1468 L E T T E R S T O T H E E D I T O R

/3,^0 AND I

1.0 .8 .6

FIG. 2. The ratio (f>p/cf>a for /x-meson-proton scattering.

Here r=#&2/V»i, yi—Ei/miy and 0= the scattering angle. The cross section <f>p (</>a) for spins initially parallel (antiparallel) is obtained by setting €ie2=— 1 ( + 1 ) . In Fig. 1 the ratio <j>P/<t>a is plotted for a number of values of the neutron velocity ph for the neutron-ju-meson system. For the neutron-electron system the center-of-mass frame is for all practical purposes identical with the neutron's rest frame; accordingly, for this system, only the case when /3i=0 is plotted.

2. Particle " 1 " is charged and has an anomalous moment; particle " 2 " is charged but has no anomalous moment. This case describes proton-electron and proton-ju-meson scattering. Equation (4) becomes

1 / eie2 \2/dti\ *(«i«0=-( ) ( - )

2\E1+E2/ \b2/

X\{2a*-2ab+bmM 1 I+&(&+*»**) (1+gi)2

I V 2mi2/

+ 6 i€ 2 ( l+g 1 )5 [6 ( l+g i )+a ( l+cos^ ) ] j , (10)

where a=pi'p2=pi-p2. The ratio <l>v/(j)a obtained from Eq. (10) is plotted in Fig. 2. I t is seen that <j>p

equals <j>a in both the nonrelativistic and extreme rela-tivistic limits. The energy at which <t>v/<j)a attains its minimum value depends on the scattering angle; it is approximately given by4

7 i » = l + ( l / * i ) . (11)

In conclusion we note that Eq. (11) provides an estimate of center-of-mass energies required to make the scattering method work as an analyzer of polariza­tion. Thus for the ju-meson-electron system (gi=0) one needs very high energies, for the /z-meson-proton system (g!~1.8) one needs intermediate energies, and for the /x-meson (or electron)-neutron system (gi= <*>) very low energies are sufficient.

The author wishes to thank Dr. M. Goldhaber for suggesting this calculation.

* Work performed under the auspices of the U. S. Atomic Energy Commission.

1 A. M. Bincer, Phys. Rev. 107, 1431 (1957). We follow here the notation and definitions of that paper.

2 Note that this implies replacing k by — k [see Eq. (3)]. 3 Recent experiments indicate that the anomalous magnetic

moment of the p meson is, like that of the electron, negligible [T. Coffin et ah, Phys. Rev. 106, 1108 (1957)].

4 Equation (11) is exact for cos0= —1 and it is valid for positive gi only. For negative g\ the situation is more complicated; for example, gi= —1 (which corresponds to a total magnetic moment zero) gives, of course, no polarization effects at any energy.

T

Polarization of Recoil Protons in Pion-Proton Scattering*

HONG-YEE CHITJ

Laboratory of Nuclear Studies, Cornell University, Ithaca, New York

(Received July 10, 1957)

WO or more Fermi-type phase shift solutions have been found at various energies above the

!T=! state resonance. The accidental character of this ambiguity has been discussed,1 and its nature is re­viewed briefly below in (c). If accurate charge-exchange differential cross sections become available, they may be able to discriminate between these solutions, but it is possible that the ambiguity will persist. The purpose of this note is to indicate that the ambiguity can be resolved by a feasible measurement of the recoil-proton polarization.

The proton polarization in the T—\ state has previ­ously been given explicitly in terms of the phase shifts by Fermi2 and Hayakawa et al? They pointed out that the polarization would remove the ambiguity arising from changing the sign of all the phase shifts simul­taneously, and also remove the two ambiguities in­dicated by Yang and Minami. There is now evidence from Coulomb interference and dispersion relations which supports only the Fermi-type phase shifts.4,6

However, the accidental ambiguity mentioned above can also be resolved by polarization measurements.

We have analyzed the pion-nucleon scattering in the 100-300 Mev region.1 The main conclusions are the following.

(a) The Fermi-type 533 has a resonance at 187±2 Mev.

(b) Below resonance 5 3 i ~ - 5 ° , 0<5 i 3 <5° , - 3 ° < 5 n < 0 , and both di and 8Z are close to the straight-line extrapolation from low-energy data.

(c) Above resonance two solutions appear. This is due to the occurrence of two roots of the quartic equa­tions which, accidentally, are approximately equal for negative and charge-exchange scattering. One solution [labeled (i) in reference 1] corresponds to Si3«53i, but

L E T T E R S TO T H E E D I T O R 1469

this requires that both 613 and 8n have opposite signs to their low-energy behavior. 81 and 53 continue on the straight-line extrapolation. The other solution [labeled (ii)] allows a mono tonic extrapolation of all the small P-wave phase shifts, and 83 follows the straight-line extrapolation. However, 81 decreases and changes sign near 220 Mev. Only the latter solution corresponds to the root obtained below resonance. Nevertheless, it is important to discriminate experimentally between the two possibilities. A polarization measurement of com­paratively poor accuracy will certainly discriminate between the two roots.

The polarization of a proton beam is measurable at energies > 7 0 Mev by scattering on a carbon target.6

The recoil energy of the proton is

Mpc2l32

Er=— {( l+cos0) - i /3 2 sin20} ( l - £ 2 ) 2

- W cos2 (P ) Mev,

where /3=cP7 r/(E7 r+Mpc2), PT is the incident-pion momentum, ET its total energy in the laboratory sys­tem, 6 is the recoil angle of the proton in the c m . system, and rj is the meson momentum in the c m . system in units of M^c. Even for an incident pion with a laboratory energy of 150 Mev, the above relation permits measurement of the polarization for 0<43°. The laboratory angle for the proton, 0', is given by

taitf' = (1 -02) * tan (§0)«tan(J0),

and the pion laboratory angle <pf is given by

sin0' tan p'—

(PJPJ) -cos0'

where Pv' is the recoil proton momentum in the labora­tory system.

As we are considering only Fermi-type solutions, we need consider only those polarization terms in which 533 appears. This an excellent approximation, as terms in which 533 is absent are cubic in the small phase shifts, but terms in which it is present are only linear in these phase shifts. Including 5, P , and D states and using the notation

/ ( l ' , r , / ' | 1,7V) = 4 s inSrv1 ' SHISTV1 sm(8T>j>l'-8Tj1)i

we have

2X-2—P+(0) = sin0{/(O,3,l 11,3,3) do)

+ (9/8)/(l,3,312,3,5)-§/(l,3,312,3,3)}

+sin20{ -f /( l ,3,311,3,1)}

+s in30{- f / ( l ,3 ,3 |2 ,3 ,5 ) - f / ( l ,3 ,3 |2 ,3 ,3 )} )

18X"2—P~(d) = 21r2 P+(0)+sin0{2/(O,l,l 11,3,3)

do) dco

+ (9/4)/(l,3,312,1,5)-/(1,3,312,1,3)}

+sin20{-37(1,3,311,1,1)}

+s in30{-f7( l ,3 ,3 |2 , l ,5 ) -37( l ,3 ,3 |2 , l ,3 )} . In the (w+p) case, the sin 20 term is proportional to

Bziy and on neglecting D states, the sin0 term is propor­tional to 8S. As 53i is only known to within a factor of 3 from the differential cross section, a rough measure­ment of the sin20 term would improve our knowledge of 53i. Similarly, from

da— da+ 18X-2 P-(0)-2X~2 P+(0),

do) do)

the sin0 and sin20 terms are proportional to 5i and 5n, respectively. The sign of 8u above resonance would remove the ambiguity mentioned before. If the sin0 term could be measured with better than 50% ac­curacy, the magnitude of 81 would also remove the ambiguity. Finally any sin30 dependence, in either the 7r+ or the 7r~ case, would indicate the presence and the magnitude of the D states. The greater sensitivity of the polarization to the small phase shifts, is due to the absence of any terms containing only 533.

At 220 Mev the expected polarization from negative pions on protons from solutions (i) or (ii) of reference l i s

da-(i) 18X-2 P-(0) = - 0 . 6 sin0-O.7 sin20,

do)

(ii) 18X-2—~-P~{e) = 0.3 sin0+O.9 sin20, do

which should be readily distinguishable. D states have been neglected. However, a theoretical indication is that the T~\ states have half the magnitude of and the opposite sign to the J P = § states. In this case their contribution to P~(6) vanishes. If, however, the D states are such as to alter significantly the sin0 term, a measurable sin30 term will probably be present.

I t is a pleasure to thank Professor H. A. Bethe for suggesting the approach followed here, and to thank Dr. E. L. Lomon for many helpful discussions.

* Supported in part by the joint program of the Office of Naval Research and the U. S. Atomic Energy Commission.

1 H. Y. Chiu and E. L. Lomon (to be published). 2 E. Fermi, Phys. Rev. 91, 947 (1953). 3Hayakawa, Kawaguchi, and Minami, Prog. Theoret. Phys.

Japan 11, 332 (1953). 4 W. Gilbert and G. R. Screaton, Phys. Rev. 104,1758 (1956). s Byfield, Kessler, and Lederman, Phys. Rev. 86, 17 (1952). 6 G. H. Stafford, Proceedings of the Seventh Annual Rochester

Conference on High-Energy Physics (Interscience Publishers, Inc., New York, 1957).