Dumbrajs, O V

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CERN LIBRARIES, GENEVA CM-P00100675 JOINT INSTITUTE FOR NUCLEAR RESEARCH, DUBNA Report P2 - 5444 COMMENT ON ELASTIC NN- AND Nd-SCATTERING O.V. Dumbrajs Dubna 1970 Translated at CERN by B. Hodge (Original: Russian) Revised by N. Mouravieff (CERN Trans. 71-22) Geneva March 1971 Elastic NN scattering is characterized by the fact that it is the only example of elastic scattering of elementary particles of one and the same isotopic multiplet that is accessible to experimental measurement. We can obtain several supplementary relations between the values which characterize elastic scattering. Let us examine the processes p + n → p + n, (1) P + P → Ρ + Ρ. (2) The amplitude of reaction (1) is * ): Α 1(θ) = ½ < 0,0 |S00|0,0 > + ½ < 1.0|S00|1,0 > = = ½ f0 + ½ √⅓ f1, (3) where f0, f1 are amplitudes corresponding to the total isotopic spin Τpn = 0 and Τpn = 1. As p and n are terms of the same isomultiplet, Α1(π-θ)= - ½ f0 + ½ √⅓ f1. (4) [see, for example, Dumbrajs and Podgoretskij1) and Dumbrajs2)]. For reaction (2) we have A2(θ) = √⅓ f1. (5) From Eqs. (3) to (5) follow the relations σ1(θ) + σ1(π-θ)= ½ |f0|2 + ½ σ2(θ), (6) cos (δ0 - δ 1 ) = σ1 (θ)-σ 1(π-θ) (7) √σ2(θ) √2σ1 (θ)+2 σ1(π-θ)-σ2(θ) *) Here, and in the remainder of the text, the influence of nuclear spins has been disregarded. - 2 -where δ0, δ1 are the phases of the amplitudes f0 and f1, respectively. Let θ = 0; then the imaginary part of amplitude f0 can, with the aid of the optical theorem, be expressed by the total cross-sections of reactions (1) and (2), and the real part by the real parts of the ampli­tudes of these reactions: Im f0 = k σ tot - k σ tot (8) 2π p n 1π p p Re f0 = 2 Re fp n (0) -Re fp  (0). (9) Finally, formulas (6) and (7) become, respectively*), the relations αpn αpp = σpp ( 0 ) , σ p n(0) - σpn (π) - 1, (10) k 2 σ tot σ tot 8π 2 p n p p cos (δ 0 (0) - δ1 (0)) = σpn ( 0 ) - σpn (π) (11) √σpp (0)√2σ p n (0)+2σpn (π)-σpp (0) where αpn = Re f p n(0) , αpp = Re fpp (0) . lm fpn(0) Im fpp (0) Expressions (10) and (11) can be used, for example, to check general agreement between the experimental and theoretical data, as well as in several other cases, in particular for investigating pd scattering. As is known4), the total scattering cross-section of p on d can be expressed in the following manner: σ tot = σ tot + σ tot - < r-2> σ tot σ tot [1-αpn α p p ] , (12) p d p n p p 4π p n p p *) A relation similar to (10) has already been obtained by Kanavets3). The author is grateful to V.A. Nikitin for drawing his attention to this paper. - 3 -where the last component is Glauber's correction, r-2 is the average value of the inverse square of the distance between the proton and the neutron in a deuteron. One of the applications of formula (12) is to calculate experimentally the value σtotpn which is difficult to measure. The product of apn app is usually assumed to be zero [see, for example, Galbraith et al.5)] since it is thought that, in the energy range for which Glauber's approximation is correct, the scattering amplitude is almost purely imaginary. Thus, yet one more approximation is introduced into Glauber's already rather inaccurate correction. By means of the optical theorem, apn app can, of course, be expres­sed by the differential cross-sections in the forward direction and the total cross-sections, but only with an accuracy of up to the sign. How­ever, when formula (10) is taken into account, this lack of accuracy is eliminated and formula (12) can be rewritten in the form: σ tot = σ tot + σ tot + <r-2> { 8 π2 [σpp (0) + p d p n p p 4 π k2 + σ p n(0)- σpn(π)]-2σ tot σ tot }. p n p p (13) A similar modified expression for Glauber's correction can be ob­tained if, instead of formula (12), Glauber's "isotopically invariant" correction is used6,7). Let us point out once more that we obtained expressions (10), (11), and (13) on the assumption that there was no spin influence; consequently, any departure from them will point to the part played by spin effects in NN and Nd interactions. I am grateful to M.I. Podgoretskij, N.M. Kuin and A.V. Tarasov for their useful comments. - 4 -REFERENCES 1) O.V. Dumbrajs and M.I. Podgoretskij, Yadernaya Fiz. 11, 232 (1970). 2) O.V. Dumbrajs, JINR communication R2-4621, Dubna 1969. 3) V.P. Kanavets, Yadernaya Fiz. 2, 931 (1965). 4) R.J. Glauber, Phys. Rev. 100, 242 (1955). 5) W. Galbraith et al., Phys. Rev. 138 B, 913 (1965). 6) C. Wilkin, Phys. Rev. Letters 17, 561 (1966). 7) R.J. Glauber and V. Franco, Phys. Rev. 156, 1685 (1967). 

Comment on elastic NN- and Nd-scattering

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