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A SectorType DoubleFocusing Magnetic SpectrometerEarl S. Rosenblum Citation: Review of Scientific Instruments 21, 586 (1950); doi: 10.1063/1.1745661 View online: http://dx.doi.org/10.1063/1.1745661 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/21/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A simple doublefocusing sector mass spectrometer with permanent magnets Rev. Sci. Instrum. 66, 4894 (1995); 10.1063/1.1146526 Highperformance doublefocusing mass spectrometer Rev. Sci. Instrum. 56, 214 (1985); 10.1063/1.1138333 A Sector-Type Beta-Ray Spectrometer: A Laboratory Experiment Am. J. Phys. 41, 242 (1973); 10.1119/1.1987183 SecondOrder Properties of DoubleFocusing Spectrometer with Nonuniform Magnetic Field Rev. Sci. Instrum. 29, 943 (1958); 10.1063/1.1716064 The Design and Construction of a DoubleFocusing BetaRay Spectrometer Rev. Sci. Instrum. 19, 771 (1948); 10.1063/1.1741157
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THE REVIEW OF SCIKNTIFIC INSTRUMENTS VOLUME 21. NUMBER 7 JULY. 1950
A Sector-Type Double-Focusing Magnetic Spectrometer
EARL S. ROSENBLUM
Case Institute of Technology, Cleveland, Ohio
(Received May 26, 1949)
An analysis is made of the focusing characteristics of a magnetic spectrometer having an annular magnetic field which varies approximately with l/r!, thus permitting axial as well as radial focusing of particles of a given energy. By considering the field angle and source location as independently variable, the analysis provides for an instrument more flexible in use than similar existing types.
An optical analogy is developed from which a first approximation to the field shape is made, and from which the source and image location, magnification, dispersion and resolution can be readily obtained. Information thus obtained, combined with the results of an investigation into the image shape based on a series expansion, through second order, of expressions for the field strength and particle coordinates, yields more detailed information on the field shape which gives optimum focusing conditions and resolution for any chosen set of initial conditions.
Comparisons are made with the homogeneous field spectrometer and with the spectrometer of Svartholm and Siegbahn.
INTRODUCTION
A MAGNETIC spectrometer which achieves axial as well as radial focusing of charged particles by
means of a magnetic field which varies radially as I/r!, with both source and image within the field, has been developed by Siegbahn1 and Svartholm. 1• 2 They showed that simultaneous radial and axial focusing is achieved after the particles have passed through a field angle of 2!1l' (254.5 0
). Shull and Dennison3 developed the path equations by standard perturbation methods, giving results through the second order of approximation, extending the theory to provide for any field whose axial component is expressible as a series representing arbitrary corrections to the field existing at a reference radius, which corrections are functions of powers of the radial displacement from the reference radius.
Snyder, Lauritson, Fowler, and Rubin4 have built a spectrometer of this general type, but with the field having an angular extent of 180 0 and with source and image outside the field boundaries, approximately equidistant from the edges of the field.
This paper proposes to develop the path equations for a similar spectrometer, considering both the field angle and source location as independently variable. A set of equations are developed representing the first-order focusing characteristics in terms of an analogous thick lens system in geometric optics.
From these equations, a first-order determination of the field shape is made, information as to the location of source and image, limits on field angle, magnification and dispersion are obtained, and a preliminary estimate made of the resolution. From the second-order equations, choices of a second-order field shape parameter are made, which will cause the instrument to have a high resolution and good focusing characteristics, as
1 N. Svartholm and K. Siegbahn, Arkiv. f. Mat. Astr. o. Fys. 33A, No. 21 (1946).
2 N. Svartholm, Arkiv. f. Mat. Astr. o. Fys. 33A, No. 24 (1946). 3 F. Shull and D. Dennison, Phys. Rev. 71, 681 (1947). 4 Snyder, Lauritsen, Fowler, and Rubin, Phys. Rev. 74, 1564
(A) (1948).
computed through second order, for a chosen set of initial conditions.
Comparisons are made, throughout the development, with the spectrometer of Svartholm and Siegbahn, and with the homogeneous field spectrometer.
THEORY
Most of the symbols used in this paper are indicated on Fig. 1. It will be noted that the coordinates r, rp, and z are used within the field, region III, while two sets of x, y, and z axes are used, single primes indicating coordinates in the object space, region I, and double primes indicating coordinates in the image space, region II.
Throughout the paper, all distances will be given in units of the reference radius, which thus does not appear explicitly in any of the equations. Using this convention, Shull and Dennison's equations3• 5 for the axial and radial components of the field become:
H z=Ho[1- (r-1)0+ (r-1)2iJ+z2(a/2-iJ)] Hr=zHo[ -a+2iJ(r-1)],
where a and iJ are arbitrary constants representing corrections to the field Ho, which is the axial field at the intersection of the reference radius and the plane z= 0, henceforth called the equilibrium orbit.
The path equations were shown by Shull and Dennison to have the following zeroth-order solutions:
r=1; z=O; r,p=vo=w=-eHolmj rp=wt.
To solve the path equations through second order, the mass, velocity, and coordinates of any particle in the field are expressed as their zeroth-order values plus first- and second-order correction terms.
r= 1+"Pl+,,2p2 Z="tl+,,2t2 ,p=w+,,{h+,,202
m=mo(1+,,'Y) V= vo(1 +"77),
6 F. Shull and D. Dennison, Phys. Rev. 72, 256 (1947).
(1)
586
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SEC TOR - T Y P E f) 0 U B L E - Foe U SIN G MAG NET Ie S PEe T ROM E T E R 587
where X is a parameter of smallness, PI, P2, S 1, S 2, {h, and (J2 are functions of the time, and", and 11 represent small variations in mass and velocity from reference values. There are no second-order corrections to m and v, as the mass and velocity of a particle remain constant within the field.
When the above values of the variables and the field components are substituted into the path equations, the solution proceeds in a manner similar to that described by Shull and Dennison, with the addition of terms in '1 and ",. It is convenient at this point to eliminate the time from the equations of motion, so that the particle coordinates rand z are given in terms of the angle cpo
Then, to extend the solutions so as to describe the motion in the image space in terms of the initial conditions at the source, it is noted from Fig. 1 that, if E', K', E", and K" are considered to be small angles,
Yl=b'-K't' Zl =d' - E't'
(2)
The asterisk is used to indicate the values of r, cp, and z and their functions at the III-II boundary. Also,
K"= (I/r*) (ar/acp)*, E"= (l/r*) (az/acp)*. (4)
Substituting Eqs. (4) and the general relationships, Eqs. (1) into Eqs. (3) give
Y" = X[p1*+X"(apl/acp )*] +X2[p2*- x" P1*(ap1/ acp)*+X"(ap2/acp)*] (5)
z" = X[s l*+X" (as 1/ acp)* +X2[S2*- x" P1*(US1/acp)*+x"(as2/acp)*]. (6)
The solutions of these equations, after substitution of the path equations within the field and their first derivatives with respect to cp, and Eqs. (2), and letting cp=<I> at the III-II boundary, are
y"= K'[ -t' cos(I-a)!<I>--_I- sin(l-a)!<I> (l-a)t
+x" {(1-a)1t' sin(I-a)!<I> -cos(l-a)t<I> ]
11+'" +--[ -cos(1-a)!<I> 1-a
+(l-a)ix" sin(I-a)!<I>+ IJ
-b'[ -cos(l-a)'!<I>+ (l-a)!x" sin(1-a)!<I>]
+ higher order terms. (7)
Z" = E'[ -t' cosa!<I> -~ sinal<I> at
+x"(a!t' sina!<I>-cosal<I» ]
-d'[ -cosa!<I>+x"a! sinal<l>]
+ higher order terms. (8)
In order to obtain a perfect first-order focus, all rays from a given point on the source must come to a common focus at the image, whose coordinates are x" = t", y" = b", and z" = d". So, for simultaneous focusing in both the y and z directions, the coefficients of the initial angles K' and E' and Eqs. (7) and (8) must be zero at the image. Equating these coefficients to zero, and solving for t" in both cases,
[t' /(I-a)!] cot(l-a)t<l>+(1/ {l-a l) t"=--------------
t' -[I/(l-a)l] cot(l-a)!<I>
(t'jat) cotat<l>+(l/a)
t' - (1/ at) cota!<I> (9)
If a is given the value of t, this equation is satisfied for any combination of t' and <1>, representing a field shape which permits flexibility in the choice of initial conditions.
Certain other values of a, within the allowable limits of O<a< 1, will satisfy this equation. However, for any such a7"'t, there is at most one value of <1><211' for each t'~O; an instrument so designed would have an inconveniently arranged geometry, would be inflexible, and its radial focusing characteristics would be different from the axial ones. Such cases will not be considered here.
m
o (a)
I m ][ Z' to.
~~PARTICL~ " t===r==l~8RIUM ""TH, \&-0) I-- ::-----l '"
(b)
FIG. 1. Path of particle between source and image. (a) Section perpendicular to axis. (b) Axial section taken along a constant radius.
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588 EARL S. ROSENBLUM
I m II y' y'
...... ;<:-----i - - f- - - - -"t" UWIT$ OF FIELD----i-- f'lK A :F N!
,'...L===~~=:;::==I>=====+==p~=FT.__,. ...... IE' b-
.... ~--- - ~-+-+--~~ h_: f-h
,---."'- PRINCIPA/"---- r / PLANES ~
PRINCIPAL FOCI ~
FIG. 2. Radial trajectories, based on optical analogy.
OPTICAL ANALOGY
Letting a=t in Eq. (9), either part of the equation, when rearranged, gives
2t cot<I>/2i= (t'ttl - 2)/ (t' + ttl) = g, (10)
where g is merely a symbol, for the moment, used to identify this equality. However, if 1'= 00 in Eq. (10), I" = g; similarly, if ttl = 00, t' = g. Therefore, g is the location, expressed as a distance from the field edges, where rays from infinity are focused, thus locating the principal foci of an analogous optical system. This parameter, and others derived below, are shown in Fig. 2, in which the equilibrium orbit is considered to be a straight line, for convenience.
The principal planes of the system can be located from the equations for the image coordinates, which are obtained from Eqs. (7) and (8), letting a=t, using the coordinates of the image, and recalling that the coefficients of K' and E' vanish at the image.
b" = b'M + 2('1]+'1')(1- M)
d"=d'M,
(11)
(12)
where M = cos<I>/2!-ttl/2! sin<I>/2!. M, then, is the magnification, and is the same in the radial and axial directions.
For unit magnification, either Eq. (11) or (12), with Eq. (10), show that there is a distance h, given by
(13)
such that h= t' = t". Therefore, h is the distance from either field edge to the principal planes of the system. Because of symmetry considerations, the nodal points lie on the principal planes.
An incidental result with t" = h is that the coefficient of '1]+'1' in Eq. (11) becomes identically zero indicating that the dispersion of the instrument is zero at the principal planes. However, the significance of this fact is limited, as explained below.
The focal length j can be determined directly from the values of g and h as, by definition,
j=g-h, from which
j= 2t csc(<I>/2t).
(14)
(15)
The optical analogy is completed by combining Eqs.
(10), (14), and (15), resulting in the familiar relations
,(2= (t' - g)(t" - g)
l/j= 1/(1' -h)+ 1/(t" - h).
(16)
(17)
Results Obtained from Optical Analogy
Several items of useful information can be obtained from examination of the equations derived above.
Although the total focusing angle is always an integral multiple of 180°, for a homogeneous field, 6 a comparable relationship does not exist for the present case, and the total focusing angle A + B+<I> (Fig. 1) is not the same for all values of I' and <I>.
Source and image move relative to one another in the same manner as for a thick optical lens; as the object moves toward the field edge, the image moves away, and vice versa.
From Eqs. (11) and (12), information on the magnification can be obtained. At ttl=g, M=O, indicating that a point image is formed at the focus, as expected. As the source moves in from infinity, the image moves out from t" = g; from the above equations M is always negative for <I> < 7r'2 t, so that the image is inverted, increasing in size directly with t". (For 7r'2 i <<I><271'2!, the image is erect.)
From Eq. (10), it is seen that object and image distances are eqU'al when
t' = t" = t= g±j.
When t= g- j, 1= h from Eq. (14), representing unit magnification at the principal planes. When t= g+ j, b" = -b' and d" = -d', representing an inverted image of unit magnification. The spectrometer of Svartholm and Siegbahn1 represents a special example of this case, with t=O.
Using standard conventions of geometrical optics, a typical path is plotted in Fig. 2 for a particle having momentum slightly greater than that necessary to follow the equilibrium orbit. Referring to Fig. 2,
b" = Yf+ f(Yf-b')/(t' - g),
while, from Eqs. (11), (16), and (17),
b" = 2('1]+'1')+ f[{2('I]+'Y)-b'} let' - g)].
Comparing the last two equations,
Yf= 2('1]+'1').
A similar substitution into Eq. (12) shows that for the same particle, Zf=O.
This indicates that particles of a given ('1]+'1') have an equilibrium orbit in the median plane, displaced from the equilibrium radius by Yf= 2('1]+1').
Herzog6 has shown that for the homogeneous field spectrometer, Yf= '1]+'1', representing only half as great a displacement as above. For further discussion of this factor, see the section on Dispersion and Resolution.
6 R. Herzog, Zeits. f. Physik 89, 447 (1934).
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SEC TOR - T Y P E D 0 U B L E - Foe U SIN G MAG NET I C S PEe T ROM E T E R 589
Limits of Optical Analogy
The analogy does not describe the motion within the field boundaries, and so is valid only for t', t" ~ o. Thus, the statement above that unit magnification is obtained at the principal planes is true only if they are outside the field, which is the case only for <I»1r21. Also, the analogy will not account for an image obtained within the field. However, such an image can be located directly from the path equations or by considering that the field is cut off at that point, and letting t" = o.
Then, from Eq. (10), additional limits can be set to the location of source and image. That equation shows that source and image distances are negative when, respectively, the image and source distances are less than g. Furthermore, for g<O, the maximum values of t' and t" are obtained for t" = 0 and t' = 0, respectively. From Eq. (10), either maximum distance is -2/g. For g ~ 0, the maxima are infinite.
Within these limits on object and image distances, the location of object and image and the magnification can be determined either graphically by the methods of geometrical optics, or by the use of the optical equations.
Dispersion and Resolution
The momentum of a particle which follows the equilibrium orbit is po=movo, and the momentum of a particle having slightly different energy is, through first order,
p=mv= po(1+17+,..)
and, letting op= p- po, op/ po= 17+,... Let fib" be the Tadial change in image coordinates, for a given change in momentum. From Eq. (11),
fib" = 2(1/+,..)(1- M), or, letting
D=2(1-M), [,b"=D(17+,..). (18)
Then ob" /(op/p) = D(17+,..)/(17+,..)=D, which defines the dispersion of the instrument. From Eqs. (18) it is seen that D varies in absolute value directly with t", so that for best dispersion t' = 0, or I' = g, whichever is greater. The dispersion is zero with the image at a
w ... .. 0 .. .. .. .. ... ... if .. if ~ u!! " g§ ..
!u .. .. ~ H it ..
U ;L
TABLE 1. Typical parameters for first-order focusing.-
t" t' yo" D Do
g 00 0 2 00
00 g 00 00 2/yo' g+J g+f -yo' 4 4/yo'
It It yo' 0 0
-2/g 0 yo' sect> /2' 2(1-sect>/2') 2
,(1-cos<I>/2') yo
0 -2/g yo' cos<I>/2' 2(1-cos<I>/2t) 2
- (1-sect>/2!) yo'
• Note: Not all of the quantities tabulated represent real images for all values of the field angle 1>.
principal plane ;D= 4 for the case where t' = t" = t= g+h. This last figure is twice the value determined by Herzog6
for the homogeneous field spectrometer with equal object and image distances, and checks Shull and Dennison's3 statement comparing the dispersion of the double-focusing type with the 1800 homogeneous field type.
The ratio of the dispersion D to the image width yo" is the reciprocal of the smallest resolvable percent change in momentum. Defining this ratio as the resolution, Do, and expressing yo" in terms of the object width yo' by means of Eq. (11),
Do=(2/yo')[(1/M)-l]. (19)
From this equation it can be shown that Do is infinite for t" = g and decreases asymptotically to a magnitude of 2/ yo' for t" = ()'J. At t" . .: h, Do= O. For the particular case where t' = t" 0 £= g-'t- j, I Do I = 4/yo', compared to the value Qf 2/yO' dete'.mined by Herzog for a comparable geometry with a homogeneous field.
'1\ computation of the resolution based on first-order image coordinates is somewhat deceptive, due to second- and higher-order defocusing; however, these values of Do are useful in obtaining a preliminary determination of optimum operating conditions.
Since it may be desirable in certain applications to design an instrument with a predetermined object distance, it will be of interest to select the field angle <I> which will give the best dispersion and resolution at fixed object distance. Expressing D and Do in terms of
... .. .. .. .. ... :. if .. ~!: u ~ .. z .. !u iii ~ ;L U ..
+----:~--+-_=:::::_----L_::::::;..I" +--+-----1,......---"=-;-_1" -t+--!=2=~~~;;;:""'-I"
tr-' q,<T1\'2"
9 < OJ 11 < -~; f> +'/"Z 1I'1l <
9 > 0; Q> h > 0 i f < -'fr
FIG. 3. Graphical summary of focusing properties. Nole: I', Yo", Do, and D are all plotted as ordinates, to arbitrary scales.
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590 EARL S. ROSENBLUM
t', and then finding the values of aD/arf> and aDo/arf> and equating them to zero, the following field angle and object distances are obtained:
rf>=2! tan-I ( -t'/2!); 1"=0,
which may be an awkward arrangement, physically. Furthermore, the second partials indicate that, while the first-order resolution is a maximum, the dispersion is a minimum. Therefore, nothing can be concluded as to the best field angle at fixed object distance, on the basis of first-order solutions only.
Summary of Parameters for First-Order Focusing
In Fig. 3, values of t', yo", D, and Do are plotted as functions of t" for several ranges of rf>. Table I gives numerical values of these quantities at certain values of t", from which the magnitude of the various ordinates plotted in Fig. 3 can be estimated.
As a special case, rf>=1I"2!. Here, g= 0() , so t" = -t', from Eq. (10). The only real solution, then, is for t'=t"=O. This is the instrument described by Siegbahn and Svartholm.1 Note that the first-order magnification, dispersion, and resolution are the same for this instrument and for the sector-type spectrometer with equal object and image distance, g+ j.
As another special case, rf>=1I"/2 i . For this case, g=O, and the curves are the same as the first illustration of
Fig. 3, with the origin of t" coinciding with g. For this angle,j=+2 i , h=-2 i , and t't"=2.
Effect of Non-Radial Field Edges An attempt has been made to determine whether or
not the performance of the instrument could be improved by shaping the field edges so that they are straight lines making arbitrary angles with the y' and y" axes, respectively (Fig. 1). However, such a modification destroys the flexibility of the instrument, in that there is no value of the field-shape parameter which will allow radial and axial focusing with various object distances. If these angles are considered as small angles, this objection disappears, but the accompanying second-order effects consist of a slight broadening or simply a shift of image position.
Furthermore, although the effects of fringing fields are not discussed quantitatively in this paper, it is probable that their defocusing effects would be as great in magnitude, or greater, than the effects due to this type of field-shape correction.
SECOND-ORDER SOLUTIONS AND CHOICE OF PARAMETER ~
If the operations indicated by Eqs. (5) and (6) are performed with a=t and if terms in 7J and 'Yare neglected, the image coordinates become, in terms of the initial conditions at the source,
[ rf> t" <P ] l- 2/1- 1 8/1- 1 3 - 4/1 3 - 8/1J
b" = b' cos-- - -- sin- + (b' _;/1')2---+ K'2_--+ (d' - f'I')2 __ + f'2 __ -21 2' 21 ,,3 6 3
[ rf> t" rf>- [ 4{)-2 4/1J[ rf> rf>]
X cos---- sin-J+ K'(b' -K't') ----/(d' -f't')- t" cos-+21 sin-21 21 21 3 3 21 21
[ 4/1+ 1 /1J r 4/1+ 1
+ (Cb' - K'1')2 -2K'2} __ - - (d' -f't')2 -U2}- [cos21rf> -t"2! ~in21rf>]+ - K'(b' -K't')---12 3 _ 3
+dd'-f't,)4/1Jf2. sin21rf>+t" COS2!rf>]+[(b' _it')2+2i2]1-4/1 3 21 4
2/1-1 I" +[(d' -f't'?+U2] __ +-[(b'-K't')2-2K'2] sin2 l¢+K'(b' -K't')t" cos2!rf>. (20)
2 2!2
[ rf> t" rf> 1 [ 4/1 6 - 16/1 [ rf> t" rf>
d"=d' cos--- sin- + -(b' -K't')(d' -f'I')-+K'f'---] cos---- sin-J 2! 21 21 3 3 21 21 21
+ 2![ - f'(b' - K't') 4/1 + K'(d' - f't') 3
-4/1][21 sin rf> +t" cos rf> 1+[ ( (b' - K't')(d' - f't') 3 3 21 2'
3-4/1J [4/1-3 4/1-3J -2K'f'I-6- [cos21rf> -2 It" sin21rf>]+2! E'(b' -K't')-6-+ K'(d' -f't')-6-
4/1-1 X[21 sin2!rf>+2t" cos21rf>]+[(b' -K't')(d' -it')+2K'f']--
2
t" t" t" +-[(b' -K't')(d' -f't') -2K'f'] sin2!¢+-[f'Cb' -K't')+K'(d' -f't')] cos2!¢+-(b'f' -d'K'). (21)
212 2 2
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SEC TOR - T Y P E DOC B L E - Foe l: SIN G MAG NET I C S PEe T ROM E T E R 591
The most satisfactory solutions to these equations depend on the uses to which a given instrument is to be put, the space and equipment available, and the economics of the project. The initial conditions and field angles selected below for investigation were based on a desire for the following characteristics, which are enumerated together with their respective somewhat contradictory effects on the instrument design.
1. Source and image should be accessible for use with auxiliary equipment and for easy removal during an experiment. If they can be located in regions where the magnetic field is relatively weak, some difficulties in the use of certain types of sources and detectors are lessened. These requirements are best satisfied by maximum separation of source and image from the field edges. However, design on this basis, besides conflicting with items listed below, may cause undue increase in the volume and complication of the vacuum system.
2. If the instrument is to be used with particles of energies so high as to make collimation of the image within the instrument difficult, it is desirable to work with an image large enough so that the detector itself can act as the exit slit. In this case, the object distance should be a minimum and the image distance a maximum, so that the dispersion will be as large as possible for the limiting image width.
3. The instrument should transmit a maximum amount of the radiation from the source. With a given pole shape and separation, this requires that the source be at the edge of, or within the field. Pole separation is limited by the problem of maintaining the field shape over a large cross section, particularly due to excessive fringing near the inner and outer radii. With a large gap, power consumption will be quite high.
4. The resolution should be as large as possible. The narrowest possible image, together with maximum dispersion make for highest resolution. First-order focusing characteristics are not very helpful here, as second-order defocusing terms may be large enough to cause extremely large changes in the resolution. It has been found that (for a given object size and fixed limitations on radial and axial angles transmitted), when the field shape parameter {3 is chosen for minimum image width at each object distance, the resolution does not vary greatly with object distance.
The resolution can be increased by decreasing the object width and radial and axial angles, but with a corresponding decrease in intensity. The optimum ratio of these quantities and object length depends somewhat on the actual size of the instrument; a radius in the 10-to 20-cm range is assumed here.
Naturally, all these considerations could not be satisfied simultaneously; one immediate result has been that the object size and other initial conditions have been limited to smaller values than those allowed by Shull and Dennison.3 In the analysis below, somewhat arbitrary limitations on the design parameters have
TABLE II. Performance data with selected design parameters.
<I> t' t" g h f D D, ~=..!.. po D,
,../21 1 2 0 -21 21 4.83 154 0.0065 ,.. 0.7 0.7 -1.08 -2.86 1.78 4.00 174 0.0057
been chosen; as one compromise combination:
WI :::;0.Q1; Id'l :::;0.05; IK'I :::;0.1; I~II :::;0.05; 0.7:::; (t', til):::; 2.
The limitation on angle can be obtained by a suitably shaped baffle placed at cp=1r/2 i ; a rectangular-shaped baffle is assumed here. The restriction on object and image distances imposes limits on the field angle which are given roughly by
1r/2t :::;<1>:::;1r.
Equations (20) and (21fhave been solved at these two limits of field angle, and;' value of {3 has been chosen to result in the narrowest possible image, for (<I>=1r/2t, t'=1, tl/=2; (3=0.35) and for (<1>=11', /'=/"=0.701, (3=0.25).
On Fig. 4 is plotted the image of half of a rectangular source for the two cases, along with a rectangle representing the object size, and sections showing the image of a point object at two positions on the source. The image width is obtained from these plots, and using that width and the optical equations, the information tabulated in Table II is obtained.
With these combinations of initial conditions, about one-fifth to one-fourth percent of the total radiation from an isotropic source is transmitted. This represents a lower intensity than that obtainable with the same gap for the case where /' = til = 0, but with other accompanying advantages in flexibility, accessibility, etc.
COMPARABLE PATH EQUATIONS FOR THE HOMOGENEOUS FIELD SPECTROMETER
With a homogeneous field, a={3=O, the path equations within the field are those given by Shull and Dennison. These path equations can be extended to describe the image coordinates with an arbitrary set of initial conditions and field angle, but this is not done
cP • if; t'. 1.0; 11'.35
1M." O' POINT toURCr. AT B
'MA.l 0' POINT .Oultel AT A
d~d"
-""-;1"-'"-'_ b', iii ... CP'1T; r.0.7; "'.25
FIG. 4. Image size with selected field shapes and initial conditions.
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592 EARL S. ROSE~BLUM
FIG. 5. Path of particle in homogeneous field spectrometer.
in the present paper. However, Stephens7 has developed an equation for the case represented by Fig. 5. where b'=d'=E'=O. That equation is
bl! = _ (K'2) (t'2+ 1)!( si~2f + si.n2/1). 2 sm/l smif;
(22)
Now, substituting the same initial conditions into the path equation gives, at the image,
b"= (~)[ 1
-----+t' sin2<l> t' sin<l> - cosP
t'2 - 1 t'2+ 1 ] +--cos2<l>--- .
2 2 (23)
When the angles if; and /I in Eq. (22) are expressed in terms of angles A and P, using Fig. 5, that equation becomes identical with Eq. (23), thus affording a check on the two approaches to the problem.
CONCLUSIONS
With suitable choices of initial conditions and fieldshape parameters, the instrument desGribed in this paper is one of high intensity ~nrl resolution, although the intensity suffers when the source :S !CJt at the field edge. However, advantages are gained in flexibility and convenience which allow the instrument to be a useful one.
It has the same advantages over spectrometers of other types as does the Siegbahn spectrometer, the
7 W. Stephens, Phys. Rev. 45, 513 (1934).
latter being a limiting case of the spectrometer described above, with object and image distance set equal to zero, resulting in a focusing angle of 1r2!. This combination transmits a higher intensity than all other combinations, except for those with zero object distance, and does not suffer from possible defocusing effects caused by fringing fields (see below), so probably has as high a resolution as any other combination. However, the source and image are located in a strong field between the poles of the magnet, thus decreasing the flexibility and accessibility of the instrument. The sector-type spectrometer is also more suitable for determining the energy of particles which must originate in other equipment, and which should cross the fringing fields as nearly as possible normal to them, to minimize defocusing.
The effects of fringing fields have not been discussed in detail, but by analogy to Coggeshall's8 analysis of the edge effects of a homogeneous field, it is probable that these effects will not cause considerable defocusing, if source and image are located either completely within the field or in regions of low field strength, and if the particles enter and leave the field nearly normal to the fringing field.
Although the resolution as computed through second order neglects the defocusing due to higher order terms, the fact that the intensity is not uniform over the image indicates that the computed values of resolution probably can be closely realized. Shull,9 and Kurie, Osaba, and Slack10 have shown that the resolution of the instrument does not vary widely with variations of {3 in the range i:S {3:S i for the 1r2t instrument; it is considered likely that the behavior will be similar for an instrument with a different field angle.
The author wishes to express his appreciation of the help of Dr. E. F. Shrader, who supervised the work of this paper, and of Dr. E. C. Crittenden, whose suggestions led to the investigation of the double-focusing spectrometer and some of its more significant properties.
8 N. Coggeshall, J. App. Phys. 18, 855 (1947). 9 F. Shull, Phys. Rev. 74, 917 (1948). 10 Kurie, Osaba, and Slack, Rev. Sci. Inst. 19, 771 (1948).
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