Antony, M

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y jEQUATIONS OF ELECTRON TRAJECTORIES UNDER THE INFLUENCE OF ORTHOGONAL ELECTRIC AND MAGNETIC FIELDS IN A SEMI­CIRCULAR SPECTROMETERM. ANTONYInstitittb o f  Okology, 1, R ue Blessig, Strasbourg Uniybbsity Strasboubg-Franob(Received December 26, 1907 ; Resubmitted April 27 and July 22, 1968)ABSTRACT. Kquations of electron trajectories under the influence of orthogonal electric and magnetic fields in a somi-circular spectrometer are derived. The influence of preaoceleration on the resolving power and the transmission of the spectrometer are discussed.I N T R O D U C T I O NReRolution less than I in 1000 can be set in a semi-circular spectrometer. Using radiographic films, and nuclear emulsion plates, a large part of the conver­sion electron specitrum (ian be surveyed and important informations about nuclear structure can b(^  obtained. However, photographic detecters have lower energy limit of d(‘tection around 7 keV. By using preaeceleration technics, electrons down to zero energy can be recorded. Tn the present article, we derive equations of electron trajectories under the influence of orthogonal electric and magnetic fields, and show that preacelcration does not seriously affect the resolving power and transmission of the spectrometer.E Q U A T I O N  O F  E L E C T R O N  T R A J E C T O R I E S  Figure Ij is a trihedral Oxyz showing the magnetic, electric and velocity vectorsFigure 1, Trihedral Oxyz showing BiEj, rB, E, and v respectively with 0 as the centre of the radioactive source emitting309electrons. The Lorcntz force for an electron under such conditions is given by e[E +  V AB] withi j  k V A B X y z 0 O ^ B3 1 0  M . AntonyThe equations of motion are :mx =  eE—eyB my ~  exB mz =  0whon^ m is the electron mass.For the initial conditions(1)(2)(3)=  .Vo =  2o- % f'OVFrom equations (3),From equations (1) and (2), we getz — v^t at time t.1  (*+»V)— (a:+»y) =  --at m mEx-\-iy =  Ae^*-\-i ^  (where A =  oe*“ , and u> is the angular velocity) B=  a(oosa4*i sin a)(oos sin wt)+iEIB a and a are defined by the initial conditions :— a cos a+ia  sinHence, we have(4)''0* a cos a, and Vqi^  ^  a sin a+E/B,Prom equation (4), we get ;»(0  =  «0x 008 ( *'0»“  g  ) (6)y(t) == ( ^  ) cos w«+t>0* sin wt^ ^  ... (6)(cos wt—\)Electron Trajectories under Electric dk Magnetic Fields 3 1 1x(t) =  sin wi-\- - ( *>,w w \1 / . JS'' \ B )z(t) =... (7)y{t)  ^ / v„„— ) sin wt— ^  (oosm;«—1)+  f  < ••• (8)W \ £> I W £>... (9)Equationw 6 to 9 define completely the trajectories of the electrons with E and B.P A R T I C U I .  A R  C A S E  O F  P L A N E  T R A J E C T O R I E S ,  O R T H O G O N A L  T O  T H E  M A G N E T I C  I N D I ^ C T I O N S  B, W I T H O U T  P R E A C C i: L E R A T I O NFor v„2 =  0, z{t) =- 0, the trajectory lies in the xy plane, and with E 0, we have :[ x -X  = sin (cos 1)w wy = r sin wt— (cosu;f—1)w w>w )F + 1 1( -/4 -H o«\ 1" V ±  V  = . 0^*\The trajectory is a circle with the centre : and radiusP = Vowi ^I eBIn figure 2, the origin is displaced by a distance 2D from the plane of the detector, thus limiting the angle o f emission in the xy plane.Suppose that andX  =  —2D+X,T ^ y .The trajectory is still a circle given by the equation :a) Suppose that the electron trajectory makes an angle with the X -a x is , and passes throaigh tho centre of the diaphragm as shown in figure 2. Let us precise the initial conditions :3 1 2  M. AntonyFigfuro 2. Showing thf* central trajectory, with initial velocity making an angle a© with O X ; the origin is displaced to X  *« x ~ 2D.% x  = "  c o s  OLqand =  Vq sin a^ .For X  =  0, the trajectory will cut the Y aids at two points Fj, and Fj defined by :i.e.,r i7 ,^  sin a« 1i“ +  (' f - - cos OCq )2 „  2=. -z?-Q| *o> / O) / <xrcos T -\ / V cos ag—4I> (D + ^ “ sina„jCO 0)2The central trajectory is given by Fj =  0, thus defining the angle of emission by the condition  ^^  j  sin a# =  —D.The point o f impact on the plate is F j =  F  ^=  2 ^  cos cos a,, whereBp IB the magnetic rigidity of the eleotron.b) Let us consider the case of a trajectory situated in a plane parellel to the median plane, but whose initial velocity makies an angle a == withOX.Electron Trajectories under Electric dk Magnetic Fields 3 1 3Figaro 3. Shuwmg the initial velocity making an angle a with the X  axis.The trajectory will cut OY at two points :cos u i  i - fI  ^ cos  ^a JUsing the relation Vq sin ao =  — we get the point of impact on the plate ; Y ^  2vq cos r j _  ^0 a j^— sin a) I ^ o> L cos^ a JY  cos a  f j _  sin a(,(sin a,)—sin a) ^ cos L2 o ix  cnxi ut-y Tcos^a J'Let us recall that Yq is the point of impact for the central trajectory. From the above discussions, wo conclude tliat a point source emitting monoenergetic electrons produces a ray of a finite width on the detector. The resolvingpower of the spectrometer is defined by the quantity  ^==cos g r J cos ao 1sin a(,(sin ao—sin a) cos  ^aThe source has a finite width being placed in the plane parallel to that of the detecter. The total resolution is given by the expression :=  £  -4- , where is half of the solid angle of emission.To 23 1 4 M . AntonyI N F L U E N C E  O F  F B E A C C E L E . R A T I O N  O N  T H E  R E S O L U T I O N  A N D  T R A N S M I S S I O N  O F  T H E  8 P E C T R O M E T E KFigure 4 shows the case of a homogeneous accelerating space between the source fixed to a plane electrode, and a grid parallely situated at a distance d•2D-Figure 4. Preaoceloration, showing tho grid*from the source. At this geometry, the motion of electrons is defined by the equations (6) to (9). To estimate the influence of preacceleration, we consider two cases :1) an electron emitted an angle with the initial velocity and energy2) an electron emitted an angle a with initial velocity Vq, energy ti>o, and ac­celerated under a potential of V Kilovolts such that on leaving the grid, its energy bears the relation TTq =  (0^ 0+  l^)keV.If T and Te are tho respective times of stay of the two electrons in the accele­rating medium.T =  . wT =  - —Fq cos ao cos 1, wTe =dcos asin wT =  wTf and cos (oT =  1 —  ^ j .At the exit of the grid, for an accelerated electron,i ( r . )  =  • »  ( 1 -k T .)  -  %  ( 1 -Electron Trajectories under Electric Magnetic Fields 3 1 5y{TB) =  v^Tb— , where =  Vqw  Iand y = eEm2 , H  CO® == 0^acceleration due to E in the direction OX,For an electron without preacceloration,Y ( T )  =  FoyT— Fo^  where Fq* ==- T"o «os a„ and Foy =  Fo sin ot  ^The relative displacement between the two images isIn order to associate the two trajectories, we consider =  0.Then, =  F,oy T e1 +y  2  ^0F 2 ^fix - jThe above equations show that we have to diminish the distance d between the source and the grid, subject to discharge conditions. The relative displacementoi^T^2  *is of the orderFor a spectrometer, where d 6 mm, radius of curvature ~  126 mm,2 2 \ r lThus, the position of a ray with preacceleration is displaced by a quantity equal to the geometrical resolution of the spectrometer in comparison to a ray without preacoeleration. However, for electrons of low energy, the resolution depends largely on the dimensions of the source. The above discussion is comparable with experimental observation obtained by studying the l48,08keV ray 1 of ThB, using Kodirex films as detectors, and applying a potential of 6KV.T R A N S N I S S I O Ntransmission is a function Ay due to the preacoeleration is negligible. Thus o^» and hence transmission remains practically unaffected.This work is a part of the thesis for the doctorate degree present^ by the author at the University of Lyon in May 1967. We thank Professor R. Chery of the Institute of Nuclear Physics, LYON, for his valuable guidance.

Equations of Electron Trajectories Under the Influence of Orthogonal Electric and Magnetic Fields in a Semicircular Spectrometer

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Equations of Electron Trajectories Under the Influence of Orthogonal Electric and Magnetic Fields in a Semicircular Spectrometer

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