed for this purpose, it often demands excessive space and a voltage equal to half of the beam acceleration voltage Consequently, the cylindrical condenser has emerged as a more favored energy spectrometer for various applications.
V b are applied to the plates so that the middle cylindrical surface lies at the potential outside the field (usually ground potential).
The field strengthE 0 =V d /din the center is chosen so that parti- cles entering with energy eV 0 on the entrance axis travel on a circular
Fig 2.2 Deflection and dispersion by cylindrical condenser
Fig 2.3 Potential distribution in a cylindrical condenser V a = −2V 0 ln (r a /r e ) ,
V b = − 2V 0 ln (r b /r e ) trajectory along the middle, which is the optic axis The electrical cen- tripetal force has to balance the centrifugal force: eE 0 = mv 0 2 r e = 2eV 0 r e , E 0 = 2V 0 r e (2.8)
In order to find the dispersive and focusing properties of the cylindrical condenser the paraxial trajectories have to be calculated.
The potential in between the plates can be written to be (Fig 2.3)
V (r) =−2V 0 ln (r/r e ) (2.9) With this, the field strength results as
E(r) =−dV (r) dr = 2V 0 r , (2.10) and for r=r e one obtains (2.8) The particle energy in the field is eV 0 +eV (r) =eV 0
Developed in a Taylor series about r = r e , (2.9) and (2.10) can be rewritten, using the coordinateρ= (r/r e )−1:
As a test, the Laplace equation (1.57) can be applied to (2.12) Since there is noz-dependence, it reads here
The inner plate surfaces are positioned at ρ a = -d/2r e and ρ b = d/2r e By substituting these values into equation (2.12), we can determine the voltages required for negative particles to ensure that the center maintains a zero potential.
The cube term of (2.12) can be neglected.
The deflection voltage across the plates is then
The equations of motion for particles outside the optic axis are m¨r=mrϕ˙ 2 −eE, (2.16) mr 2 ϕ˙ = const (2.17)
The differential equations describe oscillations around the optic axis By integrating to the first order and eliminating time, we derive the trajectory equation, which exhibits a period of π√.
By applying the boundary conditions, including the arrival energy expressed as eV = eV0 (1 + δ), we can determine the focusing and dispersive characteristics of a sector field with a specific sector angle φ In this analysis, the mass m becomes irrelevant, highlighting that only the energy eV of the particles is significant.
Fig 2.4 Cylindrical condenser acting as thick lens
The cylindrical condenser sector field functions similarly to a thick lens, featuring two principal planes The center of energy dispersion is situated at the exit principal plane P2 In the x2, y2 coordinate system, the trajectory equation for particles exiting the sector field is represented as y2 = L1α1 + x2.
It is the equation of a lens plus the dispersion term λδ The transfer matrix from the entrance principal planeP 1 to the exit principal plane
The parameters of the electrostatic sector field are: focal length f = r e
√2 , (2.21) focal distance (distance of focus from field boundary) g= r e cot√
√2 , (2.22) distance of principal planes from field boundary p=f −g= r e tan φ/√
For small sector angles φ the sector field can be considered a paral- lel plate condenser, and indeed, the dispersion factor becomes (with sin(√
2φ) λ = φ (comp (2.7); the different sign follows from the differently defined ordinate) Also, the distance of the deflection center becomes (tan(φ/√
The length of the plates in a parallel plate condenser is denoted as (= r e φ), which exhibits weak focusing properties The focal length is calculated as f = r e /2φ = L/2φ² By applying the imaging equation derived from this, we find that for y₂ = 0 and δ = 0, the image distance is represented as x₂ = L².
The magnification in an imaging system is expressed as s2/s1 = L2/L1 The theoretical energy resolution occurs when the image width s2 matches the energy dispersion y2 (δ)≡yδ, allowing the images from the entrance slit of width s1, generated by two energy beams eV0 and e(V0 + ∆V), to just touch To enhance performance, an exit slit is positioned to block particles with energy e(V0 + ∆V), thus refining the beam For optimal intensity, the exit slit should match the width of the entrance slit image; a wider exit slit compromises energy resolution, while a narrower one reduces intensity without improving resolution Consequently, the energy distribution behind the exit slit adopts a triangular shape with a full width at half maximum (FWHM) of e∆V and a base width of 2e∆V, with the peak aligning with the original energy distribution.
When an energy bandwidthe∆V is to be selected for further use, the mean pass energy should of course be set at the maximum of the original
The energy resolution of cylindrical condenser energy distribution is analyzed by scanning the pass energy across the energy distribution The theoretical energy resolution can be derived from the relationship s = 1/L^2.
The image distance L2 is eliminated, and the radius re is not included in equation (2.25), indicating that the same energy resolution can be achieved with a fixed ratio of s1/L1 using a condenser with the same sector angle φ, regardless of varying radii re However, the radius re plays a crucial role in determining angular magnification and image aberrations.
For a given ratios 1 /L 1 optimum resolution is obtained with maxi- mumλ, i.e with√
2φ= 90 ◦ ;φ= 63.6 ◦ In this case, becauseg= 0, the image position is close to the exit boundary when the object distance
L 1 is large in comparison withr e
When designing an energy analyzer, it is crucial to ensure that the image is formed outside the condenser, allowing for the placement of the exit slit This requirement can be derived from equations (2.24) and (2.23), which outline the necessary conditions for achieving this configuration.
L 1 /f −1 > p (2.26) This yields for the range 90 ◦ < √
2φ < 180 ◦ (63.6 ◦ < φ r e /2) the axial focusing is stronger than the radial one.
In case of symmetric imaging we obtain withL 1 = 2f r from (2.25) the general expression for the energy resolution:
To enhance energy resolution for a specific 1/r_e value, it is beneficial to allow c to approach 2, although this compromises the object-to-image distance along the optical axis Additionally, it is crucial to consider axial focusing to prevent intensity loss due to the axial divergence of the beam.
There is a possibility to choosec in the favourable range 1< c 2, R e < r e /2), the expression √
2−c in (2.45) becomes imaginary The lens param- eters, however, are still real and we have (with √
Such a sector field acts radially as a diverging lens and axially as a strong focusing lens (Fig 2.18).
Remembering the hyperbolic functions (Fig 2.19), very large disper- sion factors λcan be realized with such a sector field, e.g with c= 3,
R e = r e /3 and φ= 90 ◦ the dispersion factor isλ= sinhφ= 2.3 The strong axial focusing, however may be a hindrance.
Fig 2.18 Toroidal condenser with R e < r e /2 focuses axially but defocuses radially
A distinct category of toroidal condensers is identified when the radial and axial curvatures are oriented oppositely, as illustrated in Figure 2.20 In this scenario, the effective radius (R e) is considered negative, resulting in a negative value for c.
Fig 2.20 Toroidal condenser with negative axial radius of curvature: strong radial focusing, but defocusing axially
The lens parameters are then f r = r e
These sector fields act radially as focusing lenses, but axially as diverg- ing lenses.
Summary Optical parameters and dispersive properties of magnetic sector fields are discussed.
When a singly charged particle with energy eV moves into a uniform magnetic field B that is oriented perpendicular to its trajectory, it experiences a deflecting Lorentz force (F_L = evB) This force is counteracted by the centrifugal force (F_c = mv²/r), where v represents the particle's velocity and m its mass The relationship between these forces can be expressed as mv²/r = evB, leading to the equation mv/r = eB.
2emV this yields rB 2mV /e orrB = 143.6√
V [kV] For electrons (withM = 1/1,823 dalton) we have rB= 3.35√
V (3.3) with r [cm], B [G], V [V] Multiply charged ions with n elementary charges are deflected as if they had the mass M/n, if they have been accelerated by the same voltage V as singly charged ions.
Small Deflection Angles
Small deflection angles φ are obtained from φ ≈ L/r, where L is the length of the field andr is obtained from (3.2) or (3.3) The deflection
The magnetic deflection of charged particles at the end of the field can be approximated using a power series expansion of the cosine function, leading to the expression y(L) = r(1 - cosφ) ≈ rφ²/2 = L²/2r The center of deflection is located at a distance of y(L)/φ = L/2 from the end of the field, effectively positioning it in the middle of the field The degree of deflection is influenced by both the mass and energy of the particles, allowing for the definition of a mass and energy dispersion coefficient By substituting slightly altered mass (M₁ = M₀(1 + γ)) and energy (V₁ = V₀(1 + δ)) values into the established equations and expanding them into a power series, the first-order result yields r₁ = 143.6.
The dispersion factor for both mass and energy is φ 0 /2, indicating that the energy dispersion factor is half that of an electrostatic field In ion physics, magnetic deflection fields are primarily utilized for their mass dispersion, often resulting in larger deflection angles to achieve greater mass dispersion.
Fig 3.2 Optical parameters of uniform magnetic sector field
Magnetic Sector Fields
A magnetic sector field is designed to conform to the path of the beam it contains, optimizing the use of magnetic volume The optical axis consists of three sections: a straight entrance axis that is perpendicular to the field boundary, a curved axis within the field for a particle with mass M0 and energy eV0 entering the entrance axis, and a straight exit axis that is also perpendicular to the exit boundary.
In a uniform magnetic sector field, a paraxial trajectory can be easily calculated using trigonometry, unlike in an electrostatic sector field This is due to the consistent energy of the particle, which maintains a constant trajectory radius The resulting exit equation is expressed as y = L + 1α + x.
A magnetic sector field, akin to an electrostatic sector field, integrates both focusing and dispersive characteristics, represented by the terms in parentheses The dispersion component is defined by the sum of γ and δ, indicating that a particle experiences deflection similarly regardless of its mass deviation or relative energy deviation When these deviations are equal but opposite, dispersion is effectively nullified.
The two principal planesP 1 andP 2 are rotated by the sector angleφ about the intersection of the entrance and exit optic axes The transfer matrix fromP 1 and P 2 is
2 focal distance (from field boundary) g=rcotφ , distance of principal planes from field boundary p=rtanφ
From (3.5) the imaging equation can be deduced, with x=L 2 (image distance) fory= 0 and γ =δ = 0:
A real image is created when the object distance (L1) exceeds the focal length (f), resulting in a positive image distance (L2 > 0) The magnification is calculated as the ratio of the image distance to the object distance (L2/L1) The geometric position of the image can be determined using Barber’s construction, where the object point, sector center, and image point are aligned in a straight line This principle also applies to virtual objects and images.
Magnetic sector fields are primarily utilized in mass spectrometry for mass separation To determine the mass resolution of these fields, one must compare the image width of an entrance slit at the object position with the mass dispersion The mass dispersion in the image plane is defined by the equation y γ = L² νγ, where γ represents the relative mass difference (∆M/M) The image width of the entrance slit can be expressed as s₂ = s₁ L²/L₁, corresponding to ions with energy eV₀ Additionally, due to the inherent energy spread of the ions, energy dispersion must also be considered in the image plane.
Fig 3.3 Barber’s construction with real object and image points outside the field (L 1 > 0; L 2 > 0)
Fig 3.4 Barber’s construction with virtual object point (L 1 < 0) and image point outside the field (L 2 > p)
Fig 3.5 Barber’s construction with virtual object point (L 1 < 0) and image point inside the field (L 2 < p)
Fig 3.6 Barber’s construction with real object point (L 1 < f) and virtual image point (L 2 < 0)
Fig 3.7 Barber’s construction with L 1 = L 2 = 0 is, in analogy to (3.9), y δ = L 2 νδ, where δ = ∆V /V 0 is the relative energy spread of the ions The total image width is thus w=s 2 +y δ = (s 1 /L 1 +νδ)L 2 (3.10)
An exit slit is positioned in the image plane and should have a width of w for optimal intensity The theoretical mass resolution is achieved when the image width w matches the mass dispersion.
The achievable mass resolution is primarily constrained by the relative energy spread To enhance this resolution for a specific energy spread ∆V, it is necessary to increase the acceleration voltage, which subsequently demands a higher magnetic field.
When symmetric imaging is employed (L 1 = L 2 = L = 2f), we have, with (3.7), L 1 ν=r, so that
The behavior of the field does not rely on the sector angle φ; however, as φ decreases, the distance from the boundaries L−p=g+f = rcot(φ/2) causes the field boundary normal to form an angle ε with the entrance or exit optic axis This results in a field component B ε that acts parallel to the optic axis plane and normal to the trajectory, producing a deflecting force perpendicular to the optic axis plane Consequently, the fringe field functions as a thin lens that focuses perpendicularly to the plane of symmetry, with a first-order focal length given by f ε = rcot(ε).
It can be positive or negative, depending on the sign of ε Figure 3.8 shows a positive ε Oblique field boundaries, however, have an effect
Axial focusing occurs due to the fringe field created by an oblique field boundary, where the field line tangent, B t, is divided into two components: one along the trajectory and another perpendicular to it, denoted as B ε The perpendicular component, B ε, generates a focusing force directed towards the plane of the optic axis.
The optical parameters of a uniform magnetic sector field with oblique boundaries reveal that when a fringe field functions as a converging lens (positive ε), the radial focusing is diminished, whereas a diverging lens (negative ε) enhances radial focusing The key focusing parameters include f = r cos ε₁ cos ε₂ sin Ω, g₁ = r cos ε₁ cos (φ - ε₂) sin Ω, g₂ = r cos ε₂ cos (φ - ε₁) sin Ω, p₁ = f - g₁ = r cos ε₁ sin Ω [cos ε₂ - cos (φ - ε₂)], and p₂ = f - g₂ = r cos ε₂ sin Ω [cos ε₁ - cos (φ - ε₁)].
The center of mass and energy dispersion is located at a distance of p ν from the exit boundary, which differs from p 2 in this context This relationship is expressed by the equation p ν = r cot (φ/2) + tanε 2, as illustrated in Figure 3.10.
Fig 3.10 Center of dispersion of uniform magnetic sector field with oblique boundaries
The trajectory equation behind the field in thex−y ν coordinate system is y ν =xν(γ+δ) (3.16) with the dispersion factor ν = 1
In the special case where the field boundaries are parallel (Ω = 0, ε 1 +ε 2 =φ), there is no radial focusing (f =∞, see (3.14)), but always axial focusing (Fig 3.11).
The application of fringe field focusing is exemplified through a symmetric stigmatic imaging mass separator, represented by the equation ε₁ ε₂ = ε At the midpoint of the sector field (φ/2), it is essential for the beam to maintain parallelism both radially and axially By analyzing one half of the sector field, we derive that f ε = g(φ/2) Utilizing equations (3.13 and 3.14), we can express cotε as cosε cos(φ/2) sin(φ/2 - ε) and establish the relationship sin(φ/2 - ε) cos(φ/2) = sinε tanφ.
Fig 3.11 Parallel field boundaries: no radial, but axial focusing, (a) 1 = φ, 2 = 0, (b) 1 = 2 = φ/2
Fig 3.12 Symmetric stigmatic imaging by fringe field focusing, L 1 = L 2 obtained (see (3.11)) to be
This is better than in the case of normal boundaries (comp (3.12)), but at the expense of longer distances of source and image from the sector field.
3.4 Non-Uniform Magnetic Sector Fields
Another way to achieve focusing also in the axial direction, but without axial fringe field focusing, is to use sector fields which are radially non- uniform [20–23].
When the pole piece surfaces of a sector magnet are conical rather than parallel, their tangents intersect at a radial distance Rm from the optic axis in a cross section through the z-plane This configuration allows for the definition of a non-uniformity coefficient, expressed as n = rm.
The radius of the circularly curved optic axis is denoted as r_m, while B_0 represents the field strength along the optic axis In a first-order approximation, the off-axis field can be characterized by its axial and radial components.
Fig 3.13 Non-uniform magnetic sector field
The optical parameters of such a field are (comp Fig 3.2): f r = r m
1−nφ/2 distance of radial principal plane (3.24c) ν= sin√
1−n dispersion factor (3.24d) p ν =p r distance of dispersion center
√ntan√ nφ/2 distance of axial principal plane (3.24h)
Fig 3.14 Special case of non-uniform magnetic sector field: R m = 2r m
In the special case whenn= 1/2 (R m = 2r m ) (Fig 3.14), the radial and axial parameters become equal so that stigmatic imaging occurs: f r =f z =r m
In case of symmetric imaging the source and image distance becomes l=r m √
Figures 3.15 and 3.16 illustrate two examples of dispersion in the image plane, represented by the formula y γ,δ = 2f ν(γ+δ) = 2r m (γ+δ), which remains constant regardless of the sector angle φ This finding aligns with the previous example of stigmatic imaging using fringe field focusing when φ is set to 90 degrees.
Fig 3.15 Symmetric stigmatic imaging by non-uniform magnetic sector field with
Fig 3.16 Symmetric stigmatic imaging by non-uniform magnetic sector field with φ = 180 ◦ ; f = 1.78 r m ; p = 2.85 r m ; g = −1.08 r m ; l = 0.70 r m Dispersion y γ,δ =2r m (γ + δ)
Fig 3.17 Stigmatic imaging by non-uniform magnetic sector field from entrance to exit boundary
Fig 3.18 Non-uniform magnetic sector field with R m = r m
The object and image distance becomes zero with (see (3.26)) φ 2√
2×90 ◦ = 254.6 ◦ (Fig 3.17) Another special case is when n = 1 (R m =r m ) (compare the toroidal field case withc= 2) (Fig 3.18).
In this case the radius of curvature of a particle trajectory equals the distance from the z-origin, there is no radial focusing, only axial focusing (Fig 3.19).
Fig 3.19 Non-uniform magnetic sector field with R m = r m : no radial focusing, but axial focusing
The optical parameters are f r =∞, g r =∞, f z = r m sinφ, g z =r m cotφ, p z =p ν =r m tan φ
In case n >1 (R m < r m ) (Fig 3.20) the expression √
1−nin (3.24) becomes imaginary (Comp case c >2).
The lens parameters, however, are again still real: f r = −r m
Fig 3.20 Non-uniform magnetic sector field with R m < r m
The axial parameters are the same as above (3.24f–h) f z = r m
Such a sector field acts radially as a diverging lens and axially as a strong focusing lens (Fig 3.21).