New sector type double focusing beta-ray spectrometer

NUCLEAR

INSTRUMENTS

A N D M E T H O D S 65 ( I 9 6 8 ) 2 5 3 - 2 7 3 ;

© NORTH-HOLLAND

PUBLISHING

CO.

NEW SECTOR TYPE D O U B L E FOCUSING BETA-RAY S P E C T R O M E T E R H. Y A M A M O T O and K. T A K U M I

Ozenji Division, Hitachi Central Research Laboratory, Kawasaki, Kanagawa, Japan and H. I K E G A M I

Department of Physics, Tokyo Institute of Technology, Oh-okayama, Tokyo, Japan Received 13 May 1968 The design, construction and performance of a new sector type double focusing beta-ray spectrometer are described. Optimum performance has been investigated empirically. The spectrometer has a mean radius of 34 cm and a maximum transmission of 1.5% of 4 ~z. Resolution of 0.08% has been obtained with a source 1 m m wide and 10 m m high. It should be noted that a luminosity-resolution relation of the SDF spectrometer is

preferably compared with that of conventional z~/2 spectrometers. An electron trajectory in the fringing field and the focal line have been also studied both theoretically and experimentally. It has been emphasized that the spectrometer is also suited for spectrographical use with multi-detectors as well as for measurements of electron-gamma and electron-electron angular correlation.

1. Introduction

of 18 cm, had a resolution near the value estimated from the second order calculation, assuming an ideal field, even though the spectrometer has such a large gap. The fringing field did not have such a serious effect on the resolution as previously believed. However, the observed value of the maximum transmission of the INS-III spectrometer was about one half the expected one. It would be, thus, worthwhile to investigate the problem of the fringing field of the large transmission SDF spectrometer more carefully. It is, however, difficult to handle analytically the effects of the fringing field on determination of focusing point and the aberrations which will occur in an image. No reliable method of calculation has yet been established for SDF spectrometers with a large magnet gap. In designing our high-transmission SDF spectrometer, we assumed that the magnetic field falls off discontinuously at the effective field boundary which is defined so as to give equivalent deflection of electron trajectories. The effective field boundary was obtained by numerically integrating the distribution of the fringing field which was obtained by using the Schwarz-Christoffel 6) transformation method. Then, the location of the focusing point and the aberrations at the point were calculated. The reference trajectory in the fringing field region was also numerically calculated by using a digital computer H I P A C 103 and the results were compared with that of the ideal field. However, the source position and the field distribution which would give the optimum performance were finally found experimentally. Originally, the spectrometer was designed and constructed for an e-y spectro-goniometer about seven

With the recent advances in precision measurements in nuclear spectroscopy, there has been an increasing demand for spectrometers which have a high resolution and a high transmission in the measurement of internal or external conversion electron spectra and which are capable of coincidence- or angular correlation measurements. Here is the fact, that the sector-type doublefocusing spectrometers 1' 2) with nonuniform field (hereafter abbreviated as SDF spectrometers), in which a high resolution of less than 0.5% can be obtained rather easily, are considered well suited for these measurements, and especially for measuring the higher components of the Legendre polynomials in the angular correlation function. Developement of SDF spectrometers with a high transmission has seemingly lagged behind that of lens-type spectrometers, probably because of the fringing field effect. The purpose of this paper is to report a new SDF spectrometer whose transmission and resolution can be preferably compared with that of conventional rcV/2 type spectrometers. The construction of an SDF spectrometer with a large pole gap begun first around the year 1956 at INS*, and a small spectrometer (so called INS-I) was made to investigate experimentally the performance of a spectrometer having a mean pole gap more than one-half the reference radiusa). Also, experiments of electron and gamma-ray directional correlation with this spectrometer were worked out with good success*). The INS-III spectrometerS), which was constructed on the basis of these studies and which has a reference radius * Institute for Nuclear Study, University of Tokyo.

253

H. YAMAMOTO et aL

254

Unperturbed Field Region

15 X ~

i

I0 i

i

,

,

i

0

05 ,

,

'

i

i

,

,

,

~Field Boundary 05

/

~o

Fig. 1. The used coordinates and electron trjectories in the flinging field calculated by the approximations: 1, ideal field; 2, linear approximation; 3, exponential approximation; 4, the distribution obtained by eq. (7). All of the approximations have the same effective field boundary. years ago and n o w four such spectrometers* working very well.

are

Let us also assume that the fringing field can be expressed as follows:

2. Focusing properties of the SDF spectrometer

Hz(x,y) = Hoh(x,y),

2.1. THE FRINGING FIELD Few researches have been done concerning the fringing effect of n o n - u n i f o r m field magnets except for the arguments by K o f o e d - H a n s e n et al. 7' s), relating to the design o f orange-type spectrometers and by J u d d and Bludmang). F o r the sake of simplicity, the magnetic field of the S D F spectrometer is divided into the undisturbed-field region and the fringing-field region. Cylindrical coordinates and Cartesian coordinates, such as those shown in fig. 1 are used in each region. Let us assume that the field in the m a g n e t can be expressed as follows on the median plane:

h(x,y) < 1;

H~(r,O) = H o E l - a p + f l p

+...],

(I)

p - (r-ro)lro,

h(0,0) = 1,

Y = [P]x= o.

(2)

Ordinarily, it is quite difficult to find the analytical f o r m of h(x,y) for n o n - u n i f o r m field magnets. The

Mogne,,c

lZ0 g

zt X- Xo ~

Zg

J

/ / / / / 1 1 / / I / / / / / / / i

where r o is the radius o f the reference trajectory. F o r brevity t h r o u g h o u t this paper all the lengths will be given in the unit o f the reference radius ro i.e., ro = 1. * Working at Kyoto University, Tohoku University, Institute of Physical and Chemical Research and Central Research Laboratory, Hitachi Ltd.

Fig. 2. Schematie illustration of a magnet with a shield channel.

NEW SECTOR TYPE DOUBLE FOCUSING BETA-RAY SPECTROMETER following factorization a p p r o x i m a t i o n was, however, introduced by Judd and B l u d m a n 9) for continuation of the fields in the two divided regions:

h(x,y) = (1 - cry + fly 2+...)h(x),

(3)

where h(x) is the fringing field of a u n i f o r m field m a g n e t with an "equivalent m e a n m a g n e t pole gap". In general, x in the field function h(x) must, however, be normalized by the m a g n e t pole gap, which varies with y. Thus, the function h(x) must also depend on y t h r o u g h its dependence on the geometry of the m a g n e t pole and the magnetic shield channel, whereas Judd and Bludman assumed that the function h(x) itself does not depend on y. It is also, generally, preferable to use magnet pole - shield channel distance and a half height of the shield channel normalized by the half gap of the m a g n e t poles Zs(y ) (fig. 2). Then the function h(x,y) in eq. (3) m a y be represented as

As a special ease, if one m a k e s the shield channel as

a(y) = ao = constant; b(y) = bo = constant, then eq. (4) becomes

h(x,y) ~- (1 _ ~ y + fly2 + ...)x x h{(x/zo)(1-~ty+fly2+...);ao,bo}.

X/Zg = (Xo/Zg) + n - a[ln {(1 - he)/(1 + he)} + +2btg-l(behe)-l-aln{(aehe-1)/(aehe+

(5)

l)}],

(7)

where h~ = {(1 - t/t2)/(1 - t/tl)}h, h = (1 - t / q ) ( 1 - t/t2)-~(1 - t) -or, a¢ = ( 1 - t l / t 2 ) - l a ,

Here, a(y). Zg(y) and b(y). zg(y) are the half height of the shield channel and the m a g n e t pole - shield channel distance, respectively, and zg(y) is approximately expressed as follows: ...)- 1 z0,

(6)

The analytical expression of h(x/z o; a,b) for a m a g n e t having a shield channel as shown in fig. 2, is obtained by the use of the Schwarz-Christoffel transformation, assuming m a g n e t pole and shield channel of infinite permeability1 °).

h(x,y) ~- (1 - ~y + fly2 + . . . ) h {X/Zg(y); a(y), b(y)}. (4)

zg(y) _ ( 1 - ~ y + f l y 2 +

255

b¢ = (t2/tl)b,

(8a)

tl - ½[a2 + b 2 _ 1 + {(a z + b 2 - 1 ) 2 - 4 b } ½ ] ,

t2 - t2/b 2, where x 0 represents the location of the magnet pole boundary. In the case of a shield free magnet, one m a y

10

! i

4

05

Pole

I

5

i

4

3

2

X

I

0

Boundory

i

-I

-2

( in Z o U n i t )

Fig. 3. Distributions of the fringing field: Curve 1, shows an ideal field distribution; curve 2, a linear approximation; curve 3, an exponential approximation; curve 4, the distribution obtained by eq. (7); geometrical parameters a = 1.4, b = 1.0.

H. YAMAMOTO et al.

256

put b ~ oo in the formula (7) and obtain the well known expression 11)

X/Zg = (Xo/Zg)+ Tz-l[ln { ( i - h ) / ( 1 + h)} + 2/h].

(8b)

The expressions of h(x,y), eqs. (4) and (7) are, of course, not an exact form of the fringing field function but are useful to estimate the field form except for the cases of small values of the permeabilities of magnet and shield channel. By utilizing eq. (4), the displacement 3(y) of the effective field boundary is given by

6(y) =

h(x,y) dx - xo(y).

(9)

0

2.2. ELECTRONTRAJECTORYIN THE FRINGING FIELD In designing SDF spectrometers, one first estimates the locations of the source and the detector by assuming an ideal field distribution which falls off discontinuously at the effective field boundary. Since the actual

field falls off gradually in conformation with the laws of the potential theory, the trajectory deviates from that obtained when an ideal field is assumed. Concerning this displacement in a spectrometer with a non-uniform field, Judd et al. 9) used eq. (3), assuming that the fringing field drops exponentially. In order to investigate the fringing field effects on the reference trajectory, the kinematic equation was solved numerically, using a HIPAC 103 computer for the following approximate fringing fields: 1. Sharp cut off field,

6+Xo>X, hi = [1 ,

tO,

2. Linear tail field, x<0,= h2

hve F~oId ~9°Ot~

.- -- Efftec

x>6+Xo.

~

i', ½x/(6+Xo),

2(6 + x0) = > x= > 0,

x __>2(3 + Xo). 3. Exponential tail field, x_<~0, h3 = (1,

t expE-xl(6+Xo)],

Z

=

0

PLane

(a)

z

o.o

zl r ' ~

l

°=®

I

I

I i i

, I

~Z~e

(b) Fig. 4. Schematic illustration of radial and vertical focusing by sectoral field. Solid line in (a) shows a trajectory in the case of oblique entrance and exit, while dotted line shows a trajectory in the case of normal entrance and exit.

x > O.

(lo)

4. The field based on eq. (7); h4. In each case, h(x) was selected so as to give the same effective field boundary. These field distributions are shown in fig. 3 and corresponding electron trajectories calculated in fig. 1. In the case of a magnet with a shield channel, there is a general trend that the trajectory corresponding to h3 (exponential tail field) rather deviates from those corresponding to h2 (linear tail field) and h, (the most probable field). This fact can be clearly seen from figs. 1 and 3. We found that the deviation between the trajectories corresponding to hi and h4 was about 0.05 ( = 1.7 cm for an r 0 = 34 cm spectrometer) in the field free region. 2.3. FOCUSING PROPERTIES For the first order focusing property of an SDF spectrometer, reports have already been made by Judd, Svartholm 12) and Sternheimert3). Rosenblum14), and Judd and Bludman 9) have extended the discussion to the higher order properties. A possible improvement of the second order focusing properties by shaping the magnet field boundary has been discussed by Ikegami 15). A general method of treatment using a curved-

NEW SECTOR TYPE DOUBLE

FOCUSING

linear and an auxiliary local rectangular coordinate system has been s h o w n by Streib~6). Since a conventional S D F spectrometer with ~ = 1, fl = ¼, 6) ~ 180 ° and linear field boundaries is characterized as a spectrometer of wide beam aperture typeS'~5), here, we shall be concerned with such a spectrometer and discuss its focal line and the effects of slightly oblique entrance (or exit) on focusing properties. If ~ and r/ are quantities related to the oblique entrance and exit o f the reference trajectory, as s h o w n in fig. 4, then the radial focusing conditions will be expressed as follows assuming both ~ and r/are small:

BETA-RAY

1l'r~°) ~-'r

-(1)+l~l'2(¢'+q')cos(9'+(¢'l'l+rl

_ ~ L,Nr

(11) /,,r~(1) _t_~(1)

l~ =/2(1--~)~,

l~ - 11c~½,

l~ -- lzc~~,

¢'= ¢(1- ~)-"~, O' = O ( 1 - ~)~,

Lucite

,/ /

Baffle

',.... "f~,~nbe_

Pole

'i

. _ _ j Cooling Pipe

Variable Exit Slit

Coi~

Electron Detector

Source Holder Source Charnb~

Goniometer De tecltion Window o

S

0,

l; --/,(1--c~) 'r,

/

/

=

where

/

x

12)smO • t =0.

, ,

The axial focusing conditions will be as follows:

9~ 9

/

257

SPECTROMETER

Vernier

z ' - r a y Detector

Fig. 5a.

50

IOO J50 20( m m

~/' = r/(1-c0-~ , O" - O~ ~,

(12)

a. YAMAMOTO et al.

258

Coil

~ e

Pole

~

Inner Core 5~q

Diffusi~ Pump

/ ~ner Coil

J

/6oniometer

i

O

50

IOO

150

200 mm

Fig. 5b. Fig. 5. Schematical presentation of the present SDF spectrometer. A goniometer for electron-gamma ray directional correlation measurement is also illustrated. Most essential improvement of this SDF spectrometer can be seen by comparing this figure with fig. 3 and ref. 5). a. Plane view; b. Section view.

C~1) = c o s O ' - l~sinO', C (1)

~-- C O S ~ )

. .-. -. .1. 2. s l n O

,

If the momentum deviation, p = (P-Po)/Po, is not zero, the radial focusing condition is:

_,,q(x)_= sinO' + / ; c o s O ' , S(zi)

--= s i n O " +

l~cosO".

The eqs. (11) and (12) can be also rewritten in the following forms (generalized Barber's rule), tg -x {l'~/(l+l'~')}+tg -l{l'2/(l+l'2tf)}+O'=n, tg - X / ,,l + t g

(11a)

12+0 =~. (12a)

- 1 . . . .

In eq. (12), we have neglected the fringing field effect on the vertical focusing properties of the SDF spectrometer, since the vertical focusing properties are mostly due to the field in the magnet. Then, the conditions for double focusing, eqs. (11) and (12), may not be exactly satisfied. The fact is true even though one takes into account of oblique entrance and exit effects on the vertical focusing condition eq. (12). However, aberrations of the image in the axial direction do not cause much of a problem, since they do not cause any serious deterioration of the resolution. The magnification M, and the linear dispersion D are also given by the following equations:

M , = - (C~1)+ (~'+ q')l'2cos 0 ' + ~'(1 + llq') sin 0'}, (13) O = ( l - - e ) - t { ( l - C , ( 1 ) )+r/ , / t2 ( 1 - c o s O t )}.

(14)

Eqs. (13) and (14) show that it is possible to obtain a small multiplication and a large dispersion by selecting suitable values 13) of ~ and r/.

'

(~(1)-1- '~(1) J-

l'll'2(~'+q')cosO'+ +(l'~¢'+l'df)sinO'+A2sP = 0. (15)

Then, the focal line can be expessed as follows, using p as a parameter: x* = 12(P),

y* = O(p)p+ Assp z.

(16)

Hele, x* and y* are defined in the Cartesian coordinates of the image space as shown in fig. 4, and explicit forms of Az5 and Ass are presented in the appendix. In fig. 12 is shown the focal line calculated from eq. (16) for the spectrometer shown in fig. 5. The aberration of the image on the focal line changes gradually with p through the change of the arm length 12 or of the convergence angle O. The change is, however, practically negligible, for instance, less than 20% for AP/P = 0.1. Thus, one could expect a usefulness of the spectrometer as a double focusing spectrograph or as a multi-detector spectrometer.

3. Design and construction 3.1. DESIGN The most important problem on the construction of SDF spectrometers with a large magnet pole gap is how

NEW SECTOR TYPE DOUBLE FOCUSING BETA-RAY SPECTROMETER to overcome the fringing field effects. There are two kind o f fringing effects: One is a r o u n d the pole, and the other at the beam entrance and exit. W i t h respect to the former, one must correct the radial field distribution. In our S D F spectrometer, a set o f coils such as that illustrated in fig. 5b (hereafter called "distributed coil setting") was employed instead of a Rose shimt7), which has been c o m m o n l y used. This kind of arrangement of coils was already used by Wild and H u b e r et al. is) for spectrometers o f the rcV'2 type. It is, o f course, possible to get a nearly desired field distribution on a median plane by using a modified Rose shim 3' 5). The field distribution far f r o m the median plane is, however, quite disturbed by the shim (fig. 6). This would be the main reason why the actual m a x i m u m transmission of the I N S - I I [ spectrometer is depressed f r o m the expected valueS). In the case o f " d i s t r i b u t e d coil setting", each coil acts as a magnetic potential divider, and one m a y expect to get a desired

(a)

(c)

259

field distribution in almost all of the magnet space. We have verified this fact by electrolysis experiments. Some results on the typical cases are shown in fig. 6. In our S D F spectrometer, b o t h outer return yoke and source chamber act as magnetic shield. It was f o u n d that the fringing field distribution at the beam entrance or exit side was closer to the ideal one than in the case of a shield-free magnet as seen f r o m a calculation based on eq. (7). Thus, it m a y be expected that the second order calculation on the aberration at the focus point assuming an ideal field and sharp cut-off field b o u n d aries, gives rather g o o d estimates. As far as the fl-y or e-y coincidence and angular correlation experiments are concerned, it is desirable to place the source and detector outside of the field*. * Such a setting is also suitable for the pre-acceleration technique of low energy electrons since there is not any coupling effect between magnetic and electric field, which would cause certain disturbances on electron trajectories.

(b)

(d)

Fig. 6. Schematic illustration of magnetic field potentials inside the various kinds of magnet. The data are obtained by using an electrolysis method, a. Shim free magnet; b. INS-III spectrometer; c. Distributed Rose-shim; d. Present setting.

260

H. YAMAMOTO et al.

The exit slit was placed on the magnet pole boundary where the field distribution is fairly ideal as described later. For the sake of ease of fabrication, the magnet poles have a semicircular shape with conical surface so that there will be a field distribution of fl = ¼. Iron pieces were attached to the source and the detector side of the magnet pole boundaries in order to adjust the total deflection angle and the field boundary shape, if necessary. The vacuum chamber consists of the magnet poles and the return yoke as shown in fig. 5b. Since the magnet poles are supported by inner and outer return yokes, it is of no need to use distant pieces such as those which were used in the I N S - I I I spectrometerS). By this way one gets more available space inside the magnet and a larger reference radius with respect to the size of the main body of the magnet than those of the I N S - I I I spectrometer (figs. 6b and 6d). The selected dimensions were

spectrometer with fl = ¼. Such a character is desirable from the point of view of the fringing field problem and machining the pole faces. The first order perturbation calculation predicts the geometrical transmission f2 = A / ( l z + 2) sterad,

(21)

if A r 2 is defined as the maximum available crosssectional area inside the magnet. The calculated value of f2 for the geometry in fig. 5 is then 1.5% of 4n. We define the resolution R as the full width at half maximum (fwhm). The ratio of R and R o generally varies with the resolution. Now let us assume the ratio is almost ½, for brevity. Then the luminosity of the present spectrometer at the optimum condition is expected to be: L = 4.57 × 10-2R I-. (22) Here, the resolution R is expressed %.

ro = 34 cm, 3.2. CONSTRUCTION

2z0 = 24 cm. The effective field boundary was estimated from eqs. (7) and (9) and adjusted by using iron pieces attached so, that the reference trajectory will be normal to the boundary. Then, the total deflection angle and the source point are predicted as: O = 194 °, 11 = (x/Z)tan(Tz- O') = 44.7 cm, for 12 = 0 . The corresponding magnification and the linear dispersion are Mr = - cos O' = 0.733, (17) D = (1 + M r ) / ( 1 - c 0 = 3.47.

(18)

The aberration of the radial direction (appendix) will be represented as follows: y* = 0.7329yl +0.1345y 2 + 0.2039y1~0 - 0.2332¢p 2 - 0 . 5 0 0 5 z 2-1.3999zl~O-0.g650qJ 2.

(19)

Introducing the source width (2yo) and height (2Zo), the detector width w, and the defining aperture angles q~o and ~ko, we then get from eq. (19) the basic resolution, remembering the value of the dispersion: Ro = 0.4229yo + 0.2882w + 0.03881 y2 + 0.1177yoq~0 + + 0.06728tpoz + 0.1444Zo2 + 0.8077Zo~,o + 0.2496~b 2. (20) It is easily seen from eq. (20) that our spectrometer will be a wide aperture type in contrast with a rc~v/2

3.2.1. Magnet and coil assembly The magnet pole forms a conical surface and the pole gap is given by the following equation: 2Zg = zo(r + ro)/r0.

(23)

As shown in fig. 5b, the magnet pole faces are extended to the return yoke, except for the source-detector side and form a vacuum chamber together with the return yoke. The spectrometer body is of semi-circular type and its radius is 62.5 cm. Pure iron of low carbon content (less than 0.05%) is used in the poles and the yoke in order to suppress remanent magnetism. For adjusting the field distribution and for performing optimization of the focusing properties, six pieces of coils with equal ampere-turns are each arranged on the inside core and the return yoke as shown in figs. 5a and 5b. Pieces of iron for shimming are inserted between each of the coils. The corresponding iron pieces on the inside and outside yokes face each other and form a magnetic equipotential surface. The coils on the periphery of the magnet poles are bent upwards (or d o w n w a r d s ) a s illustrated in figs. 5b and 7b to form the beam entrance on the source side. Since the coils as a whole are located in the vacuum chamber, an epoxylene varnish is impregnated into the insulator between the copper wires, improving the heat conductivity within the coils. Maximum current density of the coils is 3A/mm 2 which corresponds to an electron energy of 4 MeV.

NEW SECTOR TYPE DOUBLE F O C U S I N G B E T A - R A Y SPECTROMETER

261

Ca)

(b)

Fig. 7. a. Exterior view of the present SDF spectrometer; b. Interior view of the SDF spectrometer. The upper pole piece has been removed.

H. YAMAMOTO et al.

262

3.2.2. Source chamber and slit assembly The source chamber is a cylinder of 60 m m i.d., which has a gamma-ray detection window made of a stainless steel plate 2 m m thick. The source face can be set at any angle to the reference trajectory, and the source can be inserted easily without breaking the vacuum in the apparatus. The beam-defining entrance slit is made of plastic plate 10 m m thick and its edges are tapered. The slits defining the horizontal radiation angle can be set independently to each other, while the vertical one can be set symmetrically with respect to the median plane. This is convenient for the determination of the optimum conditions, since the focusing properties generally depend on the sign of the radiation angle 9). The openings of the entrance slit are expressed in mm, and the reading accuracy is 0.1 mm. There are three baffles inside the magnet. These baffles can be set and replaced without removing the magnet pole pieces (fig. 7b). The exit slit consists of two plates made of brass, one of which has various slit widths and the other has several apertures for setting different heights. One can easily select the desired width and height without breaking the vacuum. The specifications of the two slit plates are as follows: Height: 10, 15, 20, 30 ram, Width: 0.5, 1, 2, 3. 5, 8, 15 mm. Unlike conventional slits, these slits have the advantage that they have a complete reproducibility of setting. The exit slit assembly is illustrated in fig. 8. 3.2.3. Detector arrangement The mechanism for mounting the electron detector is designed so that several detectors can be set here in the future. As shown in fig. 5a, a 70 m m × 300 m m

aperture is provided on the yoke, and a flange with a double structure is mounted here. Ordinarily, the base flange is left in its position, and the detector is mounted on this base flange. I f it is desired to mount several detectors here, all that will be necessary is merely to process this base flange suitably. As electron detector, an end-window type G M counter with a window of 1.5 mg/cm 2 thick is used and it can be replaced by a scintillation counter. An anthracene crystal of 25 m m dia. and 3 m m thick is used as scintillator, which is connected to the photomultiplier through a lightpipe. The g a m m a ray counter on the goniometer for beta-gamma coincidence can be set automatically with an accuracy of + 0.5 °. There are four settings for the angle interval: 10 °, 15 °, 22.5 ° and 30 ° . 3.2.4. Power supply and regulating device In fig. 9a is shown a block diagram of the power supply. The coil current is controlled by using three 2SB206 power transistors in parallel. Garwin's circuit 2°) was taken into consideration in designing the main parts of the circuit. The m a x i m u m output current and voltage are 60A and 30V, respectively, and the current can be varied stepwise either manually or automatically, There are four settings for the current interval: 10 -¢, 5 x 10 -4, 10 -3 and 5 x 10 -3 of each full scale: 20A, 40A and 60A. The current stability is better than 5 x 10-5/h. The value of the current can be printed out by using a digital voltmeter, as well as the counting and measuring time for performing the decay correction for counting. Since the field is simply determined as a function of the magnet current, an automatic demagnetizer has been provided to get reproducibility between the measured current and the field strength of the spectrometer.

Fig. 8. The variable exit slit assembly.

NEW SECTOR TYPE DOUBLE F O C U S I N G B E T A - R A Y SPECTROMETER

, Magnet

,i 0suoo,y

I

263

has been obtained for the K-conversion electron line of 137Cs. 3.2.5. Vacuum system

To Electric Typewriter

(a)

The evacuating system consists of an oil diffusion p u m p of 15 cm dia. and a rotary p u m p with a pumping speed of 300 1/min. A by-pass circuit is provided between the spectrometer body and the rotary p u m p ; it is used for coarse evacuation of the spectrometer. A waterlock alarm-relay is provided for the cooling system of the oil diffusion p u m p , when the water flowrate drops below a prescribed level, the heater current is automatically cut off. If the pumping system has an accident something like a power failure or trouble in the rotary pump, the magnet valve at the diffusion p u m p outlet will be closed in order to prevent counterflow of the oil. The evacuation system (rotary p u m p of 150 1/min speed) of the source chamber is separated from the main evacuation system by a sluice valve. 3.2.6. Automatic control and read-out

(b)

Fig. 9. Block diagram of the current regulating and counting system.

The dumped current which is a train of triangular current pulses oscillates more than 20 times. The amplitude of the alternating current and the period of a cycle which depends on the amplitude decrease on stepwise. The period of a demagnetization is about 6 min. A reproducibility, better than ___0.06% of the reference current,

Except for a testing period, the spectrometer and goniometer have been operated automatically. All the circuits associated with this spectrogoniometer were designed with a view to a 24-h day, seven-day week operation without manual assistance. The automatic control system has been designed so that almost any conceivable experiment; single and coincidence electron spectrum measurement, e - - v angular correlation, etc., involving the spectrogoniometer can be programmed automatically. This system can read out an information from scalers, stepping potentiometer, position angle of g a m m a counter etc. at the end of any pre-selected counting interval of 0.1, 0.2, 0.3 . . . . 1.0, 1.1, 1.2 . . . . 100 min. The timing system is basically a 100 k H z cycle quartz oscillator feeding transistorized scalers. Both the counting interval (up to 100 rain) and time elapsed from the start of any experiment are read out. After reading out the data, the stepping potentiometer (in the case of spectrum measurement) or position angle of the g a m m a counter (in the case of angular correlation experiment) is advanced to the next setting, the scalers are reset and counting at the new current or position angle setting is started. The current stabilizer and the goniometer normally settle down to the new value in 3 sec and in ~ 3 sec/10 ° respectively, except after large changes, e.g., after returning from the end to the beginning of a current sequence. The time needed to read out these data by an electric typewriter is about 5 sec.

H. YAMAMOTO et al.

264

1.3

~Yo X

Un cot r ec ted

Corrected

0

L2

n:-A ..z

I I

Lt.

09

O.B

s,=+ 4~

0.7

+70 J 8

0,6

....

Y0=34 cm

"

/

1

1

/

~

0.5 0,5

iO

1.5

2.0

( Xt 7o )

Radial Distance

I: ig. I0. Magnetic field distribution on the median plane. Cross points show the field distribution before correction a n d open circles after the correction on coil current divide, I/

," / / /

/ii/t / /

"/1'~

/

V;'

/10o!,~ k i

t ' ),.

Pole

_d_L~,'~k'___~ ~ , ~ •

i la /

. ,

Po e

LO

Yok~~

to

~ ;

----~"

.....

- Meo~,uremerff

/

/~/:~/

t/

/,::"

C,tculo,ed

=

#

/P

/

Meosurement

:

.,"

t~:~.4

/

/It ,/~

0.5

#

.,"I/so=,o I /

~"

t2 - ~ <~

,'" 11 li !

05

,~

.,,,

-"

],k.

(°=03 b=07

//

5

4

5

2

X

I

( i n Zo Unit) (a)

0

-I

-2

2

,,

g

,

o

-i

-2

X ( in Zo UnH) (b)

Fig. I 1. The distribution o f the fringing field along the reference trajectory, Curves o f b--~ oo show the distribution o f the field without the shield channel (yoke). a. On the entrance side; b, On the exit side.

NEW SECTOR TYPE DOUBLE F O C U S I N G B E T A - R A Y SPECTROMETER

265

4. Performance and its optimization

4.2. FRINGING FIELD AND EFFECTIVE FIELD BOUNDARY

4 . 1 . M A G N E T I C FIELD S H A P I N G

The fringing field distributions on the median plane on both source and detector side were measured along a normal to the inner face of the yoke. In fig. 11 is shown an example of the measurement of the fringing field on the line passing through the reference radius on the pole boundary. The unit of length is a half gap on this pole boundary. Let us first consider the fringing field on the source side. The measured distribution (fig. l l a ) is close to that of the theoretical one, with geometrical parameters

In fig. 10 are shown the results of the measurement of the magnetic field distribution in the radial direction on the median plane*. The measured field distribution fits well with the expected one, i.e. : H = H0(1 + ½p)-i = Ho[1 _ ½p + ¼p2 _ lp3 + . . . ] , p - (r-- ro) [ ro, for the pole shape given by eq. (23). As seen in fig. 10, these distributions are, however, close to that of fl = = {, 7 = 0, for r < %, while fl = ~, 7 = 0, for r > %. This is due to neglecting the higher order terms in eq. (23). In this case, it is generally possible to fit the distribution to that of fl = ¼, 7 = 0 by shifting the reference radius ro somewhat inwards. As will be described in section 4.3, the field distribution was finally corrected by adjusting the distribution of the excitation current of the inside and outside coils, so that the m a x i m u m resolution could be obtained with fixed geometrical transmission. A finally corrected field distribution such as that shown in fig. l0 fits very close to the distribution of fl = ¼, 7 = 0 for the whole region of r. * A type D-855 gauss-meter produced by the Dyna-Empire Co. was used.

a = 1.0, b = 1.4. However, the measured distribution as a whole shifts outwards from the pole boundary. This can be explained as follows: The magnetic potential of the yoke was assumed to be zero in the theoretical estimate for simplicity, but the actual potential should be finite, depending on the setting condition (positions and geometries) of the excitation coils. The effective field boundary which was obtained by integrating the actual field distribution shifted by 0.3 (in half-magnet pole-gap unit) in the outward direction from the theoretical one, resulted in an increase (0.1 rad) of the deflection angle of the reference trajectory. -25

h

-

A

-20

0 570\00493 0442

~2 ~5o \o°3o4 0"3 . \0365 I

"15

,93 ~40

0:244 °i "°:244 0242

0440

~.'293 °3~,o15,

~

0~87 • 0319 01~42 • . . . .

-15

I

i

-I0

,

,

,

,_,~

,

,

•

. . . .

, 0

,

5

,

,0797°.4°

9

0.~82 0 2 5 5 0,~ 6X~O 4

,

,

,

0.523

i Fig. 12. Focal lines on the source side and on the detector side. The right figure shows details of the focal line on the detector side; the circles and the values indicate the measuring points and the resolutions, respectively, and a solid line shows the calculated focal line with the measured effective field boundary.

266

H. YAMAMOTO et al.

The coils are located below the pole faces on the detector side, compensating the fringing effects in the manner as described in the section 3.2.1. As seen from fig. llb, the actually obtained field distribution is closer to the ideal one (sharp cut-off), than the calculated one with geometrical parameters a = 0 . 3 and b---0.7. This fact demonstrates that this is a superior method for compensating the fringing effects. The theory for field distribution, of course, cannot be applied for such a coil arrangement.

and the focal line on the detector side was measured by the same manner. These results are shown in fig. 12. There was, however, a slight difference (4,3 cm) between the experimentally obtained source distance and the predicted one from the design. This discrepancy can be reduced if one takes into account the acual effective field boundary (section 4.2); eq. (11) with 12 = 0, ~ = 0.1, O = 200 ° yields ll = 43.3 cm which may be compared with the measured value l 1 = 45 cm. The focal line on the detector side which is almost straight, as shown in fig. 12, crosses the radial direction with an angle of 42 °. The calculated focal line is, as a whole, in agreement with the measured one. The focal line corresponds to a momentum range of 15°/0 of the reference momentum. Although the values of the resolution in fig. 12 scatter somewhat, the values on the focal line are constant with _ 15%. This fact shows that the photographical method or the multidetector method are also usable in such an SDF spectrometer.

4.3. MEASUREMENT OF THE FOCAL LINE AND DETERMINATION OF THE SOURCE POSITION

At first, an electron source (t 37Cs, 2 m m dia., circular shape) was mounted in the exit position, and a detector assembly was mounted on the source side so that the GM-counter could be moved two-dimensionally to find the focal line. The point on this focal line where the best resolution was obtained, was chosen as the source position. Then the source was mounted on that position

\

i

\

2"

\

1 I t

(a) 4.4. MEASUREMENT OF THE ELECTRON TRAJECTORY IN

I

Pol e "'~,;'.c/.//.

/ " ".>,//.~-~

THE FRINGING REGION Electron Detector for Monitoring

X - r o y Film

(b) Fig. 13. a. Observed electron trajectory in the fringing field; b. Schematic illustration o f the location o f the emulsion for measuring the trajectory.

It is of interest to trace the actual reference trajectory in the fringing field region and to compare the trajectory with those calculated in course of the design. The trace was carried out by X-ray films (Sakura N-type). These were installed at the fringing region on the source side at an inclination of about 10° to the median plane. A source of x37Cs ( ~ 1 mCi) was used and the electron beams were collimated at the entrance slit into a rectangular shape of 5 mm wide and 100 mm high. The 660 keV electron beam was divided into

NEW SECTOR TYPE DOUBLE

-30 <__~<_0

FOCUSING

BETA-RAY

SPECTROMETER

267

0 ~ ,Ib -<30

- 5 ~ 5

1,00(

o

500

I

10.75

L

10.80

I

10!85 V~ 10.75 i

10.80

1().85 ~

I

10.75

I

10.80

10.8

Current Fig. 14. K - c o n v e r s i o n lines o f t h e 660 k e V g a m m a r a y o f laTCs m e a s u r e d with a n u n c o r r e c t e d distribution o f the m a g n e t i c field. T h e n o t a t i o n s q0 a n d ~p a r e m e a s u r e m e n t s o f b e a m o p e n i n g angles a n d described in the text.

/. O'

3

b

b' c'

free region is almost parallel with that corresponding to the ideal field. The separation between them was 2 cm. This value agrees well with the predicted value by using the analytical expression of the fringing field. 4.5. FOCUSING PROPERTIES AND EXPERIMENTAL OPTIMIZATION

d' e' f

[ Fig. 15. S c h e m a t i c illustration o f t h e coil a r r a n g e m e n t .

It is expected from eq. (20) and more general second order trajectory equations (appendix) that in our spectrometer the value fl < ¼ is the o p t i m u m one. In general, for a wide aperture spectrometer optimum fl-values would be, however, different for q~ > 0 and ¢p < 0 19). Then, the resolution and the image shapes were measured for the following three cases, each of which has an identical geometrical transmission: 1.

two groups by the median plane. A half part of the electrons hits the films, while another part is counted by the detector at the normal position in order to monitor the field variation during the exposure. Further exposure without the magnetic field was also performed n order to accomplish the geometrical positioning of h e electron trajectory on the films. The results are shown in fig. 13 which also shows the effective field boundaries together with the geometrical positions. Curve A shown in the figure is the trajectory for the ideal field. However, the actual trajectory in the field

2. 3.

- 3 0 < q~ < 0, 0 < ~0 < 30, -5
- 5 < ~ , < 5, -5<~k<5, - 1 5 < ~k < 15.

The radiation angles q~ and ~k are expressed in terms of the scales (mm) of the entrance slit; ~o = 30 corresponds to 0.21 rad and ~k = 15 to 0.08 rad. In fig. 14 is shown the 660 keV K-conversion line of 137Cs before the correcting field distribution. Case 1 (q~ < 0) has a resolution worse than the case 2 (q~ > 0), being consistent with the fact that effectively fl > ¼ for ~o<0.

H. Y A M A M O T O et al.

268

%

o-50<_ ~<--0 : - 5 - ~ - < 5 x 0 _~£6~-30: -5-4~-~ 5 • -5 ~ ± 5 -15-<~--~15

/

~'0

/ jl

//f

0.6

,.o

/

....

0 f.

a' f'

c.

o~

d

fl

By-Passed By-Passed

&

o.° . < . _ I , ~

. ~ , ~ ,

o e#

rY

0.2

QI

I

i

05

0

I

I0

215

r

t.5

20

A

B y - p o s s e d Currenf Fig. 16. The variation of the resolution versus the by-passed current. The notations a, b, ..., are shown in fig. 15.

-30<

c.}

~< 0

-5 < ~ <

2o0c

A I : I.SA

5 AI

AI

= Lo A

Current

" By -Passed

AI

A:OISA

15oo

O L)

IO0O

5O0

= 2.0A

AI:0

l

i ) , .J 10.70 ' Io'. 80

10.90 '

i/ /,jt I1.00

I1.10

I IP.20

11.30 '

i 11.40

p 11.50

J 1160

i 1170

i i II.BO 11.90

Current Fig. 17. The variation of the shape of K-conversion electron line with respect to the by-passed current.

i

120u

A

NEW SECTOR TYPE DOUBLE FOCUSING

It is thus useful to adjust the field shape over the pole gap so as to get an optimum focusing property. The field adjustments were carried out for the following three cases by by-passing the current around some small coils shown in fig. 15, keeping the condition that the total number of ampere-turns of the coils are the same on the inside and on the outside• a)

a, f; a', f',

by-passed,

b)

c, d; a', f',

by-passed,

c)

c, d,

by-passed.

%

• -30"--qb'-30 :-5"- @'--5

O -30~50

269

of fig. 16 and fig. 18, that the resolution is almost completely limited by ¢. Ampere-turns of the coils c and d are finally decreased by 10% compared with the other coils. As shown in fig. 10, the field distribution at this condition is close to that of fl = ¼, ~ = 0. This is also consistent with the fact that the spectrometer of O ~ 180 ° with fl = ¼ is known to be of the wide aperture type'*). 4.6. DETERMINATION OF THE SPECTROMETER TRANSMISSION

In case a) the fl-value increases for both ~o ~ 0 regions. In fig. 16 are shown variations of the resolution with respect to the current in the bypass circuit. There appears to be practically no improvement as far as the resolution is concerned. It is also shown from the results for cases 1. and 2. that the spectrometer is loosing the wide aperture property. In case b) the fl-value decreases in the region ~p < 0, while it increases in the region ~0 > 0. As shown in figs. 16 and 17, the resolution improves in the region q~ < 0 with the bypassed current (increasing the wide aperture property), while the trend is opposite in the ~p > 0 region. In case c) the fl-value decreases for q~ < 0, while the value remains unchanged for q~ > 0. Keeping the entrance slit at - 3 0 < ~0 < 30, the changes in the resolution caused by bypassing the current around the coils were examined for the two settings of - 5 < ~k<5 and - 1 5 < ~ < 1 5 . The results are shown in fig. 18. It is clearly seen from comparison of the results

0.7

BETA-RAY SPECTROMETER

-15~_qjml5

Let us define the transmission of a system with a given exit slit as follows:

T = n/N, where n: peak counts of the electron line, N: total number of electrons emitted from the source. The number N was determined for the 660 keV-line of the laTCs in the following manner. First, a gold foil was irradiated in HTR*, and its activity was absolutely measured by the beta-gamma coincidence method. Then the intensity of the 412 keV g a m m a ray of 19SAu was compared with that of the 660 keV g a m m a ray of 137Cs. Then we have N(Cs) = {Na(Au).N~(Au)/N¢oin(Au)}• {N~(Cs) / N~(Au)} {t/(Au)/t/(Cs) }~K(Cs), where t/: detection effectivity of the g a m m a rays, ~K: K-conversion coefficient of the 660 keV g a m m a ray of 137Cs. The following results are obtained at an exit slit of 15 m m x 2 0 m m : T= 1.14° R = 1.2%.

0.6

These values are reasonably, compared with the geometrical transmission of 1.5% of 4n. A resolutionluminosity relation of this SDF spectrometer is shown in fig. 19 together with those for conventional (iron core) rc~/2 spectrometer for comparison. It would be rather surprising to see essentially no difference on the relation between SDF- and nV/2-spectrometers.

0.5

0.4

03

0.2

4.7. ANGULAR RESOLUTION OF THE SPECTROMETER

In general, experimental data on angular correlation measurements are fitted by the least-squares method and normalized to the form

O.I

0

I

I

1

1

J

t

f

t

l

J

1.0 By -Passed

f 2D

A

Current

Fig. 18. The variation of the resolution vs the by-passed current.

W(O) = 1 + A'2P2(cosO)+ A~P,(cosO)+ .... * Hitachi Training Reactor.

(24)

270

Ix. YAMAMOTO

et al.

0.01%

TrY- UPPSALA,r0 = ,OCM N

Ix

0.1% ~PRESENT ..... ~ = 0

SDF SPECTROMETEI 34CM

~,~IALFOpAsADEN~SIN~~~

J 0 O3

10-7.

10-6

10-5 10-4 10-5 LUMINOSITY( CM2 )( ~ x SOURCEAREA)

0-2

i 0 -I

~[Fig. 19. Resolution-luminosityrelation. T h e n the coefficients A~ a n d A~ are corrected for the finite sizes of the detectors a n d the sources. They are represented in the form* A~, = ~'k~(t)r'(2)~'"~k. ~k

k = 2,4,6 .....

(25)

Here, the factors ~r k~(t) a n d G(k )2 are the so-called a t t e n u a t i o n factors for the Pk(COsO) term in the a n g u l a r correl a t i o n f u n c t i o n (24). The factors G~k1) a n d G~k2) are concerned with detectors 1 a n d 2, respectively. Exactly saying, one c a n n o t treat the corrections for the finite s i z e s of the source a n d detectors, separately. This situa t i o n m a y be more serious for the electron spectrometer rather t h a n for the scintillation spectrometer, because of the dependence of the resolution o n the source size. However, the correction for the finite size of the source is, i n general, small a n d can be t a k e n into a c c o u n t separately from those for a finite a n g u l a r resolution. * This factorization is not exactly applicable for the case of nonaxially symmetric detectors. However, we restrict ourselves to detectors, whose efficiency function is invariant to reflection about both horizontal and vertical axes and then, the factorization approximation is reasonably correct.

I0 ~

--Gz

~_~" t..)

Gz

u_'~ z 2~ 05 ZLU E--

G4

0

t 05

I,

' 15

2'.0

-

TRANSMISSION T (% 0F4rr) Fig. 20. Attenuation factors Ga and G4 for the angular correlation function [eqs. (24), (25)]. The solid lines give the factors for the SDF spectrometer with ~0ma~= 1.5 ~pma~. The broken lines represent the factors for a conventional lens type spectrometer with mean emission angle et = 25°. In practice, it is infeasible to measure the term A4 and those of higher order in the correlation function by using the lens type spectrometer.

NEW SECTOR TYPE DOUBLE F O C U S I N G BETA-RAY SPECTROMETER

If the detector slit is set so as to get a flat top line shape one m a y assume uniform (angular) efficiency of the s p e c t r o m e t e r at IgoI < go..... I g' I < ff .... and the expected a t t e n u a t i o n factors are calculated rather easily21). In the case of a point source the 2 na order calculation gives a resolution in the f o r m [eq. (20)] : Ro

(26)

a g o 2 + b ~ 2 + c ....

=

where, the last term comes f r o m the finite width of the detector slit and includes, partially, an effect of the source width. F o r the given resolution, eq. (26) gives an elliptic c o n t o u r for the b e a m cross-section at the (beam) defining slit. This prediction was confirmed experimentally for a similar S D F spectrometer of the T o k y o Institute of Technology. The m a x i m u m values of go and ~ are given by 2 gomax

T = 2n sin ~. &t,

(30)

G2 "~ P2(cos~), G4 ~- P4(cos ~).

2

(27)

The corresponding transmission and attenuation factors are given as follows: T ~

(28)

7~gomax" ~/¢. . . .

1,~

These f o r m u l a for G 2 and G 4 a r e also applicable to the case where the S D F spectrometer is set vertically, by exchanging gomaxto ~bmax and vice versa. In fig. 20, the factors G2 and G4 vs transmission T are illustrated for the case gomax= 1.5 ~bmax which is close to the case of the present S D F spectrometer, eqs. (20), (26) and (27). As a c o m p a r i s o n we shall be concerned with the angular resolution of a conventional lens type correlation spectrometer22). If we restrict ourselves to a small angular interval c5~ around the m e a n emission angle the expressions for T, G2 and G4 become:

(31)

= ( R o - - c ) / a,

~kmax = (R o -- c) / b.

G2

271

2

2

2

2

(gom x + 5

G , = 1 - 4(gomax+ ~/max)"

(29)

The factors G2 and G4 vs transmission T for the lens type spectrometer (~ = 25 °) are also shown in the fig. 20. It can easily be seen that the S D F spectrometer will be m o r e useful than the lens type spectrometer, for angular distribution or correlation measurements especially for measuring the higher order terms in the correlation function.

140C

/ -6- 6 0 - K 1200

I000

8O(: ¢j,

0.I% 600

400 660-L

i-

2(9(3

o

,o17o

,680

, 9o

,,'oo

ll. I0

11.20

A

Current Fig. 21. Spectrum of the conversion electrons of the 660 keV g a m m a ray of 137Cs.

rI. YAMAMOTO et al.

272

5. Conclusion The wide aperture focusing p r o p e r t y and large transmission of the present spectrometer is summarized in a few examples of instrumental performance. Fig. 21 shows the 660 keV internal conversion lines in the decay of la7Cs. The source was deposited by evaporation of an HC1 solution of 137Cs which was supplied commercially, on a film of A1 coated 9 # m thick Mylar. The deposit was 1 x 10 m m 2 in dimension and visible (yellow colored). The observed resolution (fwhm/peakcoordinate) was 0.08%. Use of the S D F spectrometer with a transmission of 0.2% of 4n and resolution of 0.2%, with a source 1 x 15 m m 2 is rather convenient. The observed m a x i m u m transmission was 1.2% of 47z. The value was, however, limited by the baffles rather than by the b e a m angle defining slit. If, one takes off the baffles one would get a m a x i m u m transmission of a b o u t 1.5% of 4n.

Appendix: y*

=

Luminosity vs resolution predicted f r o m these data is shown in fig. 19 together with those for several spectrometers. O u r results in fig. 19 are not necessarily the o p t i m u m ones, because we did not look systematically for the o p t i m u m condition. The results are, however, close to the expected values f r o m eq. (22), except one point corresponding to R = 0.08%, which would be improved with a source prepared by the v a c u u m sublimation method. One could also improve the results, especially for a large transmission setting, by shaping the field b o u n d a r y as treated by Ikegami ~5,2 s). The authors are grateful to the staff of the departm e n t of Nuclear I n s t r u m e n t Design, Hitachi Works, Hitachi Ltd., especially Mr. Y. K a z a w a for their very essential services. The assistance of Mr. S. Kishi on calculations of trajectories and aberrations was valuable.

R A D I A L ABERRATION OF THE IMAGE

AloXl +A20q)'+Asop+Alixl 2 + A12tp ,xi

_[_A22qy2+A33g2_l_A441[f,2_}_A34z11~,,q_A15Xlp_l_A55p2,

A~o =- C~l)+(~' +q')l'2cosO' +~'(l + l'2g')sinO ', A2o

- - . 1 , ~ r " r ( 1 ) ~ e O ).' l ' l l ' 2 ( ~ ' + ~ ' ) c o s O ' + ( l ' l ~ ' + l ' 2 ~ l ' ) s i n O ' + A 2 s p . , - , ,

= O,

Aso - ( 1 - ~ t ) - ' { 1 - C } l ) + ~ l l 2 ( 1 - c o s O ' ) } , A1, - (a - b) - aC~l)+ (b -21'2e)C~2)+ eS~2), A12 --

211(a-b)-211aC}l)+2aS~l)+ {211b-2(2111'2+ 1 ) e ) C ~ 2 ) - F {2b +2(/i-21'2)e}S~ 2),

A22 - (li2+l)(a-b)-{(li2+l)a-2b}C(,1)+2al'~S}l)+ [(11,2- 1)b-2{(11,2 - 1)/2, + ll}e]C~2)+ +{211b+(ll , , (2), , ,2- 1 -41112)e}S, A25 - - 21'1(a- b) (1 - a) - ' [ - d(1 - a) -1 {½(3 - a) + 2(a - b)} (1 + Ii li) + 211 a(1 - ~)-1 _

-½0'{l +4(a-b)(1-~)-l}]c}l)+[-d(1-oO-l + ½0'{1 + 4 ( a + {-2b(1-~)-1

{½(3-a)+ 2 ( a - b ) } ( l ; - l l ) + ½ { l +(l + 2 a ) ( 1 - a ) - l } +

b)(1 - ~ ) - 1)]S~l)+ { - 2b(1 - a ) - 1 1 i +2e(1 - a ) -1(1

+21'11i)}C~2)+

+ 2 e ( 1 - a ) -1 (2/2' - ll)}S;(2),

a s s --- ½ ( - 3a + (3a - c ) c 7 ) + cCV)}, As4 - 3a l'~(1 - C~1)) - c{l'~C~1)--(2K2/K1)S~ 1)} + t,4e, r(2) ±T,-,e(2)~ 4 j l~--r z },

A44 ~ ½[-3(l'~+ l)a + {3(l'12+ l)a-(l'l~- l )c}C}l)-(K2/Kt)'41'~cS~1) +(l'~2- 1)cC~(2)+ 211cS (2)], A15 ~

--2(a - b)(1 - c t ) -1 [2a(1 _ a ) - i _ l'2d(1 - o 0 - ' {½(3 - ~) + 2(a - b))]C~l)+ [d(1 - 0t)- ' {½(3 - e) + 2(a - b ) } + + ½0'{1 + 4(a - b)(1 - ~x)- 1} ] S ~ ' ) - {2b(1 - e ) - I + 21,2e}C~2)_ 2e(1 -

~)- ls!2),

NEW SECTOR TYPE DOUBLE F O C U S I N G B E T A . R A Y SPECTROMETER

273

Ass - 4(a - b) (1 - a ) - 2 [4a(1 - a ) - 2 _ l'2d(1 - ct)-I {¼ + (a - b) (1 - a ) - ' } ] C~ 1) -(1 -=)-l[{½+

2(a-b)(1-a)-l}~9'

+d{¼+(a-b)(1-a)-t}]S~l)+

+ {b(1 - a ) - 2 _ 2l'ze(1 - a ) - I }C~2) + e(1 - a ) - 1S~2), where, K 1 -= (1 - c t ) ~, K 2 = ~-,

1'1 =- K i l l ,

1'2 =- K l I 2 ,

q)' = q)/K1, ~ll" = ~ / g 2 ,

~' = ~/K1, rf = rl/K1, O t -

O"= K20 ,

If =-- K211, l~ = K212, c o s ( 2 0 ' ) - 21 i sin ( 2 0 ' ) ,

C~ 1 ) - c o s O - l ~ s i n O ,

2)

Sr¢1) _ sin (9 ' + l~ cos O ' ,

2) - s i n ( 2 0 ' ) + 211 c o s ( 2 0 ' ) ,

a =

K,O,

b -

T

Cc2) = cos ( 2 0 " ) z

S(z2) - sin (2{9") + 2l~ cos ( 2 0 " ) ,

2(3-5e+2fl)(1-ot) -1, c -

References 1) S. Rubin and D. C. Sachs, Rev. Sci. Instr. 26 (1955) 1029. 2) C. Mileikowsky, Ark. Fys. 4 (1951) 337; 7 (1953) 33 and 57. a) M. Sakai and H. Ikegami, J. Phys. Soc. Japan 13 (1958) 1076. 4) H. Ikegami, Phys. Rev. 120 (1960) 2185. 6) M. Sakai, H. Ikegami and T. Yamazaki, Nucl. Instr. and Meth. 9 (1960) 154. Now, this spectrometer is used in an e-e angular correlation experiment, combining with INS-IV Spectrometer which is just twice the INS-III Spectrometer. M. Sakai, H. Ikegami and T. Yamazaki, Nucl. Instr. and Meth. 25 (1964) 328. 6) Cf. e.g, F. Kottler, Handbuch der Physik 12 (Springer Verlag) p. 480.

7) O. Kofoed-Hansen, J. Lindhard and D. B. Nielsen, Dan. Mat. Fys. Medd. 25, no. 16 (1950). a) D. B. Nielsen and O. Kofoed-Hansen, Dan. Mat. Fys. Medd. 29, no. 6 (1955). 9) D. L. Judd and S. A. Bludrnan, Nucl. Instr. 1 (1957) 46. 10) H. Yamamoto, Private communication. A similar expression for the field distribution was obtained by Herzog in R. Herzog, Z. Physik 97 (1935) 596. 11) For example, P. M. Morse and H. Feshbach, Methods of theoretical physics (McGraw-Hill, 1953) p. 449.

21~ sin ( 2 0 " ) ,

~(1-5~) -I,

,

p2

--

!

t

p2

d = 12(12 + 1 ) -1 , e = 2/2(4/1 + 1 )

-

1

.

12) N. Svartholm, Ark. Fys. 2 (1950) 115; D. L. Judd, Rev. Sci. Instr. 21 (1950) 213. is) R. M. Sternheimer, Rev. Sci. Instr. 23 (1952) 629. 14) E. S. Rosenblum, Rev. Sci. Instr. 21 (1950) 586. 15) H. Ikegami, Rev. Sci. Instr. 29 (1958) 943. 16) j. F. Streib, HEPL-104 Report of High-Energy physics Laboratory, Stanford (1960). 17) M. E. Rose, Phys. Rev. 53 (1938) 715. 18) H. Wild and O. Huber, Helv. Phys. Acta 30 (1957) 3 ; A. A. Bartlett, R. A. Ristinen and R. P. Bird, Nucl. Instr. and Meth. 17 (1962) 188. 19) The equation of the beam trajectory has, in general, terms of ~, ~0a, ... and of cp2,~p4, .... The latter group interferes with the former one additively or destructively depending on the sign of which, in turn, result asymrnetricity of the beam concerned with the reference trajectory. 20) R. L. Garwin, Rev. Sci. Instr. 29 (1958) 233; R. L. Garwin et al., Rev. Sci. Instr. 30 (1959) 105. 31) F. Gimmi, E. Heer and P. Scherrer, Helv. Phys. Acta 29 (1956) 147; A. M. Feingold and S. Frankel, Phys. Rev. 97 (1955) 1025; H. Ikegami, Phys. Rev. 120 (1960) 2185. 23) T. R. Gerholm. R. Othaz and M. S. EI-Nesr, Ark. Fys. 21 (1961) 235. 23) A. Whaling et al., shaped magnet pole boundaries of their analyzer in accordance with Ikegami's theory and confirmed the treatment.