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 D i f f e r e n t i a l Cross Sedtion f o r the Scattering o f Polarized Neutron Beams by Protons ABSTRACT Tensor forces between two p a r t i c l e s involve a dependence upon the angle between the d i r e c t i o n o f spin quantization and the l i n e j o i n i n g the two p a r t i c l e s . The e f f e c t o f tensor forces upon the scattering of a polarized neutron beam has been investigated t h e o r e t i c a l l y . An expression has been obtained f o r the d i f f e r e n t i a l s c a t t e r ing cross section o f the t r i p l e t states as a function o f the p o l a r i z a t i o n o f both neutrons and protons. In general, t h i s cross section i s also a function o f the azimuthal angle t o the d i r e c t i o n o f propagation of the neutron beam. The Influence of Tensor Forces on the D i f f e r e n t i a l Cross Section f o r the Scattering of Polarized Neutron Beams by Protons Introduction In order to explain the e l e c t r i c quadrupole moment of the deuteron, i t has been necessary to introduce a tensor i n t e r a c t i o n potential of the form Rarita and Schvringer SV(r), where have calculated the e f f e c t s of t h i s p o t e n t i a l upon both the bound and the unbound states of the neutron-proton system. In p a r t i c u l a r , they have calculated the scattering cross section of a beam of neutrons by a proton target f o r neutrons of low energy. calculations were extended by Ashkin and Wu 3, phase s h i f t analysis. who These used a more general Recently, Rohrlich and Eisenstein ^ have solved the same problem (and obtained i d e n t i c a l results'.) by means of a method which the authors f i n d to be more s a t i s f y i n g t h e o r e t i c a l l y . In a l l of the above papers, the protons and neutrons were 1. 2. 3. 4. R a r i t a and Schwinger - Physical Review 59, 436, Rarita and Schwinger - Physical Review 59, 556, Ashkin and Wu - Physical Review 73, 973, 1948. Rohrlich andEisenstein - Physical Review 75, 705, 1941. 1941. 1949. 2 considered to be completely unpolarized; i . e . , the spins of the part i c l e s were assumed to have no p r e f e r e n t i a l d i r e c t i o n of alignment. The results of such a c a l c u l a t i o n showed the scattering cross section to be dependent only upon the polar angle to the d i r e c t i o n of propagat i o n of the neutron beam. by Dr. G.C. However, i n order to answer a question raised Laurence of Chalk River, i t was decided to investigate whether a dependence upon azimuthal angle i s introduced by c e r t a i n p o l a r i z a t i o n states o f the neutron-proton system, and to determine how the polar dependence i s modified by such states. An expression has been obtained f o r the d i f f e r e n t i a l scatt e r i n g cross s e c t i o n as a function of the azimuthal andppoiar angles and of parameters which are determined by the p o l a r i z a t i o n of the neutrons and protons. Calculation of the D i f f e r e n t i a l Cross Section . Because the tensor i n t e r a c t i o n operator y i e l d s zero when applied to any s i n g l e t spin function, only the contribtuion of the t r i p l e t scattering to the t o t a l 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 (1) where r is the vector from proton to neutron, Hk is the momentum e=£ - See Reference 3. This paper i s the s t a r t i n g point f o r 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 f o r convenience with respect t o k, and ( 7*^- o -f) are constants which depend upon the p o l a r i z a t i o n states o f neutron and proton. As a r e s u l t o f the tensor i n t e r a c t i o n , the asymptotic form of the scattered wave i s The matrix elements ^^4^ depend upon both azimuthal angles respectively; & and «p , the polar and the dependence upon ^ i s a direct result o f the asymmetry o f the tensor force. To obtain the t r i p l e t scattering cross section per unit s o l i d angle, one calculates the square modulus of the c o e f f i c i e n t of i n (2), which y i e l d s (using the orthonomality o f the t r i p l e t spin functions): An examination o f the s p e c i f i c form of the matrix elements ^% (displayed as Equations 10 and 11 i n t h i s 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 v i r t u e - See Reference 3, Page 931. The symbol "^**s*s i n t h i s paper i s i d e n t i c a l i n meaning t o the corresponding symbol i n Ashkin and wu. These elements have been written on the assumption that the z-axis i s the d i r e c t i o n of propagation of the incident beam o f neutrons. 4. of the cross-product terms. I f both neutrons and protons are completely unpolarized, the cross-products disappear i n averaging over the phases of the amplitudes ages to zero i f : a simple calculationr-shows that i- "*s , and to i f -*"s s **-s . aver- Therefore. As w i l l be seen, i f there i s a s p e c i f i c p o l a r i z a t i o n , these crossproducts do not disappear except i n s p e c i a l , symmetric cases. I t i s necessary to calculate the c o e f f i c i e n t s f o r an a r b i t r a r y but s p e c i f i c p o l a r i z a t i o n of the neutrons and protons. In the following, i t w i l l be assumed that the polar axis i s both the d i r e c t i o n of propagation and the d i r e c t i o n of reference of the spin functions. The following physical s i t u a t i o n i s considered: direction Qi , Ql i n the ( i = n,p and refers thus to neutron or proton), the r a t i o of the number of p a r t i c l e s i n a state with spin angular momentum ^ i n the positive sense t o the number having a spin of •£ negative sense i s i n the ^y^-j-e . To obtain the d i f f e r e n t i a l cross section appropriate to t h i s physical s i t u a t i o n , we may f i r s t compute the cross sections f o r four s p e c i f i c spin orientations of the neutron-proton system (protons and neutrons p a r a l l e l and a n t i - p a r a l l e l t o the directions @i (Q(} t t and then compound the r e s u l t s 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 i n t r o 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 In the above expression, the *frjy f{) t A ' are the random s t a t i s t i c a l phase factors t which must be averaged out i n a r r i v i n g at a f i n a l r e s u l t . (5) anrl tho—phooco—&"—appoap-ao a result of-tho physical oonditiona ira•pmsgd. aCi , bi represents the states of positive and negative spin i n the usual way. functions These states may be expressed i n terms of the spin referred to the z-axis by the following «ii f transformations ( 6 ) I Thus the total spin function (including both singlet and triplet states) xs: fsf where: * (A~.*~+ 6+ ^ ) - ^ //TT ^ fa/ ( f~ *> A r iXl i—77 f ) r «. - 0t Now, we have written i n (1) only the triplet part of V ^ o which i s with X , X , and A " so defined that Averaging the products of the amplitudes dj^ A^Tover the phases, one obtains: < <<*)«--•? ) ( ^ t •o-j-K'f) v , 6. + ^&tefr*-0(^ic^-]pWf) The above expressions represent the coefficients in (3); i t is now necessary to calculate the terms ^>»«sW* o r d make the Oj*-dependence of these terms explicit, we reproduce the Ashkin and Wu matrix as follows: S„, - S_„., = A -£ Si,, - *r*c c e r to 7. c-± Ztitojfim C £ ,S [ 2 ( 2 c H f i J & f i (?£l) ' t I , fr ,Xc "''°) zS 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 ° replaced by ^ .) We now define 7 ^ ^ to be u n - <? /or +• /*/* n., * a i0 ^ where ^ and § are defined by: and £ Wl - A ' then / - <? *** fe ? 7'- ^»»r**$ 1 [40* iSC* <* If expression (3) is now expanded as indicated, the a. result i s : Conclusions angle which neutrons, 1. both 2. for £p <9 the neutrons which are * = triplet ^and separately &pbecomes It For =~and ^may other scattering O protons or ,bestronger that sufficient ?)concluded cases iare scross that ,there as for aligned the after for i section scomplete is , polarization azimuthal for aan parallel dependence any that examination non-polarization. polarization there symmetry to becomes upon theare ofdirection the tCtwo the more state azimuthal conditions expression complete. in ofwhich the 9. It i s of considerable experimental interest t o note that a partiallypolarized beam o f neutrons impinging upon an unpolarized proton target would, as a r e s u l t o f the tensor i n t e r a c t i o n , be expected t o show an azimutlially asymmetric cross section. The magnitude o f t h i s asymmetry and the conditions under which the asymmetry w i l l be a maximum are t o be calculated s h o r t l y . 1- Acknowledgements I am deeply indebted: t o Professor Volkoff, f o r his generous - encouragement, assistance, and i n s p i r a t i o n , not only i n t h i s problem but i n many others also; t o Professor Opechowski, f o r several stimulating conversations and f o r h i s warm i n t e r e s t ; Laurence f o r suggesting the problem; t o Dr. G.C. to the B.C. Telephone Company f o r the bursary which made t h i s research possible; and t o the Faculty of the University o f B r i t i s h Columbia f o r bringing me t o the place where I may have the profound s a t i s f a c t i o n of carrying out t h i s and s i m i l a r researches.
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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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Title | The influence of tensor forces on the differential cross section for the scattering of polarized neutron beams by protons. |
Creator |
Lambe, Edward Bryant Dixon |
Publisher | University of British Columbia |
Date Issued | 1949 |
Description | 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 scattering cross section of the triplet states as a function of the polarization of both neutrons and protons. In general, this cross section is also a function of the azimuthal angle to the direction of propagation of the neutron beam. |
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Thesis/Dissertation |
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Text |
Language | eng |
Date Available | 2012-03-14 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0074514 |
URI | http://hdl.handle.net/2429/41398 |
Degree |
Master of Applied Science - MASc |
Program |
Engineering Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
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