A novel efficient method for measuring the polarisation of positrons

Volume 3, number 7

P HYSICS L E T T ER S

A NOVEL EFFICIENT METHOD THE POLARISATION OF

15 February 1963

FOR MEASURING POSITRONS

L. D I C K , L. F E U V R A I S *, L. MADANSKY ** and V. L. T E L E G D I *** CERN, Geneva Received 24 January 1963

In t h i s l e t t e r we w i s h to d e s c r i b e a n o v e l m e t h o d f o r m e a s u r i n g the p o l a r i s a t i o n of p o s i t r o n s . T h i s m e t h o d , a s c o m p a r e d w i t h the m e t h o d s a l r e a d y in u s e 1), h a s the a d v a n t a g e of high efficiency, i . e . , it r e q u i r e s a c o m p a r a t i v e l y s m a l l n u m b e r of p o s i t r o n s to e s t a b l i s h the s i g n (and p o s s i b l y the m a g n i t u d e ) of the p o l a r i s a t i o n to a given l e v e l of c o n f i d e n c e . F o r t h i s r e a s o n , i t m a y be of i n t e r e s t f o r h e l i c i t y m e a s u r e m e n t s in the p o s i t r o n d e c a y of e l e m e n t a r y p a r ticles, where the available source strenths are u s u a l l y r a t h e r w e a k b y the s t a n d a r d s of B - s p e c t r o scopy. T h e n o v e l m e t h o d e x p l o i t s the p r o p e r t i e s of p o s i tronium. The positrons under investigation must h e n c e b e s l o w e d down to a t o m i c v e l o c i t i e s in a m e d i u m w h e r e p o s i t r o n i u m i s a b u n d a n t l y f o r m e d . It t u r n s out t h a t t h i s h a r d l y c o n s t i t u t e s a l i m i t a t i o n even in the c a s e of v e r y e n e r g e t i c p o s i t r o n s s u c h a s t h o s e e m i t t e d in muon d e c a y . The g r o u n d s t a t e of p o s i t r o n i u m i s a d o u b l e t , c o n s i s t i n g of a 1S o ( " p a r a " ) l o w e r and of a 3S 1 ( " o r t h o " ) u p p e r m e m b e r , s e p a r a t e d b y 0.8 eV. In v a c u o , and in the a b s e n c e of an e x t e r n a l m a g n e t i c f i e l d B, the s i n g i e t a n n i h i l a t e s only into two p h o t o n s (at a r a t e A o = 0.8 X 1010 s e c - 1 ) and the t r i p l e t e x c l u s i v e l y into t h r e e p h o t o n s (at a r a t e A1 = 0.5 x 107 s e c - 1 ) . In the p r e s e n c e of B, the two m = [1 1 s u b s t a t e s of the t r i p l e t r e m a i n u n a f f e c t e d , w h i l e l i n e a r c o m b i n a t i o n s of the m = 0 s i n g l e t and t r i p l e t w a v e f u n c t i o n s b e c o m e t h e new e i g e n s t a t e s . T h e s e e i g e n s t a t e s with m = 0, to w h i c h we s h a l l r e f e r a s " p s e u d o s i n g l e t " and " p s e u d o t r i p l e t " , c a n both a n n i h i l a t e into e i t h e r two o r t h r e e p h o t o n s , and t h e i r e n e r g i e s d e p e n d q u a d r a t i c a l l y on B. If u n p o l a r i s e d p o s i t r o n s a r e u s e d (in an u n p o l a r i s e d m e d i u m ) to p r o d u c e p o s i t r o n i u m , then t h e l a t ter is formed unpolarised as well. Its four magnetic s u b s t a t e s a r e e q u a l l y p o p u l a t e d , t r i p l e t and s i n g l e t * On leave from Laboratoire Joliot-Curie de Physique NuclSaire, Orsay (France). ** Ford Foundation Fellow 1960-61; present address: Johns Hopkins University, Baltimore, Maryland. *** National Science Foundation Senior Fellow, 1959-60; present address: Enrico F e r m i Institute for Nuclear Studies, University of Chicago, Chicago, Illinois. 326

b e i n g f o r m e d in t h e r a t i o 3 : I . T h i s i s t r u e in the p r e s e n c e o r a b s e n c e of a m a g n e t i c f i e l d B. T h e m e t h o d p r o p o s e d h e r e r e s t s on the f a c t t h a t

the pseudotriplet/pseudosinglet population ratio becomes (1 - 0 / ( 1 + c) when one forms positronium from positrons polarised, at the instant of electron capture, along B. T h e p a r a m e t e r c d e p e n d s r o u g h l y l i n e a r l y on the p o s i t r o n p o l a r i s a t i o n and on B, changing sign appropriately. T h i s f a c t , and one w a y of a p p l y i n g i t to p o s i t r o n p o l a r i s a t i o n m e a s u r e m e n t s , w e r e f i r s t p o i n t e d out b y P a g e and H e i n b e r g in 1957 2). O u r m e t h o d d f f f e r s f r o m that of t h e s e a u t h o r s in t h e w a y in w h i c h the p s e u d o t r i p l e t / p s e u d o s i n g i e t p o p u l a t i o n r a t i o i s d e t e r m i n e d in p r a c t i c e . P a g e and H e i h b e r g p r o d u c e d p o s i t r o n i u m in a r g o n g a s , and e x p l o i t e d the f a c t t h a t t h e p s e u d o t r i p l e t and p s e u d o s i n g i e t a t o m s h a v e d i f f e r e n t v e l o c i t y d i s t r i b u t i o n s a t the t i m e t h e y a n n i h i l a t e into two p h o t o n s . A n n i h i l a t i o n s f r o m t h e s e two s p e c i e s of a t o m s could h e n c e b e d i s t i n g u i s h e d v i a the d i f f e r i n g a n g u l a r c o r r e l a t i o n s b e t w e e n t h e two a n n i h i l a t i o n q u a n t a ( m o r e o r l e s s s h a r p p e a k i n g a b o u t 180o). T h i s m e t h o d i s , h o w e v e r , i n e f f i c i e n t , p r i m a r i l y b e c a u s e the v e r y n a r r o w a n g u l a r c o r r e l a t i o n s i m p o s e v e r y s m a l l s o l i d a n g l e s . In t h e m e t h o d p r o p o s e d h e r e , the d i f f e r e n t time-dependence of the 2 - q u a n t u m a n n i h i l a t i o n s f r o m p s e u d o t r i p l e t and p s e u d o s i n g l e t a t o m s i s u s e d a s a " l a b e l " to d e t e r m i n e t h e i r p o p u l a t i o n r a t i o s t Such a d i f f e r e n c e a r i s e s , even in z e r o f i e l d , when p o s i t r o n i u m i s f o r m e d in c e r t a i n c o n d e n s e d m e d i a . W e s h a l l now b r i e f l y r e v i e w the b e h a v i o u r of p o s i t r o n s in t h e s e . The t i m e d i s t r i b u t i o n of the 2 - q u a n t u m a n n i h i l a t i o n s of p o s i t r o n s in m o s t a m o r p h o u s m e d i a e x h i b i t s two d i s t i n c t m e a n l i v e s : a. a f a s t c o m p o n e n t , vf = (1 - 2) × 10-10 s e c , b. a slow c o m p o n e n t , Vs - (10 - 20) Tf. T h e f a s t c o m p o n e n t i s a t t r i b u t e d to a s u p e r p o s i t i o n of the direct a n n i h i l a t i o n of p o s i t r o n s and of the a n n i h i l a t i o n of 1So p o s i t r o n i u m , w h i l e the s l o w c o r n T The application of delayed coincidence techniques instead of angular measurements to the Page-Heinberg argon experiment has already been suggested by Lundby 3).

Volume 3, number 7

PHYSICS

p o n e n t i s a t t r i b u t e d to the " p i c k - u p " a n n i h i l a t i o n of the p o s i t r o n (by t h e e l e c t r o n s in t h e m e d i u m ) in an o r i g i n a l l y 3S 1 p o s i t r o n i u m a t o m * T a b l e 1 s u m m a r i s e s t h e p o p u l a t i o n s and l i f e t i m e s of t h e v a r i o u s s o u r c e s of a n n i h i l a t i o n , u n d e r t h e a s s u m p t i o n t h a t in z e r o f i e l d direct a n n i h i l a t i o n t a k e s p l a c e a t a r a t e 1, and p o s i t r o n i u m f o r m a t i o n a t a r a t e 4n, both p e r unit t i m e .

B=o Popu- Mean lation life

Direct annihilation

1

1So

n

Tf

2n

Ts

n

Ts

3S1, m : ]11 3SI, m = 0

B~0 PopuMean lation ¢ life

~f

1 (1+

q'f ¢)n

2 (1

-

v~--vf Ts

¢)n

15 F e b r u a r y 1963

1 + x2 '

(4)

= 0.48 f o r 20 k i l o g a u s s and f o r f u l l y p o l a r i s e d e + ' s , and t h e f i e l d - i n d u c e d 2 - q u a n t u m r a t e of t h e p s e u d o triplet by Ain d ~ x 2 A o .

(5)

T h e n u m b e r n, i . e . , the f r a c t i o n of p o s i t r o n i u m f o r m a t i o n f = 4n/(4n + 1), d e p e n d s on t h e c o n d e n s e d m e d i u m u s e d and cannot b e p r e d i c t e d a p r / o r / . It i s o b v i o u s l y d e s i r a b l e to c h o o s e a m e d i u m w i t h high f and long Ts; m e d i a w i t h / a s high a s 0.5 a r e a v a i l able. U s i n g (4) and t a b l e 1, we c a n now p r e d i c t t h e m a g n i t u d e of the e f f e c t s w h i c h f u l l y p o l a r i s e d p o s i t r o n s c o u l d y i e l d . T h e c o u n t s N s ( B ) in the slow component should undergo a change

Table 1

Source

LETTERS

t

Ns(S)

Ts

- Ns(-S)

l

R s - N s ( B ) + N s ( _ 8 ) - - ~¢

(6)

i" Assuming e + polarised along B. = - 16% a t 20 k i l o g a u s s , In the a b s e n c e of an e x t e r n a l f i e l d , (1 + n) e v e n t s w i l l a n n i h i l a t e with a m e a n l i f e Tf ( a c t u a l l y a n i n distinguishable superposition of two mean lives). About 3n events w i l l annihilate into two photons at a rate A s =- 1/T s = A 1 + A p i c k _ u p .

(1)

In f a c t , a s the o b s e r v e d Ts << l / A 1 , A p i c k - u p >> A1, and t h e 3 - p h o t o n a n n i h i l a t i o n of t h e i n i t i a l l y 3S 1 a t o m s can b e n e g l e c t e d . In the p r e s e n c e of B, m a i n l y the n(1 - ¢) p s e u d o triplet atoms require special consideration. They a n n i h i l a t e at a r a t e A s' = 1 / r ~ = A 1 + A p i c k _ u p + Ain d ,

(2)

w h e r e Ain d i s the f i e l d - i n d u c e d 2 - q u a n t u m a n n i h i l a V t i o n r a t e of the p s e u d o t r i p l e t a t o m s . C l e a r l y , A s > A s and we m a y a fortioti a s s u m e t h a t e s s e n t i a l l y a l l the (1 - ¢)n p s e u d o t r i p l e t a t o m s d i s a p p e a r b y 2 - q u a n t u m annihilation. Quantitatively speaking, all field dependent effects can be expressed in terms of a mixing parameter x, given by x : 4#0

B/AE

,

(3)

w h e r e ~o = e l e c t r o n m o m e n t , AE = s i n g i e t - t r i p l e t s p l i t t i n g a t B = 0. The population differences are governed by

* An alternative explanation, namely that 3S 1 atoms get converted in the condensed medium into 1So atoms by spin-flipping collisions at a rate about 1/Tf, has been rendered implausible by the observations of Telegdi et al. 4).

w h i l e the c o u n t s Nf in the f a s t c o m p o n e n t s h o u l d give a change Nf(B) - N f ( - B ) n¢ P~ - Nf(B) + Nf(-B) - 1 + n =~¢forn

(6') =¼,f=½.

N o t i c e t h a t t h e e f f e c t s (6) and (6') have o p p o s i t e sign for fixed e+ polarisation. We have p e r f o r m e d e x p e r i m e n t s to v e r i f y the a b o v e p r e d i c t i o n s u n d e r c o n d i t i o n s of p r a c t i c a l i n t e r e s t . F i g . I s h o w s s c h e m a t i c a l l y the e x p e r i m e n t a l s e t u p we u s e d . A s m a l l s o u r c e of Na22 (~ 104 e + / s e c ) w a s d e p o s i t e d on a f a s t o r g a n i c s c i n t i l l a t o r S, e i t h e r p l a s t i c o r liquid. T h i s s c i n t i l l a t o r s e r v e d both a s an e + d e t e c t o r and a p o s i t r o n i u m - f o r m i n g m e d i u m . It i s known t h a t p l a s t i c s c i n t i l l a t o r y i e l d s a Ts c o m p o n e n t 5) ( f = 0.58); f o r l i q u i d , f : 0.45. T h i s s c i n t i l l a t o r w a s v i e w e d b y a 56AVP m u l t i p l i e r t h r o u g h a long l i g h t p i p e , a n d p r o v i d e d t h e t = 0 s i g n a l s . A n o t h e r f a s t s c i n t i l l a t o r , X, v i e w e d in a s i m i l a r w a y , s e r v e d to d e t e c t t h e 0.51 MeV p h o t o n s f r o m a n n i h i l a t i o n s in S. T h e t i m e d i s t r i b u t i o n of c o i n c i d e n c e s b e t w e e n S and X w a s r e c o r d e d b y a f a s t t i m e - c o n v e r t e r s y s t e m g a t e d in t h e u s u a l w a y b y s l o w , e n e r g y - s e l e c t e d c o i n c i d e n c e s . V e r y good t i m i n g (~ 1.8 n s l f u l l width a t half m a x i m u m , f l a n k f a l l - o f f to 1 / e in 0.2 ns) w a s a c h i e v e d b y m e a n s of s p e c i a l p u l s e - s h a p i n g c i r c u i t s 6) a t t h e i n p u t s ~o the time-converter system. This time-sorter has p r o v e d to have a s t a b i l i t y o v e r a r a n g e of m o r e than a w e e k of b e t t e r than 0.1 ns. Some t y p i c a l t i m e d i s t r i b u t i o n s o b t a i n e d w i t h a m a g n e t i c f i e l d ~ = 18.5 k G a u s s p e r p e n d i c u l a r to t h e s u r f a c e of S a r e r e p r o d u c e d in f i g s . 2 and 3. 32"/

Volume 3, number 7

PHYSICS

LETTERS

15 February 1963

Detail A

.%

i

÷ ,, 1 chonn(zl-=0.21 ns

i

÷ ~,

+

+++ S I ....

¢

w,

B I

÷

Coil LigM:Guid.

~

\

t

i0 3i •,

t

L~)~t GukW

+

I

÷++

~02

.4

RM, 56 A V P linch

,L*÷

•+ •+ •+ •il-

I

I

{

2O

Fig. I . Experimental set-up. ~

i

_

i

40

60'

80 chonnels

Fig. 3. Positrons Na 22 in sandwich plastic scintillator. ~" B-18.5OO gauss

I channel • O. 21ns

tic., . . . .

,-h . . . .

R(L} °/o

~(t)

,or

* . . . . Experimental points

~<~--(~>-.7 L (cos O> --.5

IO4

- - ~ ~,,

11 / -

~

--

0 T '~"

1

2

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4

3

IO3

t -5

"

~_ 30

"

A

_

_

40

I

±

50

60

channels

Fig. 4. R(t) = N(Bf) - N ( B O for positrons Na 22 in N(B~) + N(B~)

plastic scintillator. tO2

1"

R(t) l °l o

.__1

20

L

40

~

61 0

J__

1 ,i 80 channels

Fig. 2. Positrons Na 22 in plastic scintillator. T h e two c u r v e s shown, c o r r e s p o n d i n g to two o p p o s i t e f i e l d d i r e c t i o n s , w e r e n o r m a l i s e d to the s a m e n u m b e r of counts. F i g . 2 i s i t s e l f s e l f - e x p l a n a t o r y . A s an i n s e r t to t h i s f i g u r e we g i v e a p l o t of the quantity R ( t ) =- N ( B I ) - N ( B J ~ N(Bt) + N(BD

(BI and BI a r e d e f i n e d in fig. 1) d e f i n e d in a n a l o g y 328

•

0

--L~o

• I. • ..:,,

-2

N I

Io

I • e.e

"2

k

",J

B

_ r

i

6

ns--

•

[

40

I

i

GO

i

80 channels

Fig. 5. R(t) - N ( B ~ _ - N(B~) for positrons Na 22 in - N(B~) + N ( B | )

sandwich plastic scintillator.

PHYSICS

Volume 3, number 7

LETTERS

accepting e+ with energies > 200 keV and with initial directions of emission with respect to B uniformly distributed between 0 and 90°. Hence, barring depolarisation phenomena, we expect from (6), (6’) and (7) an effect of - 5% in the slow component and an effect of 3% in the fast component (for f = 0.5).

with eqs. (6) and (S’), to exhibit the effect more clearly (fig. 4). Fig. 3 represents similarly the result of control experiment in which the source was sandwiched between two scintillators (i.e., (a. B) = 0). The R-curve in fig. 4 has the shape expected from the effect, the time resolution and the electronic fluctuations of the equipment used *. The observed magnitude of the effect agrees to within 20% with the theoretical prediction, and the control experiment shows that no major systematic errors were indicated by the effect of reversing B. The R-curve for this last case is given in fig. 5. In comparing the magnitude of the observed effects with the predicted values (6) and (6’) predicted for unitpolarisation, one must remember that the effective polarisation is given here by Peff = (V/C> * (0.B)

15 February 1963

References 1) For a survey, see, e.g., L.Grodzins, 2) 3)

4) 5) 6)

(7)

= 0.70 x 0.5 = 0.35 ,

7)

* For the computation of such a curve see ref. 7).

Measurement of helicity, Progress in Nuclear Physics, Vol. 7 (Pergsmon Press, London, 1959). L.A. Page and M.Heinberg, Phys. Rev. 106 (1957) 1220. A. Lundby, in: Progress in elementary particle and cosmic ray physics, Vol. 5 (North-Holland Publishing Company, Amsterdam, 1960). See also the remarks on p. 219 of ref. 1). Telegdi et al., Phys. Rev. 104 (1956) 867. D. Freytag and K. Ziock, Z. Physik 153 (1958) 124. L.Dick, Mesure des intervalles de temps dans le domaine de 0.1 ns avec des df?cteteurs a scintillation. to be published. L.Dick et al., J. phys. radium 17 (1956) 583.

*****

GRAVITATIONAL

ENERGY

RADIATION

C. M@LLER Universitetets Institut for Teoretisk Fysik, and Nordita, Copenhagen Received 25 January 1963

Since the first years of Einstein’s theory of gravitation the question whether or not a system of accelerated massive bodies loses energy by emission of gravitational radiation has given rise to many controversial discussions. There are two reasons for this somewhat unusual situation in physics. First, on account of the difficulties in finding general solutions of Einstein’s non-linear field equations the discussion has mainly been based on solutions of the linearised field equations, which in many respects are poor approximations to the solutions of the exact field equations. Secondly, until recently we did not have a consistent expression for the gravitational energy current which in analogy ‘with Poynting’s theorem could be used for calculating the energy carried away by the gravitational ,waves. It is well known that the energy-momentum complex 8ik given by Einstein many years ago does not allow us to calculate the distribution of the energy and the energy flux in a physically satisfactory way since the result depends on the spatial coordinates used. Even if one is interested only in the total energy and its possible variation in time,

as in calculations of the energy emission from an insular system, the complex ek is applicable only in special systems of coordinates. In the trivial case of a completely empty space, for instance, Einstein’s expression gives an infinite value for the total energy when calculated in polar coordinates, in contrast to the correct value zero obtained if one uses Cartesian coordinates. Both of the difficulties mentioned above have now been overcome. In a recent, most interesting paper, Bondi et al. 1) have been able to give the exact form of the metric at large spatial distances from an axisymmetric, but otherwise arbitrary, insular system of matter that emits gravitational waves into the surrounding empty space. Further, in a paper from 1961 2) we arrived at an expression T$ for the energy-momentum complex that meets the objections raised against Einstein’s expression 6ik. While 6ik is a function of the metric, Tik is expressed directly in terms of tetrads connected with the metric in the usual way. In general, the energy density and current calculated by means of Tik will depend on the choice of the different tetrad fields compatible with 329