Design study of a double pass hemispherical electron energy analyser with multichannel detection

Journal of Electron Spectroscopy and Related Phenomena, 67 (1994) 2 1 1 -220 0368-2048/94/$07 .00 © 1994 - Elsevier Science B .V . All rights reserved

211

Design study of a double pass hemispherical electron energy analyser with multichannel detection A . Baraldi*° l , V .R . Dhanak Sincrotrone Trieste, Padriciano 99, 340 .12 Trieste, Italy

(First received 5 February 1993 ; in final form 19 October 1993) Abstract The characteristics of an analyser consisting of two 180° hemispherical deflection analysers placed in tandem and equipped with a multichannel detector have been investigated . The energy resolution and transit time spread for non relativistic electrons have been computed using numerical methods . The optimum size and shape of the entrance slit with respect to multichannel detection are considered . Electron energy spectra have been simulated for a multidetector with up to 100 discrete channels to show the "snap-shot" capability of the analyser .

1 . Introduction Synchrotron radiation is a powerful probe for the study of the interaction of molecules with solid state surfaces [1] . Its application in surface science has yielded important information concerning molecular identity, orientation, electronic structure and binding sites of adsorbates . The new generation of synchrotron radiation sources, such as ELETTRA, are characterised by high brilliance and a small spot size at the sample [2], providing opportunities for carrying out surface science experiments in new and different ways . This in turn calls for the development of a new generation of experimental ESCA stations . This study is part of a project to develop a novel analyser that would exploit the unique characteristics of ELETTRA to perform time resolved core level spectroscopies of catalytic systems . In a typical ESCA experiment a spectrum is accumulated over a period of few minutes using a ' Also

at Area per la Ricerca Scientifica, Padriciano 99, 34012 Trieste, Italy . ' Corresponding author .

SSDI 0368-2048(93)02038-N

spectrometer with a single channel detector . To reduce data acquisition times (by about an order of magnitude) a number of spectrometers have been developed using an analyser with a position sensitive detector [3,4] . Spectrometers with even faster data acquisition capabilities would allow new studies . For example it would be possible to follow the time evolution of adsorbed species which occurs on the time scale of typically a fraction of a second by taking "snap-shot" spectra . Such a study requires rapid data accumulation and also the ability to acquire (in parallel) a spectrum over a wide electron energy window, typically several electronvolts . In the present study we consider the extension of a 180° hemispherical deflection analyser (HDA) to a double pass configuration by placing two HDAs in tandem (Fig . 1) and consider a suitable multichannel detector that would be required for snapshot spectroscopy . The properties of a tandem arrangement of this sort with a single detector have been discussed by Mann and Linder [5] who demonstrate that the dispersion of the tandem arrangement of analysers each of radius Ro is

21 2

A . Baraldi, V.R. Dhanak/J. Electron Spectrosc. Relat. Phenom . 67 (1994) 211-220

Entrance Optics I

R2

a -rl 11111111111111H

Multichannel Detector Iv II

Fig. 1 . Schematic drawing of the double pass analyser showing the principal dispersion plane . Vj and Vz are the potentials on the inner and outer hemispheres respectively, with the mean radius R 0 . Regions I and IV are field free while in regions 11 and III a 11r 2 field is maintained by the potentials on the analyser . Electrons enter the entrance slit of width w and are collected by the detector after deflection through the hemispherical sectors .

essentially the same as that of a single analyser of radius 2R 0 . There are several advantages in using this configuration rather than an equivalent single analyser . For example, this configuration provides a straight through geometry, the peak shape and signal to noise characteristics are superior, and the analyser is lighter and more compact than the equivalent single analyser . The present study extends the work of Mann and Linder to consider multichannel detection and the time resolution of this arrangement of analysers . In Section 2 we give the relevant analytic equations to describe the dispersion characteristics of the double pass HDA configuration . Section 3 describes the results of computation of electron trajectories in the analyser . There we consider the energy resolution of the analyser, the optimum shape of the entrance slit, the analyser etendue, the optimum energy window, the required number of energy channels for multidetection and show simulated electron energy spectra . The transit time spread of the analyser is considered in Section 4. Ours is the first study of the properties of a double pass hemispherical analyser using multichannel detection . In the calculations, we only considered the ideal case in which fringing fields were neglected, and did

not consider relativistic and space charge effects which are unimportant here . The results therefore serve the purpose of comparison with ideal single HDA [6,7] . In practice, when apertures are placed at the entrance and exit planes of the HDA, they cut through the equipotentials, which can distort the electron trajectories and degrade the energy resolution and transmission of the analyser . The correction of fringing fields at the entrance and exit of the HDA has been treated in detail in the literature [8-11] and solutions given can be adapted to the double pass configuration . In the middle region where the transition from the first HDA to the second occurs, the optimisation of the fringing fields and the gap for multichannel detection purpose has been considered elsewhere, using the finite element method [12] . 2. The double pass analyser The analyser proposed here is shown schematically in Fig . 1 . It consists of two hemispherical deflectors placed in close proximity such that the exit plane of the first deflector coincides with the entrance plane of the second . The two concentric hemispheres have radii R j and R2 and potentials V1 and Vz , respectively, and produce a 1/r 2 electro-



A . Baraldi, V.R. Dhanak/J. Electron Spectrosc . Relat. Phenom . 67 (1994) 211-220

static field in the space between them . A slit of width w and length L is placed at the entrance of the analyser in the xy plane (normal to the plane of the figure), with its centre on the x axis, a distance RO = (R 1 + Rz)/2 from the centre of the first hemispherical deflector . Electrons enter the space between the hemispheres at the entrance slit and exit after being deflected by 180° in each deflector . A general trajectory in each deflector is a Kepler ellipse, confined in a plane defined by the initial velocity vector . The point of incidence in the entrance slit (see Fig. 1), with respect to the origin at 0, the centre of the first deflector, is defined by the coordinate - ri and angle a (note that the minus sign comes about from the particular choice of the origin) . In traversing the field boundary at the entrance of the analyser, the kinetic energy E and the angle of incidence a of the electron are changed to E' and a' given by

2 13

change in energy and angle . The energy change is given by i;2

= e[V(Ro ± AR) - V(R0

AR)]

4e VO RO AR (Ro - AR2 )

(3)

where AR is the distance from the central path, V(R) is the potential at radius R and VO is the potential on the central path . Note the change in sign of AR since the 1/r 2 field is in the opposite sense on either side of the plane between the two deflectors . The energy of the electron before it enters the second analyser is given by E11 = E'

+ e[V(rf1) - V(ri)]

E' + 2eR0 V0 (ri

rfi reI ri

(4)

so the energy E" in the second analyser is E ll =En

(5)

+6

E'=E+t; l

From the conservation of the angular momentum, expressed as [5]

sin a' =

cos(a')ri

/ 1 in a

/1+E1 where R0 l=2Eo --1) ri /

(1)

and E0 is the pass energy, i .e. the kinetic energy of an electron that enters the analyser at ri = R0 and follows the central path . Solving the equations of motion, one obtains an exact solution for the position r fi of the electron at the exit plane of the first deflector, given by

rfl = rii-2 1- 1_ l+sin2 a- E E EO (1-sin z a) 0

(2) In traversing the plane between the first and the second deflectors the electron undergoes a further

E7 =

cos(afl)rfl

Efl

(6)

where afl is the angle at the exit plane of the first deflector, the angle a" in the second deflector is obtained in the same way as for a' in Eq . (1), considering the potential difference at constant r given by Eq . (3) . From these relations the position r 12 of the electron in the exit plane of the second analyser is given exactly by rf2 _ - r; + 2E0 R0 X

4R0 - E(cos2 a' + cost a" (2E0 - Ecos2 a')(2E0 - Ecos a")

(7)

A Taylor series expansion of Eq, (7) and ignoring higher order terms gives z 4a2 r-=3-Ar+4-+4 y Ro RO E0 Eo where-y= E - EO and ©r = ri - R0 .

(8)

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A . Baraldi, V.R. Dhanak/J . Electron Spectrosc . Relat. Phenoin . 67 (1994) 211-220

The numerical value of 3 of the first term on the right hand side of this equation occurs because of the choice of 0 as the origin of the coordinate system. The second term of this equation shows that the magnification of the double HDA is unity . The third term shows the linear energy dispersion. The fourth term shows non-linearity in dispersion and is important for multichannel detection. The last term, often known as the trace-width, is an angular aberration term and has the effect of producing tails towards the lower electron energy side in the transmission function of the analyser . Note the absence of a term in a indicating that the double HDA is focusing to first order. Comparing this equation with the equivalent equation for the single HDA given in references [6,7,13,14] shows that the dispersion of the double pass analyser is twice that of the single analyser in agreement with Mann and Linder [5] .

choosing a is from the relation a2 = w/4RO for the single HDA [6], which means that the contribution of a to the base resolution is half that due to the entrance slit width w . For a slit width of 1 mm this gives a value of a of 2 .3° and so have used up to 3° in the computations . The choice of a will be considered again in Section 3 .3 . The energy, angle and position of the incident particle in the entrance slit is varied randomly to include the maximum range of a and all possible energies that can be transmitted, and the final position at the detector plane determined . Figure 2 shows typical trajectories in the analyser . Note the focusing (Fig. 2(a)) in the middle region and at the exit of the analyser for the trajectories with mean energy E o . The dispersive property of the analyser is demonstrated in Fig . 2(b) where an interesting aspect is the cross-over of the trajectories in the second analyser . This figure also allows a visual demonstration of the greater

3. Results of electron trajectory computations \O\\\\\\`\\\\\~~~\;" Electron trajectories were calculated in three dimensions using the Runge-Kutta single step iterative method . The trajectories were subdivided with time intervals of 0.01 ns, giving an accuracy in the central trajectory of less than 0 .08% . The details of the computation method are given in Ref. 15. This program not only allowed us to examine the properties of the analyser in the dispersive plane (i .e. by varying the entrance slit width w and acceptance angle a) but also to investigate the properties in the non dispersive direction (i .e. by varying the entrance slit length L and the acceptance angle Qi in the plane normal to the plane of Fig . 1) . For a given set of input parameters describing the analyser (Eo , R0 , w and L) and a set of electron trajectory starting parameters (E, r i (x, y), a;, Pi ), the program calculated the electron position at each At interval in the analyser . The final position r f2 in the detector plane was then displayed in the two dimensional x-y plane. In our calculations we fixed the size of the mean radius R0 at 150 mm . The usual design criterion for

Fig. 2. (a) Cross-section in the dispersion plane showing the focussing at 7r and 2a for trajectories with initial energy equal to E0 and a angles of 0°, ±3° and ±6° . (b) Trajectory with starting conditions a = 0° and energy E = E0 (central trajectory) and E0 t 0.02E0 . Note the dispersion at the rr and 27r positions and the crossover of the trajectories in the second analyser .



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A . Baraldi, V.R. Dhanak/J. Electron Spectrosc . Relat . Phenom . 67 (1994) 21 1 -220

(b)

Image distortion Dotactor Plan

1,

000 1r

Dispersion direction

Fig. 3. (a) Entrance and exit plane parameters used for optimising the entrance slit geometry, The vertical plane cutting the horizontal line through O'-O" is the dispersion plane . The circles of radii R o are the centre path corresponding to each analyser . The line A-D is through the centre of the entrance slit of length L . The heavy curved lines E-B and C-F are the images of A-D at their and 2,r positions, respectively, of the analyser . The shaded areas with widths AS, and AS2 in the dispersion plane illustrate the energy degradation due to the curving of the images . (b) Ray trace image at the exit plane of an entrance slit of finite width and length L = 40 mm for R o = 150 mm . The initial starting conditions of the electrons were a = 0 0 , E = Eo and r; = Ra .

dispersion at the exit plane compared to the it position corresponding to a single HDA. Using the form of Eq . (8) we have parameterised the computed values of r f2 and obtained the numerical constants using a least squares method . We find 2

Or 'Y = 3-3 .85a 2 + 3 .95 + 4 .1 Re Eo Re (TO-) rf2

(9)

which is in close agreement with Eq . (8) . The small discrepancy in the numerical values is attributed to errors in neglecting higher order terms in deriving Eq. (8) . 3.1. The choice of entrance slit length

Ideally for multichannel detection with a rectangular entrance slit one would like to have a regular array of rectangular anodes (channels) . However, the image at the exit of a HDA of a rectangular entrance slit is curved into a "smile" which can overlap into an adjacent channel, degrading the energy resolution . The origin of this curvature can be seen by referring to Fig . 3, which schematically shows the plane of the double analyser (top view) in the section containing the

entrance slit and exit planes . Our analysis extends the results of Hadjarab and Erskine [6] who considered the entrance slit geometry for the single HDA . The line through O'-O" (Fig. 3(a)) is in the plane of the analyser passing through the centres of the entrance slit and detector plane . The vertical line through A-D represents the centre of the entrance slit and Re is the centre radius of each HDA . From Eq . (8) and the similar equation for the single HDA given in Ref. 6, an electron that enters the analyser at point A (or D) (Fig . 3) with energy E0 must arrive at point B (or E) inside the circle described by Re after deflection in the first HDA . Likewise, the electron entering the second HDA at B (or E) must arrive at C (or F), hence the entrance slit line A-D is curves into a "smile" C-F . This image distortion is shown schematically in the ray tracing results in Fig . 3(b) for a finite slit width and length . From geometrical considerations the spread AS 2 at the exit of the analyser and the length AL 2 are given by AS2 = 2Ro (2 - cos ~' - cos (P") (10) AL2 = Re tan 4)" [2(cos 4/ + cos V') - 3]



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A . Baraldi, V.R . Dhanak/J . Electron Spectrosc . Relat . Phenom . 67 (1994) 211-220

where aretan

Vn = arctan

(2R0 2 cos ~' - 1 tan L3-2 cosh'

Note that the image curvature is independent of the pass energy . If a straight entrance slit is used then the width of a detector channel should not exceed the value of AS2 corresponding to the slit length L . For example, for the double HDA of mean radius 150 mm, and a slit length L = 10 mm, AS2 = 0.33 mm .

In order to derive a numerical expression for the energy resolution of the double HDA we consider the entrance slit width w, length L and angle a separately in our trajectory calculations . Treating these parameters separately for the energy resolution consideration is justified because they are not correlated . We used Eq. (9) and parameterised AEb, base width, using the usual formula

E0

E } + Bat + co/2 Ro '

mpmL

r

- Ba t E0 CEb

l AR

/

showing that the maximum value of F depends on a m and w . To obtain a m we differentiate Eq . (13) and set

3 .2 . Energy resolution

AEb = A

angle and it is possible to consider it as a figure of merit for the spectrometer [6] . This can be expressed as r = 4a m,(Im wL where am/3m is the product of the maximum angles in the dispersive and non dispersive directions respectively, at the entrance plane of the analyser and w and L are the entrance slit width and length, respectively . From Eq. (12), and considering that the V contribution is small, we can substitute for w so that r can be expressed as

(12)

where V is given by Eq . (11), and used a least squares fitting to derive the numerical constants . Our results are A = 0 .27, 8 = 1 .07 and C = 0.53 . In Eq . (12) the last term can be made small by choosing an appropriately curved entrance slit or by using a value of L defined by some criterion relating the other terms in the equation . The usual way is to make the second order terms equal to some fraction of the first term so that the slit width dominates the energy resolution . We will consider this in the next section . 3 .3 . The analyser etendue The etendue of the analyser is the product of the entrance slit area and the entrance solid

ar - 0

darn

(for maximum etendue) . Substituting from Eq . (12) we find ( 14 )

2 - Aw am 2BRo

which is the same result obtained by others for a single HDA (with appropriate values for A and B) and using fixed entrance and exit slits. Using Eq . (14) and substituting the value for the numerical constant A, we write the base resolution AEb

E0 =

0.267

(15) 2RO

with 2

0 .267

am - ( 1 .07

w

(16)

2R0

In the above we have neglected the contribution to the resolution arising from the use of a rectangular entrance slit of finite length . We now choose the value of L which meets the criterion that the contribution of L to the resolution term is half that of the contribution of the a term . From



217

A . Baraldi, V.R. DhanakJJ. Electron Spectrosc. Relat . Phenorn . 67 (1994) 211-220

Eqs . (12) and (16) we find that this criterion gives a value of L defined by Aw

L = 2Ro tan I

4CR0

Substituting for the numerical constants A and C, and using the values R o = 150mm and w = I mm, we get a m = 1 .65° and L = 8 .7 mm . In practice we expect L to be fixed below 10 mm . 20

40

60

80

100

Number of channels

3.4 . Energy window and the number of energy channels

From Eq . (15), we can write the effective dispersion D in eV mm -1 as D .,, 8R°

(18)

0

The total energy window for a given pass energy E0 is simply the Eq . (18) multiplied by the length of detector in the dispersion direction . The choice of detector length will depend on the available size of the gap between the hemispheres and the nonlinearity of the analyser . The size of gap (R 2 - R 1 ) is limited by the size of hemispheres (mean radius R0 ) and the effect of fringing fields. For Ro = 150 mm, which is a well established size for single hemispherical analysers used in ESCA experiments, the total gap R 2 - R1 is typically about 80 mm but in practice the useful gap is much smaller due to the necessity of proper field termination at the exit plane . We therefore limit ourselves to a detector length of 50 mm . In a typical ESCA experiment an energy window of 5-10eV width is usually acquired to show details in core level spectra . To get this information in a single shot would therefore require operating our analyser at a pass energy of between 40 and 80 eV . To determine the number of channels required we note that 5 or 6 points are generally required to fit a gaussian shape . Ideally, one would like to use this number of points per energy resolution of the analyser . However, since our energy window of interest is quite large, using this criterion makes

Fig. 4 . Non linearity in the dispersion expressed as the ratio of the difference in energy between the first two channels and the last two channels to the mean base resolution (centre channel width) as a function of the number of channels .

the number of energy channels impractially large . Fortunately for our purposes, we are interested in core level spectroscopies where a typical line width is of the order of 0 .5 eV . For a 10 eV total window then, we require 100--120 energy channels, which corresponds to about 3 channels per unit of energy resolution of the analyser . The non-linearity in the dispersion is described by Eq. (9) . This equation shows that when a detector is placed symmetrically at the centre of the exit plane (the usual case) then the energy separation between adjacent channels at the edges will not be constant . The difference is greater when fewer channels are used for the same detector length . This is shown in Fig . 4 where we have plotted the difference between the energy separation of the first two and the last two channels (normalized to the width of the energy channel) as a function of the number of channels in a 50 mm detector for our analyser . In the figure we only considered the non linear terms in y in Eq. (9). The essential features are the same if the non linearity in angle is also considered . From the figure it is seen that the error converges after 90 or so channels, indicating that our choice for 100-120 channels is adequate . We will consider further the choice of the number of channels in the following section where we have used our ray tracing program to simulate typical spectra .



218

A . Baraldi, V.R. Dhanak/J. Electron Spectrosc . Relat . Phenom . 67 (1994) 211-220

If we choose 100 channels in a detector of length 50 mm, then it is reasonable to have a channel width of about 0 .45 mm and a gap between channels of 0 .05 mm so that the detector dead area is about 10% . For a slit width w = 1 mm the length of the slit L and hence the length of the channel element should be about 8 mm . This choice satisfies the requirement that the slit image distortion measured by OS2 is much less than the channel width. 3.5. Simulation of the operator of the analyser with a multichannel detector

In our ray tracing program it is possible to define an initial energy distribution profile for the electrons at the entrance slit and examine the profiles at the exit of the analyser . The exit plane can be fitted with a detector of variable length and divided into any number of discrete channels . The resultant profiles (spectra) can be determined for any specified parameters relevant to the analyser and trajectory starting conditions . To demonstrate the advantages of the multichannel detector and to compare different number of channels for a specified detector length we have considered different examples for which we fixed the mean radius Ro = 150mm, entrance slit width w = 1 mm, entrance half angle a = 2° (note that this is larger than the maximum half angle a m we optimised in the previous section, allowing for a worse case of angular aberration) and pass energy Eo = 50 eV. A detector of total length 50 mm was placed symmetrically in the centre of the gap at the exit plane and divided into N discrete channels so that the dead area in each case was 10% . Here we show one example which serves to illustrate the versatility of the computer program . We consider an experiment from catalysis, specifically the methanation reaction on a nickel surface . Exposing Ni to CO results in the build up of carbon as seen by an increase in the Cis XPS peak centred at 283 .4 eV and then with increasing coverage, a second peak at 284 .6 eV [16] . The two carbon species are referred to as

282

283

284

285

286

Energy (eV) Fig . 5. Carbon deposition on Ni(111) . Two species of carbon are populated by increasing exposure to CO . Simulation of the peaks for the double pass HDA with 100 discrete detectors in 50 mm length at exit . The input was defined by two gaussians centred at 283 .4 and 284 .6 eV, each with a FWHM of 0.86 eV. The FWHM was chosen from the C l s peak from a pure graphite crystal [16] .

carbidic and graphitic carbon, respectively, based on the relative activity of the two species to hydrogen . During the catalytic reaction the two species co-exist and it becomes important to distinguish them in the spectrum . In our simulation we used an input profile consisting of two gaussians centred at 283 .4 and 284.6 eV, respectively, each with a FWHM of 0 .85 eV, which is the observed FWHM of Cls from graphite [16] . The results of the simulation for the double pass configuration with a 100 channels detector is shown in Fig . 5, demonstrating that the entire energy window of interest can be seen in a single shot at a pass energy of 50 eV and with adequate resolution to clearly distinguish the two species of carbon . In this spectrum the non linearities of the double analyser are not important . It should be noted that the snap-shot mode is advantageous provided that the gain across the detector is sufficiently linear and the excitation source is sufficiently bright to provide sufficient counting statistics in the snap shot . As far as the statistical quality is concerned, we estimate that using standard analyser entrance optics and a non-monochromatic laboratory X-ray source



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A . Baraldi. V.R. DhanaklJ. Electron Spectrosc . Relat . Phenom . 67 (1994) 211-220

operating at 300 W, a 1 s snap-shot of the silver 3d peaks would have about 450 counts in the 3d 512 peak, with a FWHM of about 1 eV . With the expected brilliance of ELETTRA, the statistics would be several orders of magnitude better .

100 a _ 0°

• 80

e

a

60 40

4. Transit time spread

. 20

Hemispherical deflection analysers are increasingly used for coincidence experiments where it is important to know the variation of transit times of the electrons through the analyser [17] . Here we present an analysis of the transit time properties of the double pass analyser and show results from trajectory calculations . The transit time in the double pass analyser can be calculated by considering separately the transit times T l and T2 in the first and second analyser respectively with appropriate correction for the energy change at the midplane where the first analyser meets the second . . We can calculate Tt and T2 by considering the equations of motion of an electron confined to a Kepler ellipse dA _ 1

2 mr dO =1dt

dt

2

d0

(19)

2 r dt

where 1 is the angular momentum and A, the area defined by the coordinate r and angle 0, is given by

fa _ 1 I2 2 1 (20)

.

0 40

0

80

120

160

Energy Eo (eV)

Fig. 6. Difference A T in transit time between two electrons with initial energies E 0 f AE/2(a = 0), as a function of E 0, where E0 is the mean energy and AE corresponds to the energy window at Ep across a detector of 50mm length .

From ray tracing calculations the accuracy of Eq . (21) is determined to be about 0 .1% . Electron trajectory calculations were performed for electrons entering the 150 mm mean radius double HDA analyser with a range of energies E and angles a . Figure 6 shows the difference AT in transit time between two electrons with initial energies E0 + AE/2 (a = 0), where E 0 is the mean energy and DE corresponds to the energy window at E 0 across a detector of 50 mm length. In Fig . 7 we show the results for the difference in the transit time for two electrons with initial energy E0 and angle ±a . A linear curve fitting of these

A 2 rmk) Jo (1 + e cos O)2 de

where E is the eccentricity of the orbit . The total transit time Ttot = Tt + T2 is given by 7

_

12

Tot - mk

[~(I'3+I"')+I7r(E

1

r3

+

s2

1

n3)

+ .. _

1-

where we have used a binomial expansion of the integrand . In Eq. (21)1" and e" are the momentum and eccentricity in the second analyser. For an electron entering the analyser at the centre of the entrance slit with energy E and a = 0, Eq . (21) can be simplified to

C

E0

2+ 3

EE+ 0

3 (AE)2] 0

(22)

∎

a

•

5o •

0 0

•

100

0

50 eV

0 0

0

0

0

0

0 0 0 o p 0

0

0

2

4

6

8

10

12

a

Difference in the transit time AT as a function of angle a for two electrons with initial angle ±a and energy E 0 = 10 eV, 20eV and 50eV. Fig . 7.

Ttot = FRO 4

o 20 eV

∎

]

(21)

∎ 10 eV'

∎

150

220

A . Baraldi, V.R. DhanakfJ. Electron Spectrose . Relat. Phenom . 67 (1994) 211-220

results yields the relation

AT 7, = 4 .15a

(23)

where AT is the base width and T is given by Eq. (22), with AE = 0 . It may be compared with a single HDA, where the numerical constant in the equivalent equation is 2 .1 [7] . Two comments may be made about these results . Firstly, it is seen that the requirements for good transit time spread and energy resolution are in conflict, which is in agreement with results for a single HDA [7] . Secondly, for a 300 m circumference storage ring such as ELETTRA the separation between two synchrotron light pulses in the single bunch mode is about 1 ps which is greater than the time scale for an electron to traverse through the analyser at low pass energies (i.e. in the high energy resolution mode) . It would therefore appear that the analyser distinguishes photoelectrons corresponding to a particular source pulse . However, in order to exploit this capability, each pulse would be required to have sufficient photon counts to give rise to a statistically meaningful number of photoelectrons and the detector electronics would have to be fast enough to separate the photoelectron pulses in time. 5. Conclusions

show that time resolved experiments are possible . Expressions for the energy resolution, energy window, and transit time spread have been given . The optimum acceptance angle and entrance slit size are discussed . Acknowledgements The authors wish to thank Drs . K .C . Prince and G.C. King for many useful discussions and critical reading of the manuscript . References 1 2 3

4 5 6

7 8 9 10 11 12

Electron optical characteristics of the double pass hemispherical analyser have been derived analytically and by numerical calculation of electron trajectories . The results show that the dispersion of this analyser is twice that of a single hemispherical analyser, in agreement with published results . We have extended these results to multichannel detection considerations and

13 14 15 16 17

G. Margaritondo, Introduction to Synchrotron Radiation, Oxford University Press, 1988, and references cited therein . Sincrotrone Trieste, Scientific Division - First Phase, Internal Report, 1990 . J . Gelius, L . Asplund, E . Basilier, S . Hedman, K . Helennelund and K . Siegbahn . Nucl . Instrum . Methods B, 1 (1984) 85-117 . L .J. Richter, W .D. Mieher, L .J . Whitman, W .A . Noonan and W . Ho, Rev . Sci . Instrum., 60 (1989) 12 . A. Mann and F. Linder, J . Phys. E, 21 (1988) 805-509 . F. Hadjarab and J.L. Erskine, J . Electron Spectrosc . Relat . Phenom ., 36 (1985) 227, and references cited therein . R.E. Imhof, A . Adams and G .C. King, J. Phys. E, 9 (1976) 138 . R . Herzog, Z . Phys., 97 (1935) 596; 41 (1940) 18 . K, Jost, J . Phys . E, 12 (1979) 1001-1005 . B.A. Gurney, W. Ho, Lee J . Richter and J .S . Villarubia, Rev . Sci . Instrum ., 59 (1988) 22-44 . Eue-Jin Jeong and J .L. Erskine, Rev . Sci . Instrum ., 60 (1989) 3130 . A . Baraldi, V.R . Dhanak and G .C . King, Meas . Sci . Technol ., 3 (1992) 778-779 . C.E. Kuyatt and J .A . Simpson, Rev. Sci . Instrum ., 38 (1967) 103 . H.D . Polaschegg, Appl . Phys., 9 (1976) 223 . A . Baraldi, Laurea Thesis, University of Trieste, 1991 . A .A . Dost, V .R. Dhanak and S . Buckingham, J . Catal., 89 (1984) 159 . U. Amaldi, A . Egidi, R. Marconero and G. Pizzella, Rev . Sci . Instrum ., 40 (1969) 1001-4 .