THE REVIEW OF SCIENTIFIC INSTRUMENTS
VOLUME 38. NUMBER 1
JANUARY 1967
Electron Monochromator Design*
C. E. KUYATT AND J. AROL SIMPSON
National Bureau of Standards, Washington, D. C.20234
(Received 20 July 1966; and in final form, 8 September 1966)
A study has been made of all the known factors which limit the performance of high resolution (0.07 to 0.01 eV
monochromators. These limiting factors have been incorporated into design equations for the optimum
(ma~mum current output) monochromator. The conclusions are tested by performance measurements on a prototype Illstrume~t. The results require the introduction into the design equation of a new limiting factor, an anomalous
energy spread III dense electron beams, which is empirically determined.
F~M)
A monochromator based on 180 0 electrostatic deflection
between concentric spheres has been reported by Meyer,
Skerbele, and Lassettre,6 with mean radius of 12.7 cm and
entrance and exit slits 0.0062 cm wide. A gun utilizing an
oxide coated cathode furnishes electrons of 100 to 400 eV
at the entrance slit. The exit beam is about 10-7 A at 400
eV and has a FWHM of 0.14 eV, as measured with an
identical energy analyzer.
The two types of monochromators so far described
operate at very different energies, 1 and 200 eV, the former
giving approximately the same current with a slightly
smaller energy FWHM. No reasons were given for the
choice of electron energy or for any of the other parameters
of the devices, such as the radius of the electron path in
the electrostatic deflectors. A third type of monochromator, described by Boersch, Geiger, and Hellwig,7 operates
near 20 eV, and is based on the Wien filter. Electrons are
decelerated from 20 keY to 20 eV and pass through crossed
electric and magnetic fields adjusted so that electrons of
the desired energy are not deflected. The energy selected
electrons are then reaccelerated to 20 keY. A fairly complete analysis of the device was given. The effect of space
charge was not discussed, but we have determined that
this monochromator is limited by cathode brightness at
currents sufficiently small to have negligible space charge
effects. The analysis of Boersch, Geiger, and StickelS results
in specification of aperture sizes and other parameters
which give maximum current for a given energy FWHM.
The aperture sizes found best in practice deviate greatly
from the sizes specified by the analysis. The extreme deceleration ratios and consequent strong focusing action at
the entrance and exit slits result in nonstandard operation
of the Wien filter, and therefore the analysis given is not
completely applicable to the monochromator as operated.
The current furnished by the monochromator is not stated
by the authors; only energy FWHM are given. When
measured with an identical energy analyzer, FWHM's as
small as 0.007 eV have been obtained. A typical FWHM
used in energy loss measurements was about 0.04 eV, with
I. INTRODUCTION
THElessproduction
of electron beams with energy spreads
than 0.1 eV and with current sufficient for measurement of elastic and inelastic scattering in gases has
led recently to the discovery of resonance effects in both
elastic! and inelastic2 electron scattering from atoms and
molecules, and of new energy levels in the rare gases. 3
Further and more detailed study of these processes would
benefit greatly from the availability of monoenergetic
electron beams with higher current. Unfortunately, the
electron monochromators in current use are already, for
the most part, "second generation" devices of considerable
sophistication, and major improvements may not be easy.
Perhaps the best known of the monochromators is that
of Marmet and Kerwin. 4 It is an improvement of a device
first built by Clarke,5 and consists of a 127 0 cylindrical
electrostatic deflector with 1.25 cm mean radius and 0.5
mm wide slits. The cylindrical electrodes are constructed
of fine wires, and exterior positive electrodes collect those
electrons which pass between the wires of the inner electrodes. A tungsten ribbon filament placed 1 mm from the
entrance slit is used to provide electrons for the monochromator, using a small accelerating potential. Electrons
with a mean energy of about 1 eV traverse the cylindrical
deflector and emerge from the exit slit, producing an electron beam of up to lX 10-7 A, .and with an energy distribution having a FWHM of 0.06 to 0.08 eV. The energy
distribution was measured with an energy analyzer identical in construction to the monochromator. Since the energy
distributions encountered are close to Gaussian, the
FWHM of the electron beam from the monochromator is
approximately 0.7 of the FWHM as observed with the
analyzer. (Not 0.5, as sometimes claimed.)
* Supporte~ in part by Project Defender, sponsored by Advanced
R;search Projects Agency, Department of Defense.
G.
Schulz, Phys. Rev. Letters 10, 104 (1963); C. E. Kuyatt,
J. A. SImpson, and S. R. Mielczarek, Phys. Rev. 138 A385 (1965)'
J. Chern. Phys. 44,437 (1966).
"
2 G. J. Schulz and J. W. Philbrick, Phys. Rev. Letters 13, 477
(1964) ; G. !'-. Chamberlain, Phys. Rev. Letters 14, 581 (1965); G. E.
rl::65)~rlalll and H. G. M. Heideman, Phys. Rev. Letters 15, 337
r
a J. A. Simpson, G. E. Chamberlain, and S. R. Mielczarek Phys.
Rev. 139, A1039 (1965).
'
4 P. Marmet and L. Kerwin, Can. J. Phys. 38,787 (1960).
6 E. M. Clarke, Can. J. Phys. 32, 764 (1954).
6 V. D. Meyer, A. Skerbele, and E. W. Lassettre J Chern Phys
43,805 (1965).
,.
.
.
7 H. Boersch, J. Geiger, and H. Hellwig, Phys. Letters 3 64 (1962)
8 H. Boersch, J. Geiger, and W. Stickel, Z. Physik 180, 415 (1964)"
103
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C.
104
E.
KUYATT AND ]. A.
SIMPSON
EXIT
SLIT
ENTRANCE
SLIT
oD
CATHODE
ILLUMINATING
OPTICS
IMAGING
OPTICS
0
ENERGY
DISPERSOR
IMAGING
OPTICS
BEAM
FORMING
OPTICS
OUTPUT
FIG. 1. Block diagram of the electron monochromator.
some recent measurements made with a FWHM about 0.01
eV. A similar monochromator with the Wien filter modified
for two dimensional focusing has been described by Legler.9
The fourth monochromator is that described by
Simpson1o ; it is based on the 180 0 spherical electrostatic
deflector, but incorporates several unique features. One is
that the electrostatic deflectors are slitless; that is, there
are no physical apertures in the entrance and exit planes.
Instead, these planes are imaged by electron lenses onto
physical apertures at much higher potentials than exist in
the spherical deflector. Two main purposes are served by
this arrangement. First, electrons at very low energy do not
have to pass through the actual slits; and second, any
secondary electrons produced at the slits are prevented
from reaching the electrostatic deflector. A second important feature is that the electron beam entering the
monochromator is carefully collimated to match the characteristics of the electrostatic deflector. Hence unwanted
electrons are reduced to a minimum, and no surface
coatings or grids were needed to reduce the deleterious
effects of large numbers of extraneous electrons. This
monochromator utilized a mean radius of 2.54 cm and
equivalent apertures of 0.25 mm diam. When operated at
a mean energy of ",4 eV, it produced a beam of about 10-7
A with a FWHM of 0.07 eV. This monochromator has
proved capable of producing usable beams with FWHM's
down to 0.0035 eV, and has shown its versatility in a wide
range of measurements of elastic and inelastic scattering
from gases.
Despite the successes of these electron spectrometers,
the problem of producing high current electron beams of
energy FWHM less than 0.1 eV cannot be considered
solved. A simple calculation based on the known emission
properties of tungsten shows that it should be possible to
produce an electron beam with an energy width of 0.02
eV, cross sectional area of 1 mm2, and a current of 100 p.A.
This value is several orders of magnitude greater than the
best reported monochromator performance. Moreover, the
amount of monoenergetic current available from a tungsten'
9 W. Legler, Z. Physik
10
A. Simpson, Rev.
J.
171,424 (1963).
Sci. Instr. 35, 1698 (1964).
emitter should be approximately proportional to the energy
FWHM. In practice, the current actually obtained from
monochromators decreases very rapidly as the energy halfwidth is decreased. lO
We have therefore made a study of all of the known
factors which limit the performance of electron monochromators. We discuss these limiting factors and incorporate them into equations for the optimum electron monochromator. We have focused our attention on the system
shown in diagrammatic form in Fig. 1. In this design, electrons emitted by the thermionic cathode are accelerated
and focused onto the entrance slit by the illuminating
optics (electron gun). This slit is then focused by the
imaging optics onto the focal plane of the energy disperser.
The exit plane of the energy disperser, where the "spectrum" of energies is spread out, is focused onto the plane
of the exit slit where electrons in the desired energy band
pass through the physical slit. The beam forming optics
then transform the electrons emerging from the slit into a
beam with the properties desiredll for an experiment. We
have allowed for as many 'changes of energy during the
focusing operations as were necessary or desirable, and
have placed no arbitrary limit on the number of electron
optical elements used.
II. FUNDAMENTAL ELECTRON-OPTICAL
RELATIONS
In any electron beam system, there are three basic restrictions which must be satisfied. These are the law of
Helmholtz and Lagrange, Langmuir's equation which is derived from it, and the effect of space charge in high current
electron beams.
A. Law of Helmholtz and Lagrange
This law states that along any electron beam path where
current is conserved (and there are no energy dispersing
devices), the energy E, differential solid angle dO, and the
differential area dA at any two cross sections 1 and 2 are
11 The problem of choosing the proper beam parameters to optimize
a particular scattering experiment involves the design of an optimized
analyzer, and will be dealt with in subsequent papers.
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105
ELECTRON MONOCHROMATOR
related byI2,13
~-----------L------------~
(1)
The law of Helmholtz and Lagrange might be subtitled as
the conservation of electron "brightness". "Brightness,"
as we use the term, is called Richtstrah.lwert in German, and
has been called current intensity by Sturrock. To avoid
confusion Richtstrahlwerl is used here, and is defined by
R=dlldAdQ,
(2)
where dI is the current through a differential area dA, and
dQ is the solid angle sub tended by the electrons. Conservation of current together with Eq. (1) leads to
dII E 1dQ 1dA I = dII E 2dQ 2dA 2.
(3)
Use of Eq. (2) then gives
RliEI = R21E 2•
(4)
Hence we see that Richtstrahlwert, when divided by the
electron energy, is a conserved quantity.
The law of Helmholtz and Lagrange can be cast in yet
another form. Consider the case where planes 1 and 2 are
conjugate; that is, one is imaged' on the other. Let two
electron rays originate at a point in plane 1 and be imaged
to a point in plane 2, and let fit and O2 be the angles between
the rays at the two planes. Then
(5)
where M is the linear magnification from plane 1 to plane 2.
Equation (5) is often called the Abbe-Helmholtz sine law,
and its relation to Eq. (1) can be easily seen by squaring
Eq. (5). Equation (1), however, is valid between non.:conjugate planes.
B. Langmuir's Equation
This equation relates the maximum achievable current
density Jmax in an electron beam to the current density
and other parameters at a thermionic cathode. By application of the law of Helmholtz and Lagrange between the
cathode and any other plane in an electron-optical system,
one obtains the Langmuir equation,
where dQ 2 has been expressed in terms of the convergence
angle 82 in the plane under consideration, k is the Boltzmann constant, and T is the cathode temperature.14
See P. A, Sturrock, Static and Dynamic Electron Optics(Cambridge University Press, Cambridge, England, 1955), p. 93, et seq. of
a complete treatment.
13 W. Glaser, Grundlagen der Elektrontmoptik (Springer-Verlag,
Vienna, 1952), p. 276.
14 The maximum value of current density is achieved only at very
low current transmission efficiency [See J. R. Pierce, Theory and
Design of Electron Beams (Van Nostrand Books Co., New York,
1934), 2nd. ed., Ch. 8]. In practice, the maximum value can be
closely approached.
La
d
2.35 = 2.35
FIG. 2. Geometrical parameters of a fully space charge limited beam.
Although the full derivation of Eq. (6) is somewhat involved, an approximate form restricted to an image plane
can easily be derived from Eq. (5). Let M be the magnification between the cathode and the plane of interest. Then
J 2 = (l/M2)Jcathode' From Eq. (5), M2=EI sin281lE2 sin282.
If we take EI = kT as the characteristic energy for a
cathode, and remember that sinOl = 1, then there results
Equations (7) and (6) are exactly the same.
C. Space Charge
When the mutual repulsion of electrons in a beam is
taken into account, it is found that there is a maximum
amount of current which can be passed through any
volume, and that this current is always proportional to the
three-halves power of the electron energy.l. When the
volume is defined by two apertures of diameter d separated
by a distance 1 [Fig. 2J, the maximum amount of current
which can be transmitted is given byI6
I max= 38.5E! (d2ll2) ,
(8)
where I max is in p.A and E is in eV. To obtain this maximum
current, the electron beam must enter the volume so that
in the absence of space charge the beam would focus to a
point at tbe center of the volume. In the presence of space
charge, the beam profile assumes a shape as shown in
Fig. 2, with a minimum diameter17 which is d/2.35.
Another form of Eq. (8) is often convenient. This is obtained by replacing dll with a,
(9)
12
15 H. F. Ivy, Advances in Electronics and Electron Physics,
L. Marton, Ed. (Academic Press Inc., New York, 1954), Vol. VI,
p.137.
16 See Pierce14 (Cb. 9).
17 This model is somewhat simplified, in that a point focus and
laminar flow are assumed. When the focus has a finite size, the electron trajectories can cross the axis, and a more general treatment is
required. See Glaser13 (p. 66). The main features, however, remain
approximately the same.
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C.
106
E.
KUYATT AND
J. A. SIMPSON
given by
(10)
The potential of the inner sphere is V o[3-2(Ro/Rl)], and
the potential of the outer sphere is V o[3-2(R o/R 2 )], and
in general the two dividing resistors used to establish the
potential of the incident electron beam with respect to the
spheres are unequal.
Let Xl be the radial distance of an incident electron
measured from the path of radius R o, and let a be the angle
the incident electron makes with the path of radius RD.
Let X2 be the radial distance of the outgoing electron measured from the path of radius RD. Then
where /lE = E - Eo. The absence of a term linear in a shows
that the spherical deflector has first order angle focusing.
To obtain the energy resolution of the spherical deflector,
one must calculate the transmission of electrons as a function of energy, taking into account the distribution of the
incident electrons over space and angle. With entrance and
exit slits of equal width w (or with virtual slits), such a
transmission function, neglecting the a 2 term, would be a
triangle whose width at half height /lEI; is given by
--...-.-
(12)
X2
FIG. 3. Schematic of a spherical deflector.
m.
ENERGY DISPERSING ELEMENT
The heart of any electron monochromator is the energy
dispersing element. Our choice of the 180° spherical electrostatic deflector was based on two considerations. First,
devices containing magnetic fields were ruled out because
they are difficult to work with at low electron energies, and
the problem of shielding other parts of the apparatus from
magnetic fields is severe; and second, the device must
provide focusing in two directions so that lenses of axial
symmetry can be used for beam transport and control, and
so that a well defined electron beam is available for angular
scattering measurements.
The spherical deflector has been described by Purcell,18
and the version we have used is shown schematically in
Fig. 3. A 1/r2 electrostatic field is produced by a difference
of potential between two concentric hemispherical surfaces
of radii R2> R I • Electrons enter the deflector near the
center of the space between the spheres, and exit after
being deflected by 180°. If Eo=eV o is the energy in electron volts of electrons which Itravel in a circle of radius R o,
then the potential difference V between the spheres is
18
E. M. Purcell, Phys. Rev. 54, 818 (1938).
Use of a round entrance aperture instead of a slit would
change the shape of the transmission function (line shape)
without significantly changing /lEi. Addition of the a2
term also has the effect of changing the shape of the transmission function without appreciably changing /lEI, as
long as a 2 <w/2R o. We have chosen equal effective entrance
and exit apertures, and to reduce the tailing of the line
Col
FOCAL
FOCAL
PLANE
PLANE
(b)
~-(~
I
4
~Ro
4
,[
FIG. 4. Models of space charge limited deflector (a) without internal focusing, (b) with internal focusing concentrated in two thin
lenses located as shown.
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ELECTRON MONOCHROMATOR
AE
~----------~~------------------aEo
I
I
1-'
I~
I ~
considered only in the regions between the slits and the
thin lenses, using the universal space charge curve. 19
The results of this model calculation are shown in Fig. 5,
where the effective energy resolution t1E of the deflector is
plotted as a function of the injected current I. Up to a
critical value Icrit( '" 1.5 f.J.A) of injected current, the energy
resolution remains virtually constant. The rapid increase
of t1E at higher currents is due to the point of smallest
diameter on the exit side moving out of the focal plane.
The critical value of current is given by
(15)
lu
11-1
I
FIG. 5. Effect of space charge on the resolution !>f a. deflector with
concentrated internal focusing. flEe is the reS?lU~lOn ill the absence
of space charge, as given by Eq. (12). IcritioaliS glven by Eq. (15).
shape have chosen
a 2 =w/4Ro.
(13)
A final property of the spherical deflector which is of
interest is the maximum deviation Wm of the beam in the
deflector from the central path of radius R o,
Wm _
Ro
t1E +[a2+(~+ t1E)2J!
Eo
107
(14)
2Ro Eo
Equations (12) and (13), together with Eq. (9), are the
basic equations needed to design the optimum monochroma tor; optimum, that is, in the sense that it provides
the maximum possible current for a given t1EI' Having
chosen t1EI and the size of the spherical deflector, the
geometry of the electron beam at the spherical deflector is
fixed, but its energy is not. Before discussing the choice
of energy which gives maximum current, the effect of
space charge on the spherical deflector must be considered.
Space Charge in Spherical Deflector
Because no detailed treatment of the spherical deflector
is available, one is reduced to model calculations. These
calculations vary in degree of sophistication, and the ultimate choice must be based on experimental verification of
their predictions.
The first and simplest of these is shown in Fig. 4(a).
The analyzer (shown straightened out) is considered to be
a tube whose length is that of the beam path in the analyzer and whose width is equal to that of the slit. This model
predicts the optimum operation to be at very low energies. 10
This obviously oversimplified model ignores the internal
focusing of the deflector, and can be improved as shown in
Fig. 4 (b). In this model, the focal properties are considered
to be concentrated in two thin lenses located as shown. The
electron beam is injected so that after space charge spreading the beam fills the entrance slit. Space charge effects are
Since this model relates to many other analyzers, the choice
of slit width wand electron energy E for maximum current
at a given t1E in this model is discussed below.
In the final model of space charge in the spherical deflector, a more sophisticated point of view is taken. In this
point of view, cognizance is taken of the fact that, in the
system sketched in Fig. 1, there is no slit in the focal plane
of the dispersing element, and that what is important is
the image of this plane at the physical exit slit, where by
accelerating or demagnifying, or both, space charge
spreading can be made negligibly small. To a good approxima tion, space charge spreading can be considered to be the
consequence of a negative lens in the region of minimum
beam cross section,20 in particular at the exit focal plane
of the spherical deflector. If the electron beam is converging
toward a very small spot at minimum beam cross section,
the effect of the negative lens produces a beam which, after
space charge spreading occurs, appears to come from a very
small spot. This point of view has a profound effect on the
monochromator problem, as is seen below.
The classical "laminar flow" space charge th eory20.21
predicts that such a space charge lens has vanishing aberrations. Although this theory is not exact, the fact that
aberrations affect spot size as the aperture angle to the
third power, while the aberration term of the spherical deflector is a second power term, suggests that, even if the
aberration constants are large, for small angles such a
space charge lens would form a satisfactory image.
IV. OPTIMUM MONOCHROMATORS
In the earlier work/o it was pointed out that the most
successful monochromators operated at comparatively low
energy, and the simple theory of a focusing dispersing
element in the presence of space charge called for operation
at E"-' t1E. This conclusion is considerably modified in the
more realistic treatment of the focusing dispersing element
which is given below.
19 K. R. Spangenberg, Vacuum Tubes (McGraw-Hill Book Company, Inc., New York, 1948), p. 444.
20 O. Klemperer, Electron Optics (Cambridge University Press,
Cambridge, England, 1953), 2nd ed., p. 207 et seq.
21 V. K. Zworykin et al., Electron Optics and the Electron Microscope
Gohn Wiley & Sons, Inc., New York, 1945), p. 541.
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108
C. E. KUYATT AND
SPACE CHARGE LENS ~
W
0-
,. .... /'
~.......
""....
WITH ANOMALOUS
.... "'"
...... ....
i.S
ENERGY SPREAD
I
/
-...--, --.,-,~--.::...
... "
''''-~
'"oc
v"
....
~
.lEK ""2.5kT.) We find that
,,;
'"wa;
.... ;,.
!tj8
"o
i
go
"
~:~:
:
~~
wi
~::~
01
>1
01
(;)1
,
0;'
-"-....,.
~I
,
',,-
...
w,
>,,
,
liE \12 : Q,02eV
0,'
I t!.E= 350E-l(AEi )3/.lEK ; E?:.18AEt
(21a)
= 19.3E!(.lE,)2/.lEK i E:=:;18.lEt
(2Ib)
= 82(.lEi W.lE K ; E=18.lEt .
(2Ic)
0
o0;
:r
J. A. SIMPSON
10
100
ELECTRON ENERGY 'N MONOCHROMATOR.eV
FIG. 6. Predicted pelformance of electron monochromator. Calculated performance of the "real slit" and space cbarge lens model are
shown. The effect of anomalous energy spread on the space charge
lens model is to be noted. Two experimental points corrected for
transmission defect are shown.
A. Monochromator with Real Apertures
For the case of real apertures at the entrance and exit
planes of the spherical deflector, we must take as the slit
size the actual size of the space charge limited beam at the
entrance plane of the spherical deflector. Otherwise, not
all of the incident electron beam would be usable. This
model of space charge in the spherical deflector is applicable to the case of real apertures. Referring to Fig. 4(b), we
see that the electron beam diameter after space charge
spreading is t7rRoa, and therefore the minimum beam
diameter w is given by
w= (1/2.35) (7rRoa/2):::;:'2Roa/3.
(16)
The basic monochromator Eqs. (12) and (13) then become
(17)
whichever is greater. The two terms in Eq. (17) containing
are equal whena=i. Fora<t, the right hand term can
be ignored to an accuracy of a few per cent. Solving the
remaining equation for a and substituting into Eq. (15),
the total space charge limited current It entering the spherical deflector is found to be
a
(18)
As before, electron energy is in eV and current is in ",A.
For a>t the center term in Eq. (17) can be ignored, and
we find
(19)
For a=t, where .lE/E=!a=1/18, the corresponding
current is
(20)
To obtain the monochromatized electron current I t!.E,
Eqs. (18)-(20) must be multiplied by .lEt/.lEK , where
AEK is the' effective width of the electron beam entering
the deflector. (For a thermionic cathode operating at TOK,
Equation (21) is plotted as a solid line in Fig. 6 for the case
.lEi=O.02 eV and .lEK =0.25 eV. The largest monoenergetic electron beam is produced when E= I8AEj.
Part of the curve is dotted because operation would be required with an impractically large aperture in relation to
the mean radius of the deflector.
B .. Monochromator with Virtual Apertures
The use of real apertures at a much higher energy than
in the deflector, together with accelerating and decelerating
electrostatic lenses to image the real apertures in the exit
and entrance planes of the deflector, has been successful
in the past for other reasons, and is indicated schematically in Fig. 1. Such virtual apertures can play an
even more important role when attempting to exploit
the fact that, to first approximation, space charge acts
as a negative lens. After space charge spreading, the electron beam appears to come from a source smaller than the
actual minimum diameter of the beam. If such a beam is
accelerated and/or demagnified, thus reducing the degree
of space charge, a real image of the apparent small source
should be obtainable. The net result is to decouple the
effective size of the electron ,beam (due to space charge)
from its angular spread, so that one is free to choose the
ejJective aperture size w to satisfy Eq. (12). Under these
conditions, we can always choose a2= .lEt /2Eo, which,
when substituted into Eq. (9), gives the total space charge
limited current lin entering the spherical deflector,
Iin= 19.3Et.lEi;.
(22)
The monochromatized electron current I "'E is then
(23)
Equations (22) and (23) are identical to Eqs. (19) and
C21b), respectively, but with one important differencethere is no longer an upper limit for the electron energy E
in the monochromator. The dashed curve in Fig. 6 shows
the extension of Eq. (23) to large values of E. Note that
according to this model of space charge in the spherical
deflector, the amount of monochromatized current increases indefinitely as the mean energy in the deflector is
increased, while at the same time the effective aperture
size wand entrance angle a must be reduced.
V. TEST OF OPTIMUM MONOCHROMATOR
MODELS
Models II and III predict optimum operating conditions
for the monochromator which differ markedly, with the,
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ELECTRON MONOCHROMATOR
109
ELECTRON SPECTROMETER
(- 30· to +90· SCAN)
FIG. 7. Cross section of prototype electron monochromator and analyzer used to test
monochromator model calculations.
space charge lens theory calling for high energy at high
dispersion with correspondingly small slits, while the real
slit theory calls for low energy and larger slits. In practice,
the real slit theory is easily realized, while the space charge
lens theory soon calls for cathodes of impossible brightness
and extremely small slits, or dispersing elements of impracticably large dimensions. For a given energy resolution, the higher the energy of operation of the monochromator, the more sensitive it becomes to stray magnetic
fields. This apparent paradox arises since, for a fixed
apparatus size, the slit width is inversely proportional to
the energy, while the magnetic rigidity of the beam increases only as the square root of the energy.
In order to determine the vaiidity of the space charge
models, a series of experimental studies were made on two
monochromators (which we call versions A and B) whose
parameters were chosen to give operation in the region
where the space charge lens model and the real slit model
differ appreciably, as shown in Fig. 6. The design parameters used are listed in Table I together with the predicted
and measured currents for .:lEt = 0.02 eV.
A. Experimental Arrangement
The apparatus used is sketched in Fig. 7. The monochromator was mounted on a hinged support, allowing the
monqduomatized current to be directed either to a
Faraday cup, which, when biased 30 V positive, collected
essentially all of the beam, or onto the entrance aperture
of a high resolution analyzer which sampled .....,8% of the
beatn./The apparatus was enclosed in a mercury pumped
stainless steel vacuum system with an operating vacuum
;S2X1Q-7 Torr.
Measurements were made of the current to the Faraday
sup as a func,tion of the amount of current injected into
the spherical deflector. The latter current was determined
by measuring the current collected on the outer sphere
with the deflecting voltage turned off and 30 V positive
applied to the outer sphere to retain low energy secondary
electrons. The current into the spherical deflector was
varied by changing the initial diode stage voltage. Energy
resolution of the monochromator was checked by moving
the beam onto the analyzer, and remained close to the
nominal value throughout the measurements (after correction for energy resolution of the analyzer).
B. Experimental Results
The results of the monochromator measurements are
shown in Fig. 8, where current into the Faraday cup is
plotted as a function of current into the monochromator
spheres. For a monochromator which obeys the space
charge lens model, one expects a linear rise in monochromator output current as the input current is increased
up to the space charge limited current, after which monochromator performance rapidly deteriorates. If the real
slit model is valid, the critical current would be considerably lower.
TABLE
1. Parameters of prototype monoehromators.
Monochromator mean radius Ro
Effective entrance aperture w
Effective exit slitwidth
Energy in deflector
Energy resolution t..E,
Current (real slit theory)
Current (space charge lens theory)
Measured current
Measured "efficiency"
(Measured current) I (efficiency)
Current (anomalous energy
spread theory)
A
B
2.54 cm
0.50 mm diam
0.50mm
2eV
0.02 eV
8 X 10-9 A
4.5 XlO- s A
1.1 XlO- s A
0.3
3.67XI0-s
2.85 X lO-s
2.54 cm
0.177 min diam
0.177 mm
7 eV
0.016 eV
2.15XlO-9 A
5.4 X1O-s A
6 XIQ-9 A
0.45
1.33X1O-s
1.57XIQ-s
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C.
110
E.
KUYATT AND J.
rJ)
W
0::
W
a.
,,
..
::;;
I
«
1
100%
SPACE
CHARGE
I
I
I
3 12
0::
W
I
I
;;jl
m
8'
:./
/
~IO
J:
u
VERSION A
12'
'-'"
'"z
ii:
8
w
100%
SPACE
CHARGE
I
--'I
'-'"
"'I
I
~I
ff-
I
u,
::t::1
<l
U
UJ
6
1
,-",
u,
!;;;
~I
f-
"'/
0
1
=>
~ 4
=>
/
I
/
0::
I
0
I
2
!;i
::;;
0
0::
3
0
0
0
z
2
4
6
8
MONOCHRPMATOR INPUT CURRENT,10- 7 AMPERES
0
::;;
FIG. 8. Measured performance of prototype electron monochromator. The deviation from the space charge lens theory is due to anomalous energy spread in the dense electron beam entering the instrument.
Operating conditions-version A, 2 eV, Mi""O.02 eV; version B,
7 eV, Mt""O.02 eV.
A.
SIMPSON
in versions A and B, and should not be applied to other
devices. There may be geometrical factors implicit in
Eq. (24) which would change in a different geometry.
There is one other factor which causes a decrease in output current measured at the scattering chamber as the
input is increased. Some current is lost in the electronoptical system between the monochromator and the scattering chamber. The fraction transmitted varies from about
i to t as the input current is increased from zero to the
space charge limit. Since the other measurements clearly
indicate that the monochromator focusing in the energy
dispersing plane is not affected by space charge, we conclude that space charge produces an astigmatic focus;
i.e., the best focus perpendicular to the energy dispersing
plane is not at the monochromator output slit, and the
result is the loss of electrons by interception on the electron
lenses or collimating apertures. This effect is not unexpected, since there are no converging forces within the
spheres in the nondispersing plane.
VI. OPTIMUM MONOCHROMATOR IN PRESENCE
OF ANOMALOUS ENERGY SPREAD
Instead of the expected linear increase of output current,
we find a strong bending over in both A and B versions.
In the A version, the bending over has proceeded to the
point of apparent saturation of output current. The dashed
line indicates the prediction of the space charge lens model.
Both versions A and B give output curves which are nearly
tangent to the predicted curve at low input currents, but
quickly deviate from this curve as the input current is
increased.
An extensive series of measurements were made to determine. the cause of the nonlinearity and saturation of
monochromator output. These measurements22 lead to the
conclusion that, as has been reported previously,23 dense
electron beams exhibit energy widths which are current
dependent, and are much greater than those characterized
by the cathode temperature.
Measurements of the anomalous energy spread at several
energies on both versions A and B can be fit with a single
energy spread equation,
The anomalous energy spread given in Eq. (24) must
now be incorporated into an optimum monochromator
model. Since monochromator focusing has been clearly
demonstrated to be independent of space charge up to the
space charge limited current, the anomalous energy spread
logically should be incorporated into the space charge lens
model.
..,
en
....,,.'"
<t
,.:
..,
Z
~
u
10-7
...
~
::J
o
'"
,.~
where tJ.E in is the FWHM in eVof the input energy distribution at current I, tJ.EK is the FWHM of the energy
distribution at vanishing current (the cathode energy distribution), J is the current density in p.A/cm2 at the
entrance to the spheres, and E is the mean energy in eVof
the electron beam in the spheres. It must be emphasized
that Eq. (24) applies only to the special conditions existing
&!r
~
,.o
10-8
10
ELE~TRON ENERGY IN MONOCHROMATOR,eV
J. A. Simpson and C. E. Kuyatt, J. Appl. Phys.
37, 3805 (1966).
H. Boersch, Z. Physik 151,519 (1958) ; D. Hartwig and K. Ulmer,
Z. Physik 173, 294 (1963).
22
21
100
FIG. 9. Relationship between monochromatized current and energy
within monochromator for various values of MI in the presence of.
anomalous energy spread.
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111
ELECTRON MONOCHROMATOR
10-6
given by
I
q'
I
I
1&.1
I
a:
It1E=19.3
I
,,0/
1&.1
CL
~
"
.
z
1&.1
a::
0:
::;)
0
~
::;)
0..
ler'
I
~
:::)
0
I
E= 116AEKAERo2,
I
'/
,
/
lu max =31.2[RoCAE)!/(AEK)lJ.
10-·
100
10
AE
(28)
where r is in em, and the corresponding maximum current
I u max is given by
/
5
(27)
AEK+8.6X Io-S(E/Ro2AE)
For the special case AE=0.02 eV and AEK=0.25 eV, lu
is plotted as a function of E in Fig. 6 (labeled "anomalous
energy spread"). Figure 9 shows several such curves. It is
seen that there is an optimum value of E, the energy in the
monochromator, which gives maximum monochromator
output current. It is easily found from Eq. (27) that this
optimum E is given by
,'/.
C
~
(26)
E!(AE)2
I,'/
tn
= 6.I4(E!/rAE).
Equation (25) then becomes
I /
I
J =lin/!W= 19.3ElAE/l-Jr(2RrAE/E)~
mY
FIG. 10. Comparison of prototype performance with predictions of
calculations. The log of the monochromatized current is plotted
against log of l!.Et to illustrate (LlEt)6/2 current dependence. The pre·
diction of the real slit model as applied to version A is shown as a
broken line. The corresponding prediction of the space charge lens
model with anomalous energy spread is shown as a. solid line. The
points represent measured values without transmission correction. If
corrected they would lie slightly above the solid line.
The fraction of the input current transmitted by the
monochromator is now AE/AE in, rather than AE/AEK.
Hence the output current is given by
Since J is proportional to lin, Eq. (25) has the fonn
CJ/(A+BJ), a monotonically increasing function. Hence
I AE is a maximum when lin is given by the space charge
limited current from Eq. (22). The current density J is
(29)
It is interesting to note that the dependence on AE in
Eq. (29) is the same as in Eq. (2Ic) for the real slit model.
We also find that the effective electron beam size at the
monochromator is given by
w= 2r(AE/E) = 1/58RrAEK.
(30)
We have omitted from consideration here the transmission
factor of the optics after the monochromator itself.The
transmission could be raised to the theoretical value by
modifying the external electron optics.
It can be seen from Figs. 6 and 9 that both versions A
and B are close to optimum. The choice can be made on
grounds of convenience. We have chosen version A because
of lower operating voltages, and because this version appeared to be easier to adjust and to have better stability.
Figure 10 shows how the final optimum monochromator
model fits the measured performance of version A over the
AE range of 0.02 to 0.07 eV FWHM of the monochromator
'alone. Because of the efficiency factor discussed above, the
measured currents fall below the model, but have the predicted dependence on' AE. The currents obtained are
clearly greater than predicted by the real slit model, even
without correction for the efficiency factor.
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