64-587 Applications of Electrons, Ions and Atomic Beams
Dr T J Reddish
Space Charge
Consider a long cylindrical beam of electrons moving in the z direction at a constant speed; then:
v=
dz
2eV
=
dt
m
(1)
where V is the electric potential in the region of space. There is a radial Coulomb force, repelling
the electrons outwards, due to the electrons enclosed in an inner cylindrical volume of charge.
This radial equation of motion is then given by:
d 2r e
= Er
dt 2 m
(2)
The electric field at the surface of the cylindrical beam of electrons is, using Gauss’ theorem:
Er =
ξ
2πε 0 r
(3)
where ξ is the linear charge density, in C m-1, inside the beam, which we assume – for simplicity
– to be independent of radial position. The linear charge density is simply related to the electron
beam current by:
ξ=
I
=
v
I
2eV / m
(4)
Thus the equation of radial motion becomes:
d 2r e
I
=
2
m 2πε 0 r 2eV / m
dt
(5)
As we have assumed a constant axial speed, then – from equation (1):
d
=
dt
2eV d
m dz
or
d2
=
dt 2
2eV d 2
m dz 2
Hence:
d 2r
1
I
=
2
3/ 2
dz
4πε 0 r 2e / m V
(6)
(Note: I /V 3 / 2 is the perveance.) Substituting:
β=
1
I
I
≅ 30329.5
2πε 0 2e / m V 3 / 2
V 3/ 2
and multiplying both sides of equation (6) by dr / dz gives:
2
d 2 r dr β dr
=
dz 2 dz r dz
When integrated this yields:
1
(7)
64-587 Applications of Electrons, Ions and Atomic Beams
Dr T J Reddish
⎛r⎞
⎛ dr ⎞
2
⎜ ⎟ − α = β ln⎜⎜ ⎟⎟
⎝ dz ⎠
⎝ r0 ⎠
2
(8)
Where r0 is the initial beam radius and α is the initial beam convergence angle. Although the
initially parallel beam in our example will have α = 0, we will keep it in our expressions and
consider it as a special case of a more general solution which is still valid in the paraxial
approximation.
Rearranging equation (8) gives the functional form of the slope angle for a given perveance and
initial beam size.
dr
=
dz
( β)
⎛ r ⎞ α2
ln⎜⎜ ⎟⎟ +
⎝ r0 ⎠ β
(9)
Note that as the slope angle depends on r / r0 , this is a general expression that scales with
physical size.
In order to obtain an expression for the beam radius one can make the following substitutions:
⎛ r ⎞ α2
η2 = ln⎜⎜ ⎟⎟ +
⎝ r0 ⎠ β
(10)
2
2
r = r0 e η e − α / β
(11)
Hence:
and
2
2
dr
dη
= 2ηr0 e η e − α / β
dz
dz
(12)
Comparing equations (9) and (12) results in:
2
2
dη
β = 2r0 e η e − α / β
dz
(13)
which when integrated over the appropriate limits gives:
2/β
2r
z − z0 = 0 e − α
β
κ2
∫
e η dη
2
(14)
κ1
and κ i = ln(ri / r0) + α 2 / β from equation (10). Tables of the ‘Dawson function’, D(κ), exist
where:
2
64-587 Applications of Electrons, Ions and Atomic Beams
κ
Dr T J Reddish
D ( κ ) = e η dη
∫
2
(15)
0
or this integral can be evaluated numerically using standard computational packages, as shown in
Figure 1.
With z0 = 0 defining the origin, z/r0 then provides a relative length scale in equation (14),
resulting in a general – or normalized - beam spread curve for electrons that is simply a function
of the perveance.
Returning to our original parallel beam of electrons, (i.e. α = 0, z0 = 0, κ1 = 0, κ 2 = ln (r / r0 ) )
simplifies equations (9)-(15) significantly giving:
z
2
=
D ( ln(r / r0 ))
r0
β
(16)
For a given initial radius of the electron beam, r0, and its β value (see Equation (7)), the
corresponding D value can be calculated for a specific distance down stream, z. From the graph
shown in Figure 1, the corresponding κ value can be determined, and hence the find the factor by
which the beam has been broadened, r/r0. Alternatively, one can use Figure 2, which shows the
shows the characteristic beam expansion curve directly.
20
10
18
0.0146
16
8
14
12
6
D(κ)
10
r/r0
8
4
6
4
2.35
2
2
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4 1.6 1.8
2
κ =(ln(r/r0))1/2
0
0
0.01
0.02
0.03
0.04
(I1/2V-3/4 )(z/r0)
Figure 1: Graph of the Dawson Function.
Figure 2: The normalised beam expansion curve for an
initially parallel beam of electrons, obtained from equation
(16).
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64-587 Applications of Electrons, Ions and Atomic Beams
Dr T J Reddish
These expressions predict how an initially parallel beam of electrons will expand due solely to
Coulomb repulsion. However, the time reversal is also true so that an initially converging beam
will create a narrow waist of finite size and zero slope before re-expanding. We can consider this
problem directly with the above formalism by considering an initially converging beam with a
finite α.
r
Figure 3: Schematic diagram of an
electron beam, initially converging the
optic axis at zF, being expanded by
space charge repulsion within the beam
to a minimum ‘waist’ of radius rm at zm
from the origin (z0 = 0).
r0
α
rm
zF
z
zm
We can evaluate an expression for the size at the ‘waist’ of the beam (see Figure 3), where the
slope will be zero, from equation (9), namely:
2
rm = r0 e − α / β
(17)
Note that only space charge effects are considered here and, in practice, lens aberrations (both
imaging and ‘chromatic’) will also have a contribution to the spot size. The integration of
equation (13) has different limits resulting in:
2/β
2r
z m − z0 = 0 e − α
β
0
∫
e η dη
2
(18)
α/ β
Since this is a converging beam, α < 0, thus the integral can be expressed as:
2/β
2r
z m − z0 = 0 e − α
β
α/ β
∫
e η dη
2
(19)
0
After the converging beam has reached its waist position at zm, the beam re-expands following
equation (16); explicitly:
z − zm =
2rm
β
D( ln(r / rm ) )
(20)
The values of the integral (20) can be obtained graphically as before. To evaluate (19) one can,
following Pierce, make the substitution:
F ( x) = e − x D( x)
2
giving:
4
(21)
64-587 Applications of Electrons, Ions and Atomic Beams
Dr T J Reddish
zm
2
=
F ( α / β)
r0
β
(22)
The values of F(x) are shown graphically in Figure 4, below.
1
F(x)
0.1
0.01
0.01
0.1
1
x
Figure 4: Plot of the Function F(x) Note that for small x,
10
100
F ( x) ≅ x , and for large x, F ( x) ≅ 1 / 2 x
If space charge effects were absent, the beam would converge to a point at zF, as shown in figure
3. Within the paraxial approximation:
r
z F − z0 = 0
α
(23)
Taking z0 = 0 as before, then using equation (22) one obtains:
z m 2α
=
F ( α / β)
zF
β
(24)
Which is shown in Figure 5. Regarding the functional form of Figure 5, consider the idealized
situation were r0 and α are constant and we increase β ( ∝ I /V 3 / 2 - see equation (7)) to reduce x.
Initially zm/zF = 1, then increases to approximately 1.2847 at x ≈ 1.502 and then steadily reduces
for very small x. Space charge effects decrease with large x, i.e. as β decreases (low current and
high voltage), as anticipated. Whereas the beam barely converges (before diverging again) for
small x, i.e. as large β, with high current and low voltage - (see figure below).
5
64-587 Applications of Electrons, Ions and Atomic Beams
Dr T J Reddish
1.4
1.2
1
0.8
zm/zF
0.6
0.4
0.2
0
0.1
1
10
x
100
Figure 5. The ratio of zm/zF, given by equation (24), as function of the characteristic parameter x, defined in the text.
There is a further unique value when zm/zF = 1, which graphically occurs at x ≈ 0.9242. At this
condition equation (24) can be expressed as:
β = 4α 2 ( F ( x)) 2
(25)
F(0.9242) = 0.541, from Figure 2 – where it can be seen that this x value corresponds to the peak
of the F(x) function. Substituting β from equation (7) and recognizing α = r0 / z F from equation
(23) gives the well-known expression:
I
⎞
⎜ z ⎟⎟
⎝ F⎠
⎛r
= 3.86 x10 − 5 V 3 / 2 ⎜ 0
2
(26)
From equation (17):
2
rm
= e − x = 0.4256
r0
(27)
Physically this corresponds to the maximum current, I, that can be transmitted through a tube of
length 2zF, diameter 2r0 and at a potential V, as shown in Figure 6. The electron beam must enter
the tube with an angle α and the beam will converge to a waist of diameter 2r0/2.35 half way
along the tube before diverging to its original diameter as it leaves the tube. If more than the
maximum allowable current enters the tube the space charge spreading will increase and less
current will emerge from the cylinder. Note that as this scales with the r0/zF ratio, the current
density can scale with geometrical size even if the current itself is limited.
6
64-587 Applications of Electrons, Ions and Atomic Beams
Figure 6: Schematic diagram of
optimum space charge electron
flow through a cylindrical tube,
length 2zF, diameter 2r0 and at a
potential V. The electrons, with
energy eV, must be launched
towards the tube center with an
initial angle α in order to exit the
tube (see text).
r0
Dr T J Reddish
α
zF
It is worth mentioning before leaving this section that we have also assumed throughout that the
potential difference arising from the space charge between the edge and the middle of the beam is
small compared to V. (L. Jacob An Introduction to Electron Optics Methuen (1951), states that
for a 100eV beam with I = 1.5mA, the δV ~ 1.5V – (p135))
Below is a summary plot of beam spreading profiles from Klemperer and Barnett (1971);
compare with Fig 5 above.
See Givet Electron Optics (1965) Pergamon Press p277-283
See Klemperer and Barnett Electron Optics (1971) Chapter 8.
See also Read et al (1974) J Elec Spec & Rel Phenom 4 293-312
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64-587 Applications of Electrons, Ions and Atomic Beams
Dr T J Reddish
Read et al (1974) obtained semi-empirical expressions for Imax, namely:
38.5 x10 −6 3 / 2 ⎛ r0 ⎞
⎜⎜ ⎟⎟
I max ≈
V
ln( r0 / 2rm )
⎝ zm ⎠
2
(28)
where all quantities are in SI units, and the lengths are given in Fig 3. This approximation is
correct to ±20% for rm / r0 = 0.0035 to 0.27. An alternative expression, also accurate to ±20%,
applies when rm / r0 = 0.04 to 1.0:
I max ≈ 38.5 x10
−6 ⎛
⎛r ⎞
r ⎞
⎜⎜ 0.3 + 3.7 m ⎟⎟V 3 / 2 ⎜⎜ 0 ⎟⎟
r0 ⎠
⎝
⎝ zm ⎠
2
(29)
The value of the above expressions is in the applicability for a broad range in rm / r0 .
Example:
Let θ = 4° (and letting tanθ = r0/zm), r2 = 0.5 mm and r0 = 3 mm.
Using (29), one can show I max ≈ 173nA V 3 / 2 (ignore the units of V!)
Thus: for a 1eV the maximum current is 173 nA
for a 10eV the maximum current is 5.45 μA
for a 100eV the maximum current is 173 μA
This is not a major problem unless θ and V are small; typically the worse scenario is at the
entrance of the monochromator (having a low pass energy to obtain high energy resolution).
8
64-587 Applications of Electrons, Ions and Atomic Beams
Dr T J Reddish
Space Charge and Energy Distributions.
In the previous section we considered the limitation on current passing through an aperture of a
given size due to the radial component of the Coulomb force, Here we will examine the change
in the energy profile of the electron beam as it converges to a focal ‘point’ within a field-free
region. Historically this energy spreading was first noted by Boersch (1954) and has been known
as the ‘anomalous energy spread’ and the ‘Boersch effect’.
Loeffler (1969) studied the energy spread in the case of an initially mono-energetic, paraxial
beam, converging to a ‘point’. In the cross over plane, the current density within the beam is also
assumed to be constant. He shows that there is (also) a net longitudinal repulsive force which, in
essence, speeds up the front electrons while slowing down the following electrons within the
beam. Consequently, this effect will be most significant at regions of high current densities and
long interaction times – i.e. focal points in lens systems of high currents and low beam energies.
His analytical expressions giving the energy spread (in eV) are given below:
ΔE B =
1
r0α 0
χ(
Ir0
)
V 1/ 2
(1)
where r0 is the radius of the beam in the cross-over plane, I is the current, α0 is the beam pencil
angle and V is the potential in the region (i.e. the energy of the beam in eV). All units in equation
(1) are SI except for ΔEB, which is in eV. The forms of χ are given as:
B
χ(
Ir0
πη
) = C0
1/ 2
V
2
η << 1
for
(2)
and
1/ 2
1
⎤
⎡
) = C0 η ⎢3 + ln(2) + 2 ln(η) + ln 2 (η)⎥
χ(
1/ 2
4
⎦
⎣
V
Ir0
for
η >1
(3)
where C 0 = e / 4πε 0 = 1.438 x10 −9 and
η =8
Ir
m Ir0
≅ 8.44 x1013 10/ 2
3
1/ 2
2e V
V
(4)
It is clear that for Ir0 /V 1 / 2 < ~10-14 then χ is essentially linear in Ir0 /V 1 / 2 , whereas when
Ir0 /V 1 / 2 > ~10-12 then χ varies approximately as
Ir0 /V 1 / 2 . A plot of χ ( Ir0 / V 1 / 2 ) as a function
of Ir0 /V 1 / 2 is shown in Figure 1. Loeffler’s analysis has been examined by Zimmerman (1970),
who considers it to be an overestimate by between 20-40% from his own calculations. Even so,
this is quite acceptable for practical purposes.
Consider a 1 eV electron beam energy with r0 = 0.5 mm, α0 = 1/10 and I = 10-7 A. Ir0 /V 1 / 2 =
5x10-11, χ ≈ 5.74 x 10-7, hence from (1) ΔEB ≈ 11.5 meV.
B
1
64-587 Applications of Electrons, Ions and Atomic Beams
Dr T J Reddish
While this effect is present at the entrance to an electron ‘monochromator’, the additional energy
spread is negligible on comparison to the thermal spread from a cathode. However, at the exit of
the energy selector, and at other cross over points towards the target, this effect will degrade the
energy resolution in the beam. This is a fundamental and limiting problem for high energyresolution electron spectroscopy, with no analogue in light optics. It is, however, not relevant in
electron analyzers after the target as the electron number density in the beam is so low (1-10 nA)
and can limit the resolution at the 1-10meV range – depending on energy and angles.
Another area when it is an issue is virtual parallel (collimated beams) of ~50 μA and V ~1keV, as
the energy spreading is of ~0.2-0.5 eV, in which case using a monochromator has little value.
1 .10
5
1 .10
6
1 .10
7
χ(Iro/V1/2)
1 .10
8
1 .10
9
1 .10
10
1 .10
11
1 .10
16
1 .10
15
1 .10
14
1 .10
13
1 .10
12
1 .10
11
1 .10
10
1 .10
9
1 .10
8
Iro/V1/2
Figure 1. The variation of χ as a function of Ir0 /V
1/ 2
(in SI units), as given by Loeffler’s expressions: equations
(2-4).
2