Relativistic effects in the calibration of electrostatic electron analyzers. I. Toroidal analyzers

Journal of Electron Spectroscopy and Related Phenomena, 13 (1978) 107-l 12 @ Elsevler Sclentlfic Pubhshmg Company, Amsterdam - Prmted m The Netherlands

RELATIVISTIC EFFECTS ELECTRON ANALYZERS

0

IN THE CALIBRATION OF I TOROIDAL ANALYZERS*

ELECTROSTATIC

KESKGRAHKONEN

Laboratory

M 0

of Physzcs,

HeIsrnkr

Umversrty

of Technology,

02150 Es-00

15 (FMand)

KRAUSE

Transuramum

Research

Laboratory,

Oak

Ridge

Natronal

Laboratory,

Oak

Rx&e,

Tenessee

37830

(USA)

(Fnxt received 23 May 1977, m final form 31 August 1977)

ABSTRACT

Relatlvlstlc correctron terms up to the second order are derived for the lunetlc energy of an electron travellmg along the circular central traJectory of a toroldal analyzer Furthermore, a practical energy cahbratIon equatron of the spherrcal sector plate analyzer IS written for the varrable-plate-voltage recordmg mode Accurate measurements with a sphencal analyzer performed using kmetlc energies from 600 to 2100 eV are In good agreement with this theory showing our approxlmatlon (neglect of frmgmg fields, and source and detector geometry) IS reahstlc enough for actual cahbratlon purposes INTRODUCTION

Dunng the last ten years electrostatic electron spectrometers have reached a high level of sophrstlcatlon’P ’ and It has been possible to perform accurate energy deterwnatlons3 However, rn general httle attention has been pald to the relatlvlstic correction terms of the callbratron equation A standard formulal’ 3-6, specified to apply “to electrostatic Instruments”, has been presented but to our knowledge no experlmental evidence of rts valrdlty has been published In ths paper we re-derive the equation for the relatlvlstic hnetic energy T, as a function of the nonrelatlvlstlc kmetic energy Tfor the fatly of torolda1 electrostatic analyzers Thrs equation 1s expanded m powers of z = T/(m,c’) where m,c2 1s the rest energy of the electron Experimental results which we obtamed with a spherical * Part of this work was sponsored by the U S Energy Research and Development Admmlstratlon under contract with the Umon Carbrde Corporation

108 sector plate analyzer, with the theory DEFINliTION

a special case of toroldal

analyzers,

showed

good

agreement

OF THE PROBLEM

Electrostatic toroldal analyzers7 are formed by two concentric sector plates kept at potentials V+ and V- (Fig 1) The radu of curvature of the tori m the plane of the paper are R, and R, and perpendicular to the plane of the paper rl and r2 respectively This family includes as special cases spherlcal (R, = rl, R2 = rZ) and cylmdrlcal (rl = r2 = co) analyzers A narrow pencil of rays admltted from the source S through the analyzer IS focused at the point I m the nonrelatlvlstrc (NR) hmlt, but with higher energy the focus departs from L say to I’ At the same time the focus becomes astlgmatlc8, ’ and the posltlon of I’ 1s determmed as a point of least confusion for that energy This defocusing deteriorates the resolutron of the analyzer to some extent, but up to the first order the mean energy of the pencil 1s still given by the energy of the central raylo For real analyzers the exit sht IS fixed at the point I and thus outslde the focus for nonzero energies In the first approxlmatlon the entrance and exit slits can be taken as points Therefore we have to calculate the relatlvlstlc kmetlc energy T, of an electron ray pencil whose central traJectory crosses the point I We assume the relation between the plate voltages and the NR kinetic energy IS given by T=

T(V+,

I’-)

Figure 1 Toroldal electrostatic Defimtlon of symbols

(1)

analyzer cut along the plane contammg

the central

trajectory

109 For small z, T’ + T, and the relation T, = 7’,(T) can be expanded Into a power series T, = T[l

-I- clz + c$

+

]

(2)

where we have to calculate the coefficients c1 and c2 TOROIDAL

CALIBRATION

EQUATION

Wlthm the toroldal field the central trajectory 1s an arc of a circle along which the electric field has a constant value K, This field exerts a force eKo on the electron (e > 0) which balances the centrifugal force6’ 11’ I2 eK, = mvi/r,

(3)

Here v0 IS the velocity and m the mass of the electron and r. the radius of the central trajectory The kmetrc energy IS given nonrelatlvlstlcally T = i m,vi

(4)

and relativIstlcally (PO = vo/c) T, = m,c2 [(1 -

pz)-*

-

1]

(5)

Equations (3) and (4) yield the NR cahbratlon equation T=

eKoro

(6)

2

whereas eqns (3), (5) and (6) yield If solved for T, T, = T -

m,c2 + (T’

+ (m,c2)2)*

This can be expanded m power series 1 + ;

T + 0(r3)

1

(8)

Thus in eqn (2) for a circular central traJectory c1 = 3 and c2 = 0 However, the values of cl and c2 depend generally on geometry for analyzers with noncn-cular central traJectorres as shown m the followmg paper13 T, and Trefer to kmetlc energies wlthrn the field of the condenser assuming an ideal sharp cut-off of the field at the plate edges If the potential outside the condenser IS assumed to be zero, electrons entenng the condenser at the radius r0 lose (or gain) energy by the amount U,,, = U,,, (ro, V+, V-) Exphclt calculations of field strength K0 and potential at r0 yield equations for T and U,,, for a spherlcal sector condenser 12, 14.15

(9)

110

u r=t

RI R,

-

e

RZ -

RI [ -

V+ - vr.

and for a cyhndrlcal condenser’

;ep+-

T=

+

v+ --_ R,

1

vR,

(10)

’ 4Dl6

(-KR2)

v-)/h

(11)

[v+ln(~) + v-ln(.y],ln(~)

+=

(12)

The numerrcal determmatlon of T and U,,,, which cannot be expressed m closed form for general torordal analyzers, IS described by Wollmk et al ’ It was found recentIy I7 that a general toroldal field allows optlmlzatlon of the analyzer so that It IS m some regards supenor to exlstmg analyzers COMPARISON

WITH EXPERIMENT

FOR A SPHERICAL

SECTOR ANALYZER

The vahdlty of eqn (8) was tested experlmentaIly for a spherical sector plate analyzer built by Krause1 8 X-rays from an alummlum target were converted Into photoelectrons through the Ne 1s and 2s levels Usmg a preacceleratlon sht structure described elsewhere1 ‘, electrons with kmetlc energies from 600 to 2100 eV could be created The difference u of the kmetlc energies of electrons resultmg from the process under study and a reference process can be wntten19 u t= hv -

EB -

U -

U,,, - E, - (hv, - E; -

U, -

U,o,, - E,O) = T, -

T,O (13)

where hv IS the X-ray energy, E, the bmdmg energy of the converter level, - U the preacceleratlon energy and E, the recoil energy of the converter atom If an electron IS elected perpendicular to the dn-ectron of the Impmgmg photon E, 1s approximately E, = m,(hv

- E,)jM

(14)

where MIS the mass of the converter atom From eqn (10) the preretardatlon electron entermg the analyzer at r. = 3 (R, + R,) 1s U,,, =

-

e(V+R1

+

V-R,MR,

+ R,)

(15)

Substltutmg these two equations Into eqn eqn (8) yields (hv = hv,)

u= + R2(V-

(1- 2) -

V,)l

Q% -

Ei) -

=f(V

-

v,)

of the

(U

-

(13) and expressmg T, -

U,) + R,

1

R2

[R,(V+

-

q

by

V,*)+

(16)

111

whereT=fV=+f(V+a form

V-)

Denoting

V -

V, by v this equation

can be put mto

(17) which m the (v, U/V) coordmate

system describes a straight he

Photoelectron spectra were measured for the Ne 1s and 2s levels as a function of preacceleratlon energy by adjusting the analyzer plate voltages such that photopeaks appeared m about the same channel m the multichannel analyzer operated m the multlscahng mode’ ’ The acceleratron and plate voltages were measured with a dlfferentlal voltmeter to an accuracy of 1 x lo- 5, and the peak posltlons were determmed by fittmg Gausslan shapes to the photopeaks Smce the Al Ku 1, Z lme IS asymmetric the posltlon of the peak obtained from our fittmg procedure depends on resolution and thus on the kmetlc energy Model calculations showed, however, this effect was smaller than 1 meV m our experiment and thus neghglble The energy of the and the bmdmg energies of the Ne Is and Al KEI, 2 lme was taken from Bearden” 2s levels were averaged from the data of refs 3 Data are plotted m Fig 2 m terms of u/v and v as suggested by eqn (171, and are well represented by a strarght hne The least-squares fit yields the followmg values f = 4 0429(8) eV/V andf2/2m,c2 = 1 5( 1) x lo- ’ eV/V2 Using the experlmental value off, the relativlstlc coefficient f ‘/2m,c2 becomes 1 60 x lo- 5 eV/V’, and shows good agreement with our theory The theoretical value offcalculated from the dlmenslons of the analyzer IS 3 94(l) usmg eqn (19) The devlatlon of 3 % from the experimental value IS at least partly due to the neglect of the frmgmg fields For practical apphcatlons the experlmental value off should be used

LU~Sd”““.“l”“““““““““‘....’ 100

I 200

300

LOO

Figure 2. Ekperlmental venficatlon of the vahdlty of eqn (17) for a spherical sector plate The error bars gwen are maxlmum errors

analyzer

112 CONCLUSIONS

The analysis made shows that the energy-voltage relation of a toroldal energy analyzer m the variable-plate-voltage recordrng mode exhlblts relatlvlstlc nonlmearlty given by eqn (8) for the whole family For accurate callbratlon one must either make use of eqn (8) with pertinent formulae for T and U,,, or measure the “effective” cahbratlon factor with a standard kmetlc energy close to that of the lme to be measured Measurements with a spherlcal sector plate analyzer show that our ldeahzed treatment IS reallstlc and eqn (16) provides a convenient correction formula for practical work ACKNOWLEDGEMENTS

One of us (0 K-R ) gratefully acknowledges financial support from the ASLA/Fulbrlght-Hays foundation and the Umverslty of Tenessee under contract AT-(40-I)-4447 for hl s s t ay at the Oak Ridge National Laboratory m 1974 REFERENCES 1

2 3

4

5 6 7 a 9 10 11 12 13 14 15

16 17 18 19 20

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