Use of iron lenses for the transport of muon beams

NUCLEAR

INSTRUMENTS

AND

METHODS

141

(1977)

81-86;

~

NORTH-HOLLAND

PUBLISHING

CO.

USE OF I R O N LENSES FOR T H E T R A N S P O R T OF M U O N BEAMS J, L E B R I T T O N , F. L O B K O W I C Z , A . C . M E L I S S I N O S and W. M E T C A L F

Department of Physics and Astronomy, University o/' Rochester, Rochester, New York, U.S.A. Received 17 September 1976 It is shown that axial current lenses made of iron can be used efficiently for focusing and transporting high energy muon beams. A model lens was constructed and its focal properties as obtained by measuring the deflection of high energy muons, are presented.

1. Introduction Since high energy muons have a large penetrating power through matter it is possible to use magnetized ron to deflect or focus muons. Indeed, magnetized iron either in dipole or toroidal configurations, is routinely used to momentum analyze high energy muons. The advantage is the low cost and low power consumption of these devices as compared to conventional magnets: on the other hand the multiple scattering in the iron limits the accuracy of the momentum measurement. It is also possible to use magnetized iron to form focusing lenses for muons. For instance, wedge-shaped

saturated toroids have focusing properties~). Another approach towards achieving an iron fens is to excite the iron with an axial current 2) as shown in fig. la. We constructed such a device and measured its focal properties by using the 6 - 1 2 G e V muon beam at Brookhaven National Laboratory. We discuss these results and in particular the linearity of the lens. In section 4 we present the transport characteristics of a channel made of such iron lenses. 2. The axial current lens It is well known that an axially symmetric current of uniform density J results in focusing of charged particles moving in the direction of the current. The azimuthal magnetic field is a linear function of the radius B(r) = ½/toJ r = Gr.

o

(G)

I

~"

I I

I

0/2

"

L I

(b) CROSS-SECTION

Fig. I. (a) Schematic principle of the axial current lens. (b) Schematic of the current density distribution in a ring-fed axial current lens if r ~ ¢ t .

(1)

However, the required current densities are high and can be established only with pulsed devices 3) such as the so-called "plasma lenses". The magnetic field can be increased by having the axial current flow through iron or any other ferromagnetic material. High magnetic fields can be achieved with minimal power consumption. The complication arising from the introduction of the iron is the non-linear behavior of the magnetic permeability /x with the magnetizing field H. One simple approach to correct for this effect is to feed the current close to the periphery of the lens. Consider an iron cylinder of radius R and length a, into which the current Io is fed in an axially symmetric way from a thin copper ring of radius r o. A second ring electrode of the same radius r 0 is at the other end of the cylinder. In an (r, z) coordinate system with origin at the cylinder's center the current distribution has a z-component

j= = Io

+ ~ n=l

a.~.lo(c~.r) cos h(c~.z) ,

(2a)

82

J. L E B R I T T O N et al.

where c~. is the nth solution to the equation 1, (~. R) = 0 and the coefficients a. are given by a. =

2ro 1o(c(.ro) ~,, R2[lo(o:,,R)] 2 cos h(7,a/2)"

Fig. 4 shows the prototype lens which had dimensions R = a = 10 cm. The current was fed by a brass ring o f 20 cm outer diameter and 16 cm inner diameter, 2 cm

(2b) 20

The azimuthal magnetizing field H~ can be easily obtained from eq. (2a) as H~ = I 0

r

+ ~ ?l =

1

a . l l ( ~ . r ) cosh(ct.z ) . ]

R=lOcm a = lOcm

1 = 4 . 7 kA /

%,d" 9.7 cm

//4.o 2.7

(2c)

The integral .[ B4,dz = ~ i~H,~dz, which determines the focussing property of this short lens, has to be determined by numerical integration using the permeability curve o f the iron used for the lens. The current distribution for a . ~ 2 R is shown in fig. lb. By varying the ratio a/R, one can change the radial dependence o f the current distribution and with it the radial dependence of the integral ~ Bodz along the length of the cylinder. If one uses the permeabillity of cold-rolled steel as shown in fig. 2, and if one chooses r o = 0.95 R, then one can have an approximately linear dependence o f Bdz with r for a = R as shown in fig. 3. To maintain the behavior shown in fig. 3 one must scale a, R and I o by the same factor. Several lenses can be connected in series (with a feed ring between them) if one wishes to construct a device of the same radius but greater overall length.

20-

2.0

I

I

1

I

I

2

4

6

8

I0

r (cm)

Fig. 3. Calculated magnitude of the average magnetic field as a function of radius for the experimental iron lens for different values of the excitation current. In principle the field has only a tangential component. The dimensions of the lens are: width a = 1 0 c m , radius R = 1 0 c m and the current is fed at a mean radius r() = 0.95 R.

-= le m

I

50

I

tO0 gauss

tD

H

Fig. 2. The magnetic permeability of cold roiled steel used for the ,'onstruction of the lens.

Fig. 4. Photograph of the prototype iron lens used for these measurements. The 0.5" copper plates were added so as to facilitate the connection of the current leads.

TRANSPORT

OF M U O N

TABLE 1 Resolutio n of trajectory m e a s u r e m e n t s before and after the lens.

dx

60x 6y 50,. 15p/p

Before lens

After lens

-i: [.5 m m :1:1.4 mrad ~0.7 mm =t=0.9 mrad 4- 1.5 %

1.0 1.0 1.0 1.0

mm mrad mm mrad

BEAMS

83

with a set of proportional chambers which are part o f a spectrometer. The resolution before and after the lens is shown in table I. To excite the lens it was most convenient to connect it in series with a standard magnet which was used as a load for the power supply. In this way excitation currents from 2.5 to 4.0 kA were used, whereas the voltage drop across the lens was 40 mV at 4 kA, ( towords origin )

MeV

SATURATED

70

thick; the rings were silver-soldered onto the two flat faces of the lens. The copper plates shown in the figure were bolted onto the rings so as to facilitate the connection o f the current leads. According to fig. 3 one expects typical gradients of G = 1.5-1.7 k G / c m and therefore a transverse momentum kick o f p z = 4 5 - 5 0 M e V at r = 1 0 c m . For this particular thin lens the r.m.s, multiple scattering transverse m o m e n t u m is of the order of 36 MeV. However, as the length of the lens is increased, the ratio of transverse kick to multiple scattering improves as the square root of the overall length. For reference purposes we give the focal length of the lens Lf =

/~TOROID 2 2 k G

60

z.,fBEST LINEAR FIT T//" 2.4 kG/cm

5O 40 50 I=4kA

20 I0

20

40

60 80 --~ r [mrn)

~0o

(3)

pc ' 3 Ga

(towords

with f in m, pc in GeV, G in k G / c m and a in m. Namely, for 6 GeV/c muons the focal length of the prototype w a s , / ' = 12.5 m.

origin )

MeV 60 ~

I = 4.'?

5o

3. Results

CALCULATION

The lens was placed at the target location of Brookhaven's m u o n beam as shown in fig. 5. The incident muons are m o m e n t u m analyzed and their angle and position are measured with a set of hodoscopes. The direction and angle of the outgoing muons is measured

4o 3o 2o

I=27kA IO

LENSEl I I II1~

f

'~

T t

TT f

,Y .

Y

Y

V Y Xj

.O0O~COPES

X

' 20 4 0 cm

I

ZO

I[I

!

0

XYV XY Y

p~vc

Fig. 5. Schematic layout of the spectrometer used to measure the angle of deflection o f / t - m e s o n s passing through the lens.

40

(b) 6~0

: " 8 ~0

L I00

r (m m ) Fig. 6. Plot of the me a n radial c o m p o n e n t of the transverse m o m e n t u m kick i mpa rt e d to the muons by the lens, as a function of the distance Ir[ from the origin. The me a n tangential c omponent is zero (within error) for all values o f r a nd 0. (a) Exciting current I = 4 kA. The best linear fit to the da t a is s how n as well as the calculated curve; the dashed line is the be ha vi or of a s a t ura t e d toroid (at 22 kG). (b) Exciting current 1 = 2.7 kA: the calculated curves for various excitations are shown.

84

J. LEBRITTON et al.

resulting in a power-consumption of 160 W, much less than the power dissipated in the feeder cables. To analyze the data we measured the change in angle in the horizontal and vertical planes and multiplied by the m o m e n t u m to record the transverse kicks p~_ and p~ as a Function of position. For single events, the multiple scattering exceeds the kick provided by the lens at small radii; nevertheless by accumulating sufficient events in bins of given .v and y the mean of the distribution gives an accurate measure ofp_~ and PC. The error assigned to the mean was obtained from the r.m.s, deviation in the bin divided by \.."(n-I), where n is the number of events in the bin (typically 200). D a t a were obtained with both Focusing and defocusing current configurations and yielded symmetrical results. Therefore to eliminate systematic effects we present the mean

and fig. 7b for 1 = 2.7 kA as a function of y; R is the outer radius o f the lens. Fig. 7c shows again p~_ but for + 0 . 1 5 R < . v < + 0 . 4 5 R For l = 3 . 2 k A . Similar grapY (MeV) 60

(20kG}

• 40

I= 4kA =-O.45R

20 20 +

P

!- 0

]--~---I

-60

40

60

80

Y (rnm)

,

-40

_

-20 -ZO-

~\\\ : kk :

(a)

1 '

-40

T

-60 ~

2.5 kG/cm

-

± = ½[pl(focus) - p l ( d e l 0 c u s ) ] . The systematic difference between the Focusing and defocusing configuration was of the order of 10-20% and arises from the resolution of the angular measurement combined with the convergence of the muon beam onto the lens. Measurements with no current through the lens were also performed and are in good agreement with zero mean kick. Hysterisis effects were detected at the 10% level. We first calculate the radiai component of the transverse m o m e n t u m kick p~_ imparted to the muons by the lens. Namely for every position r, 0 with respect to the center of the lens we form -(p.v)/r (i.e. positive values of pr imply focusing). This is shown in fig. 6a for an excitation of 1 = 4 kA and in fig. 6b for I = 2.7 kA. Also shown are the calculated values ofpS_ as obtained for various currents from the data of fig. 3. We note that for 1 = 4 kA the measured values exceed the calculation and the best linear fit to the field yields a gradient of 2.4 kG/cm ; this indicates that for 1 = 4 kA, the lens must reach saturation close to the periphery. On the other hand the 2.7 kA data (Fig. 6b) while still exceeding the calculated value shows smaller gradients and suggests that the lens is linear over the entire radius. The tangential c o m p o n e n t of the transverse m o m e n t u m has a mean value of zero within errors for any given 0, and for all values of r as it should for an ideal lens. Finally, the dashed line in fig. 6a shows the transverse kick produced by a saturated toroid (at 22 kG) and contrasts with the behavior o f the lens. To further demonstrate the axial symmetry o f the lens we plot the transverse m o m e n t u m kick in the vertical plane for a fixed band in the horizontal plane• Fig. 7a showsp~ for - 0 . 6 R <.v < - 0.3 R for I = 4 kA

p (MeV) Y (20 kG)

60 40

I = 2.5 kA :-O,45R

ZO

,

[ -80

20 i

[

*

-60

I

-40

,_210

40

60

80

Y(mm)

i

-2(

(b)

-40.

L "~1.9

kG/cm

-60

y p~ {MeV) ......

~

~'~

T

60

I=3.2 WA

,t "

~

-8~0 '-6}0 '-4Jo '-2'0

1 20 '

-60

~

'

40 I

60 '

~

t

~

80

Y(mm)

'

I

"

_

SATURATED -- " ~ T O R O I D 2 2 kG

Fig. 7. P l o t o f t h e m e a n v a l u e o f t h e t r a n s v e r s e kick in t h e y d i r e c t i o n f o r fixed x v a l u e s (a) l = 4 k A , x =-0.45R; (b) l = 2 . 7 k A , x = -0.45R; (c) I = 3 . 2 k A , x = + 0 . 3 R. T h e b e s t l i n e a r fit to t h e d a t a is s h o w n ; t h e d a s h e d c u r v e r e p r e s e n t s t h e b e h a v i o r o f a s a t u r a t e d t o r o i d at 22 k G .

TRANSPORT

OF M U O N BEAMS

dients as before are obtained and again there is some indication of saturation at large radii for I = 4 kA. The dashed curve represents as before the behavior of a saturated toroid (at 22 kG). In order to show the effect of multiple scattering we have plotted in fig. 8 the magnitude of the transverse m o m e n t u m kick as a function of radius, in this case one expects IP,I = [(p~)2 q._(p[!_)2]" and indeed at r = 0 where p~_ = 0 one measures [(pT~)2]~ ~ 40 MeV as it should be for 10 cm of iron plus 2.5 cm of copper4). We conclude that the device behaves as a linear lens focusing in all planes as contrasted to a saturated toroid or to a quadrupole lens.

{~5

element with Z0 = I a ° 1 ,

LaoJ its displacement at the exit of the ruth element of the lattice is given by a,, = Uo costa 0 +

(2ao J - a o )

E(4/~)-

sinm 05,

(4)

I] +

where cos 05 = I - ½~. The phase space acceptance ellipse is given by a good a p p r o x i m a t i o n by rcR 2

+: =

4. Applications

(2 - ½~).

(5)

I

Iron lenses can be used most effectively when it is desired to transport a muon beam. We speak of a F O F O channel and consider a lattice of "'iron lenses" of equal strength spaced at a distance / from one another5). In the thin lens approximation the transfer matrix for one element of the lattice (i.e. FO), is

We can also consider the case where we approximate a continuous distribution. Then /-+0, /---,oo as / / ' = c o n s t a n t . In this limit ~---,0 (the system remains stable) and eq. (4) reads x ( m ) = Xo cosm05 + [Xo \.'(!f)

-

½Xo ,~'~] sinm05. (4')

II)'

Ill

Further, since m = z/I and 05 ---, x:~ one obtains where./' is the focal length. The lattice is stable if

,705 = z/,,(U).

0<~

A continuous distribution has the oscillatory solution

= t/{<4,

with the acceptance being reasonable only for ~ < 2. If we consider an arbitrary ray which enters the first

x ( z ) = XoCOS(2rcz/2) + X'o()+j2rr)sin(2rcz/2 ) .

(4")

C o m p a r i n g the two equations (4') and (4"), we see that in the limit ~---,0 the F O F O channel has a betatron wavelength given by

;~ = 2~ \ ' ( ! f ) ,

(6)

I00 T

and an acceptance ellipse

.~ 1

L

80

] P,] (MeV)

T

]

.//

/

/

2re 2 R 2 =

6C

t

r

i

-

-

7rR 2

-

(5)

,,/(If) '

/

+I

Finally the total length of N distributed lenses required to perform one complete betatron oscillation is given by

2O

4rr 2 pc

L

A

20

40

dO

dO

L I00

.--

A = Nu =----,

2

(7)

3G

r (ram)

Fig. 8. Plot of the mean of the magnitude of the transverse kick for a m u o n passing through the lens. This arises from a combination in quadrature o f the multiple scattering and of the focusing effects. The dashed curve is the expected dependence of ]PLI for a multiple scattering of 40 MeV combined with a resolution of 24 MeV and a gradient o f 2.5 kG/cm.

in the same units as eq. (3) and with 2 in m ; a represents the length of one individual lens. Another application of such lenses is at the position where the hadronic component of a m u o n beam is absorbed. For high energy muon beams the x-/a decay angle is small so that the emittance of the beam is

86

J. L E B R I T T O N et al.

increased m a i n l y d u e to the m u l t i p l e s c a t t e r i n g at the absorber. Therefore a focusing absorber presents a d v a n t a g e s in spite o f the relatively h i g h Z o f iron. W e are i n d e b t e d to H. F o e l s c h e , W. W a l k e r a n d the A G S S t a f f for m a k i n g these m e a s u r e m e n t s possible. W e also t h a n k J . C . S a n d e r s , J. S c h a r e n g u i v e l a n d M. Miller for assistance in setting up a n d p e r f o r m i n g these m e a s u r e m e n t s .

References ~) See, for instance, M. Strovnick, Fermilab Summer Study

2)

~)

'~) 5)

SS-73/215; W. A. Wenzel, Note BeV.3066, Lawrence Berkeley Laboratory. After our original proposal to use axial current lenses (see Fermilab proposal P-314 and F Lobkowicz and A. C. Melissinos 12/I 1/74), we were informed that similar devices are used in the Serpukhov muon beam. The axial plasma configuration is unstable due to the focusing effect that is sought. Thus, even if it were possible to provide the power required for dc operation, the plasma would "'pinch" in a short time. Including an r.m.s, resolution of 24 MeV/c. For a discussion of FOFO channels see, for instance, S. V. Pepper and A. C. Melissin~s, University of Rochester preprint [also BNL report 7957 (1962) p. 40] and A. Citron, M. Morpurgo and H. Overas, CERN report 63-35 (1963).