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The influence of tensor forces on the differential cross section for the scattering of polarized neutron.. Lambe, Edward Bryant Dixon 1949

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THE INFLUENCE OF TENSOR FORCES ON THE DIFFERENTIAL CROSS SECTION FOR THE SCATTERING OF POLARIZED NEUTRON BEAMS BY PROTONS by EDWARD BRYANT DIXON IAMBE A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of MASTER OF APPLIED SCIENCE in the Department of The Influence of Tensor Forces on the Differential Cross Sedtion for the Scattering of Polarized Neutron Beams by Protons ABSTRACT Tensor forces between two particles involve a dependence upon the angle between the direction of spin quantization and the line joining the two particles. The effect of tensor forces upon the scattering of a polarized neutron beam has been investigated theoretically. An expression has been obtained for the differential scatter- ing cross section of the t r i p l e t states as a function of the polarization of both neutrons and protons. In general, this cross section i s also a function of the azimuthal angle to the direction of propagation of the neutron beam. The Influence of Tensor Forces on the Differential Cross Section for the Scattering of Polarized Neutron Beams by Protons Introduction In order to explain the electric quadrupole moment of the deuteron, i t has been necessary to introduce a tensor interaction potential of the form SV(r), where Rarita and Schvringer have calculated the effects of this potential upon both the bound and the unbound states of the neutron-proton system. In particular, they have calculated the scattering cross section of a beam of neutrons by a proton target for neutrons of low energy. These calculations were extended by Ashkin and Wu 3, who used a more general phase shift analysis. Recently, Rohrlich and Eisenstein ^ have solved the same problem (and obtained identical results'.) by means of a method which the authors find to be more satisfying theoretically. In a l l of the above papers, the protons and neutrons were 1. Rarita and Schwinger - Physical Review 59, 436, 1941. 2. Rarita and Schwinger - Physical Review 59, 556, 1941. 3. Ashkin and Wu - Physical Review 73, 973, 1948. 4. Rohrlich andEisenstein - Physical Review 75, 705, 1949. 2 considered to be completely unpolarized; i.e., the spins of the par- t i c l e s were assumed to have no preferential direction of alignment. The results of such a calculation showed the scattering cross section to be dependent only upon the polar angle to the direction of propaga- tion of the neutron beam. However, i n order to answer a question raised by Dr. G.C. Laurence of Chalk River, i t was decided to investigate whether a dependence upon azimuthal angle i s introduced by certain polarization states of the neutron-proton system, and to determine how the polar dependence i s modified by such states. tering cross section as a function of the azimuthal andppoiar angles and of parameters which are determined by the polarization of the neutrons and protons. Calculation of the Differential Cross Section . applied to any singlet spin function, only the contribtuion of the t r i p l e t scattering to the total cross section w i l l be considered. In the centre of mass co-ordinate system, the i n i t i a l incident wave i s represented by the expression where r is the vector from proton to neutron, Hk is the momentum e=£ An expression has been obtained for the differential scat- Because the tensor interaction operator yields zero when (1) - See Reference 3. This paper i s the starting point for the c a l - culations which follow. 3. of the incident neutron i n the centre of mass co-ordinate system, X* 5 (f<s- /, O,~/) are the three t r i p l e t spin functions, defined for convenience with respect to k, and ( 7*^- o -f) are constants which depend upon the polarization states of neutron and proton. As a result of the tensor interaction, the asymptotic form of the scattered wave i s The matrix elements ^^4^ depend upon both & and «p , the polar and azimuthal angles respectively; the dependence upon ^ i s a direct result of the asymmetry of the tensor force. To obtain the t r i p l e t scattering cross section per unit solid angle, one calculates the square modulus of the coefficient of in (2), which yields (using the orthonomality of the t r i p l e t spin functions): An examination of the specific form of the matrix elements ^% (displayed as Equations 10 and 11 i n this paper) leads one immediately to the conclusion that ^-dependence disappears completely from the f i r s t term i n (3a), and may enter the f i n a l expression only by virtue - See Reference 3, Page 931. The symbol "^**s*s i n this paper i s identical i n meaning to the corresponding symbol i n Ashkin and wu. These elements have been written on the assumption that the z-axis i s the direction of propagation of the incident beam of neutrons. 4. of the cross-product terms. If both neutrons and protons are completely unpolarized, the cross-products disappear i n averaging over the phases of the amplitudes : a simple calculationr-shows that aver- ages to zero i f i- "*s , and to i f -*"s s **-s . Therefore. As w i l l be seen, i f there i s a specific polarization, these cross- products do not disappear except i n special, symmetric cases. It i s necessary to calculate the coefficients for an arbitrary but specific polarization of the neutrons and protons. In the following, i t w i l l be assumed that the polar axis i s both the direc- tion of propagation and the direction of reference of the spin functions. The following physical situation i s considered: i n the direction Qi , Ql ( i = n,p and refers thus to neutron or proton), the ratio of the number of particles i n a state with spin angular momentum ^ i n the positive sense to the number having a spin of •£ i n the negative sense i s ŷ̂ -j-e . To obtain the differential cross section appropriate to this physical situation, we may f i r s t compute the cross sections for four specific spin orientations of the neutron-proton system (protons and neutrons parallel and anti-parallel to the directions @it(Q(}t and then compound the results by means of a s t a t i s t i c a l argument. However, i t may be shown that i t i s also possible to proceed by intro- ducing at the outset a wave function which refers to an assembly of neutrons and protons i n the state described above: / . (/f ̂ 4 AjTf *frjyt f{) (5) In the above expression, the At' are the random s t a t i s t i c a l phase factors which must be averaged out i n arriving at a f i n a l result. anrl tho—phooco—&"—appoap-ao a result of-tho physical oonditiona ira- •pmsgd. aCi , bi represents the states of positive and negative spin in the usual way. These states may be expressed in terms of the spin functions «iif referred to the z-axis by the following transformations ( 6 ) I Thus the total spin function (including both singlet and triplet states) xs: fsf * (A~.*~+ 6+ ̂ ) (Af~ *>r f r ) where: - ^ //TT ^ fa/ iXl i—77 «. - 0t Now, we have written in (1) only the triplet part of V^o which is with X , X , and A " so defined that Averaging the products of the amplitudes dj^ A^Tover the phases, , one obtains: < <<*)«--•? ) ( ^ v t •o-j-K'f) 6. + ^&tefr*-0(^ic^-]pWf) The above expressions represent the coefficients in (3); i t is now necessary to calculate the terms ^ > » « s W * o r d e r to make the Oj*-dependence of these terms explicit, we reproduce the Ashkin and Wu matrix as follows: S„, - S_„., = A Si,, - *r*c - £ c 7. c-± Ztitojfim J i & ' t I £ ,S C [ 2 ( 2 c H f f i (?£l) , fr ,XczS"''°) In the above expressions, ̂  Id)are the normalized associated Legendre polynomials, and <\ are the phase shifts of the scattered wave. (It is to be remarked that each of A, B, and C is also equal to a similar cf/ c?-' expression with ° u replaced by ^ .) We now define 7^ ̂  to be ^»»r**$ ' then ni0 - <? /or +• /*/* - A / n., * a ^ - <? *** where ^ and § are defined by: fe1? [40* iSC* and £ <- *  W l7'- If expression (3) is now expanded as indicated, the a. result i s : Conclusions It may be concluded after an examination of the expression for the triplet scattering cross sectio  th t there are wo conditionswhich are separately sufficient for azimuth l symmetry t C 1. <9K= &p ~ O or ?) t that i , f r any pol rization state in which b th neutrons and p oto s ar  align d parallel to the direction of t eneutrons, a d 2 £p * ^ = ̂  , that is , for complete non-polarization. Fo  other a e  there is a d pe de c  upon h  zimuthal angle which b com s s o ge  the p iz i  b c mes mor  comp ete. 9 . It i s of considerable experimental interest to note that a partially- polarized beam of neutrons impinging upon an unpolarized proton target would, as a result of the tensor interaction, be expected to show an azimutlially asymmetric cross section. The magnitude of this asymmetry and the conditions under which the asymmetry w i l l be a maximum are to be calculated shortly. 1- Acknowledgements I am deeply indebted: to Professor Volkoff, for his generous - encouragement, assistance, and inspiration, not only i n this problem but i n many others also; to Professor Opechowski, for several stimulating conversations and for his warm interest; to Dr. G.C. Laurence for suggesting the problem; to the B.C. Telephone Company for the bursary which made this research possible; and to the Faculty of the University of British Columbia for bringing me to the place where I may have the profound satisfaction of carrying out this and similar researches.


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