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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 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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