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On the derivation of the nuclear resonance scattering formula. Lawson, Robert Davis 1949

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l-K % ft]  m  THE DERIVATION Og THE NUCLEAR RESONANCE SCATTERING FORMULA  by ROBERT DAVIS LAWSGH  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE IN THE DEPARTMENT OF PHYSICS  THE UNIVERSITY OF BRITISH COLUMBI AUGUST, 1949  iJ  ABSTRACT In t h i s theBis detailed calculations are given showing the equivalence of Siegert's derivation of the nuclear resonance scattering formula, and Hu's derivation of the same formula.  Although at f i r s t glance i t appears that Hu  has given a solution to the problem using an entirely different formalism, we have shown that no matter what the f i n a l expression for the resonance scattering cross section may be, i t must be the same i n the case of Siegert's calculation and that of Mng Hu, provided,of course, that no more or less arbitrary approximations are introduced into the calculations.  AQOOWLEDGEMBHT  I wish to express my sinoere a p p r e c i a t i o n t o P r o f e s s o r W. Opeohowski o f the Department o f P h y s i o s , who o r i g i n a l l y suggested the problem, and under whose guidance t h i s work has been o a r r i e d on.  Thanks are a l s o due t o the N a t i o n a l  Research C o u n c i l o f Canada f o r a Bursary, d u r i n g the tenure of which, t h i s work has been done.  TABLE OF CONTENTS  Introduction  . . . . . . . .  I - Preliminary Considerations 1. Siegert's Calculation  • . . . . . . . . . . . . . . . . . . .  I 1 1  8. Some General Properties of the Scattering Matrix 3. Hu's Calculation II *- Proof of the Equivalence of Siegert's and Ning Hu's Derivations • I I I - A Further Discussion of the Relation between Siegert's Formulae and those of Ning Hu  5 8. 10 13  I  IHTRODUCTIOH  There are several d e r i v a t i o n s o f the Breit-Wigner d i s p e r s i o n formula f o r nuclear resonance s c a t t e r i n g .  These  d e r i v a t i o n s can be d i v i d e d i n t o two groups: those using time-dependent wave functions; and those using timeindependent wave f u n c t i o n s .  Among the d e r i v a t i o n s belonging  to the l a t t e r group we mention, i n p a r t i c u l a r , those by S i e g e r t * * , B r e i t * * , Wigner^ *, Feshbaoh and c o l l a b o r a t o r s 1  2  3  f 4 )  ang King H u ^ . Although the interdependence o f the above d e r i v a t i o n s has been p a r t l y cleared up by the authors themselves, there are c e r t a i n points whioh seem t o require f u r t h e r i n v e s t i g a t i o n . I n t h i s t h e s i s d e t a i l e d c a l c u l a t i o n s are given of a o r i t i o a l comparison of King Hu's and S i e g e r t ' s work.  A  o r l t i c a l i n v e s t i g a t i o n o f King Hu's r e s u l t s seemed p a r t i c u l a r l y d e s i r a b l e beoause h i s resonanoe s c a t t e r i n g (1) - Siegert (2) - B r e i t  -  P h y s i o a l Review, 56, 760, 1939 P h y s i c a l Review, 58, 506, 1940 58,1068, 1940 (3) - Wigner - P h y s i o a l Review, 70, 15, 1946 (4) - Feshbaoh, Peaslee, and Welsskopf - P h y s i o a l Review, 71, 145, 1947 (5) - King Hu - P h y s i o a l Review, 74, 131, 1948  II formula differed from the usual one, and beoause the author himself attributed t h i s difference to the fact that h i s "formulae rest on a much more s o l i d basis than other theoretical derivations hitherto given". It turns out: fa) that the difference i s due to a t r i v i a l error i n the calculation. fb) that King Hu's derivation i s exactly equivalent*" to Siegert's derivation, which i s , at f i r s t view, not obvious. After our work had been oompleted a b r i e f "Erratum" was published by King Hu i n Physical Review^ ) i n which he 6  concedes more or less statement fa) and retracts h i s above quoted sentence.  However, he does not make any new statement  about the r e l a t i o n of his derivation to the previously published ones. In Section I of t h i s thesis we s h a l l give a b r i e f outline of the calculations presented by Siegert and Hu together with a short note on some general properties of the "scattering matrix" used by King Hu i n his c a l c u l a t i o n . The equivalence of Siegert's derivation and Hu's derivation i s proved i n Section I I .  In Section I I I we discuss more  e x p l i c i t l y the r e l a t i o n of Hu's expression for the "scattering matrix" to Siegert's formulae. *  - By stating that Hu's derivation i s exactly equivalent to S i e g e r t ' s , we do not say that Hu's paper does not go beyond Siegert's results i n other respects. This i s so even i f we ignore the fact that Hu's oaloulation i s made for arbitrary 1 (angular momentum), whereas Siegert oonfines his attention to the case 1 = 0 . (6) - King Hu - Physical Review - 75, 1449, 1949  1  ON THE DERIVATION OF THE NUCLEAR RESONANCE SCATTERING FORMULA  I  PRELIMINARY CONSIDERATIONS  In t h i s s e c t i o n we confine our a t t e n t i o n t o those aspects o f S i e g e r t ' s and Hu's c a l c u l a t i o n s which are o f d i r e c t i n t e r e s t i n Sections I I and I I I , F o r convenience we have a l s o added a short note on some general p r o p e r t i e s o f the S c a t t e r i n g matrix". (1) S i e g e r t ' s C a l c u l a t i o n I f <t> i s the s o l u t i o n o f the r a d i a l part o f the Sohroedinger Wave Equation f o r the case 1=0,  where 1 i s  the angular momentum o f the inoident p a r t i o l e , then p  e  must be a s o l u t i o n o f the  /  'OS -  where the primes on p  e  equation  =  0  E i s the energy o f the inoident p a r t i c l e for r <a  (1)  denote d i f f e r e n t i a t i o n w i t h respeot  to r ,  and V = V(r)  s r if  2  V • 0  for r > a  where a may be regarded as the "nuclear r a d i u s " . The asymptotic s o l u t i o n of the wave equation may be w r i t t e n as P  s  e  I sin kr /  R e  (2)  i l E r  k" where R and I are funotions o f k and k  =  2  2mE/n  (3)  2  The s c a t t e r i n g cross s e c t i o n i s then given by <T  =  4ir|R/l\  (4)  2  p  R/I may be expressed i n terms of the wave f u n c t i o n ,  et  evaluated at the nuolear r a d i u s , a, R/I  =  p() &  Q  0 0 8  k  PH -)  *  a  6  s  i  n  k  a  /* e  p (a) e  -  -ika ( ) 5  ikp (a) e  The author then l o o k s f o r s i n g u l a r i t i e s o f the cross s e c t i o n a r i s i n g from the v a n i s h i n g o f the denominator.  The  eigenvalues o f the wave equation and hence the energy values f o r which the denominator vanishes, are given by the s o l u t i o n s o f the equation n  2  A  / < n - V) w i t h the boundary c o n d i t i o n s p  n  w  0  n  r 0 at r s 0  pl - i k j 3 p  n  =  0  at r = a  = o  (6)  ) )  (7)  )  where P (r) i s the wave funotion corresponding t o the energy n  W  n  o f the compound nucleus c h a r a c t e r i z e d by k = 2mWN/E2 2  (8)  3  To o b t a i n R/I i n the neighbourhood o f a s i n g u l a r i t y , one m u l t i p l i e s equation f l ) by p  W, n  and equation (6) by p  n  e  and on s u b t r a c t i o n o b t a i n s §J (We  -  PSPn>  < n - >PnPe  /  E  w  0  s  (9)  I n t e g r a t i n g (9) from 0 t o a and u s i n g the boundary c o n d i t i o n s (7), the author obtains fl'(a) - ifcCL(a)  s (W - E)  (  n  How assuming the eigenvalue W  i s not degenerate, then  n  W, 0  i n the l i m i t as E  e  n  —^  p  n  P — • =  *»  / ( a ,  )  1  l  «  W **n  • . .dl) 1  For the numerator o f R/I we have i n the l i m i t as E —>-W  n  P (a) cos ka e  P (a)  s i n ka /k  e  (oos kjja  n  n  P (a)  p() n &  e  -  l k  n  - i s i n k^a)  a ( 1 Z )  Thus i n the l i m i t , R/I becomes R/I  e  1  n /2m ft (a) e ~ 8  ^  2  2 1 k  2k  n  a  / f ( B ) , . . . (13)  n  where f(E) i s a regular f u n o t i o n i n the surrounding of W  n  and gives r i s e t o t h a " p o t e n t i a l s c a t t e r i n g " . I n order to express the s c a t t e r i n g oross s e c t i o n i n more f a m i l i a r terms, S i e g e r t d e r i v e s , from the Sohroedinger Wave  4 Equation, (1), and i t s complex conjugate, the r e l a t i o n  JU >  2m  where W  n  8  =  AH  Y  r*i ,* i  J«  I I LI  *n 79  a  2  4  (  R  - ijfn 2  1 4 )  (15)  with E and >« both r e a l . n  How for s u f f i c i e n t l y small values of tf„we can multiply p  by a suitable oonstant, A, of modulus one, so as to make  n  Ap  n  r e a l near r a a. Then i f we assume that^ihe major  contribution to the i n t e g r a l i n the denominator of (13) occurs for r very nearly equal to a, we may write /l 0 dr 2  2  W*)  (16)  = jfyafar «  |fa(aj| e ^ W  V> 8  ( 1 7 )  where S « i s a phase determined e n t i r e l y from the properties of the oompound state. This assumption means that the nuclear eigenfunotion takes on values appreciably different from zero, only near the nuclear radius.  The v a l i d i t y of suoh an assumption i s ,  of oourse, questionable. Prom equations (13), (16), and (17) i t follows that R/I «"  1  H /2m 2  |p (af  e *" 1  n  2  / f(E) (18)  *n  and by virtue of equation (14) the second term i n the denominator i s very much less than the f i r s t end hence one obtains the well known one l e v e l formula  5  4"i»  /  e  2  f(E)  (19)  2 That the eigenvalues and eigenfunotions given by the boundary oondition characterize a long l i v e d oompound nucleus may be shown by the following arguement:  I f the  l e v e l s are narrow, and therefore the escape of a p a r t i c l e from the nucleus i s a rare event, then the state of the oompound nuoleus w i l l undergo very l i t t l e change i f we prevent the escape of the p a r t i c l e altogether.  It i s  obvious that (7) i s equivalent to preventing the escape of the p a r t i c l e , since i n the l i m i t as k  k , I n  and hence we have no stream of incident p a r t i c l e s .  0, Therefore,  there can be no scattered beam, so that there actually are no p a r t i c l e s escaping from the nuoleus. (2) Some General Properties of the Scattering Matrix The "scattering matrix", o r i g i n a l l y introduced by (rt \  Wheeler'  has been used by several authors - Wheeler,  Wigner, B r e i t , Heisenberg, and others.  The f i r s t three  named physioists have used the matrix only to solve c o l l i s i o n problems.  Heisenberg, however, has attempted,  through use of the "S-Matrix", to set up a future, divergence free, theory of elementary p a r t i c l e s .  According to h i s idea  (7) - Wheeler, Physical Review, 52, 1107, 1937  6 the "S" function should play a role i n the future theory analogous to the part played by the Hamiltonian i n the present quantum theory.  It has been shown by Heisenberg,  Kramers, M i l l e r , and others, that from the "scattering matrix" one may obtain a l l observable quantities.  However, i n  obtaining the energy l e v e l s of the system from the "scattering matrix", one must proceed with the utmost caution, since for a long range potential we are l e d , i n some cases, to redundant energy values. The r e l a t i o n of the "scattering matrix" (which i n the considered case reduces to a single element) to the asymptotic form of the wave funotion i s as follows:  Consider  a n o n - r e l a t i v i s t i o p a r t i c l e i n a central f i e l d of force. The Sohroedinger Wave Equation i s , i n t h i s oase, 1**^)  - X(A j 1 U /  / kV  r  V  W  0 . . (20)  2  where X i s the angular momentum of the pajrblole.  p  -  I f we set  a r ^ , then the asymptotio solution of the wave equation  i s given by p  =  sin  (kr  { \t(lL)  (21)  )  where <yO*Hs "tae phase shift due to the interaction p o t e n t i a l . Equation (21) may be rewritten, aside from a factor, as  p where  *  e  S^(k) = e  2 1  i  k  -  r  ^ ^ 1  S^(k) e  i k r  (22) (23)  i s the "scattering matrix". (8) Jost, Helvetica Physica Acta, 20, 256, 1947 - In t h i s paper Jost discusses the conditions under which redundant energy values may occur.  7  Since flrO^is a r e a l and odd f u n c t i o n o f k - t h i s i s r e a d i l y seen from equations (20) and (21) - we have the relations S(k)S(-k) = 1  (24)  S(k)S*(k) = 1  (25)  For the c a s e X r 0, the expression f o r the s c a t t e r i n g cross s e c t i o n o f the p a r t i c l e by the c e n t r a l f i e l d o f force f o l l o w s immediately from equation (22): <r  =  ^ls(k) - i l I 2ik \  2 (  2  6  )  According to the suggestion o f Kramers and Heisenberg, we may continue the wave equation, and henee the "S" f u n c t i o n , i n t o the complex "k-plane".  The s t a t i o n a r y s t a t e s o f the  system are then given by the negative imaginairy values of k which make S(k) s 0. I n t h i s case, 0, defined by (22) becomes P -  e-Unlr  _  S^(-Hk \) e ^ r n  .  and i f S (-i|k |) = 0, p c e r t a i n l y s a t i s f i e s the c o n d i t i o n n  necessary f o r i t t o represent a c l o s e d state (provided p does not vanish i d e n t i c a l l y ) . There are, o f oourse, many other general p r o p e r t i e s o f the " s c a t t e r i n g matrix" which could be quoted, however, since t h i s t h e s i s deals only w i t h a n o n - r e l a t i v i s t i o s c a t t e r i n g problem, f u r t h e r general c o n s i d e r a t i o n s w i l l not be necessary.  8  (S) - Hu's Calculation As Hu has pointed out, i f the S function has simple poles, i t follows that we may write for S(k)  (k - k ) ( k /  4 )  n  where k = k  n  i s a singularity of S(k) and i s related to the  nuclear energy l e v e l , W , by W n  n  a  n .2 2m* *  s E - i  2  n  n  2  and f(k) i s a regular funotion of k. Substituting (27) into (26) one obtains the usual one l e v e l formula ^  =  4 1 T  ( E - E) - i i«  •  n  ' / *Oc)( (28) kn / k£ 1  2 which i s the same as Siegert's formula, equation (19). That S ( k ) , i n a oertain approximation, has the above form may be shown as follows:  Prom equation (24) one sees that  i f k s -E i s a pole of the "S" function, then k = K i s a zero of S(k).  How i f we can show that (dS/dk)jj. -  K  f 0,  i t follows' at once that the s i n g u l a r i t y of the "S" funotion at k s - K i s a pole of the f i r s t  order.  Consider the asymptotic form of the wave function* P = b(e- S(k) e ) (29) i l c r  i k r  where b i s a funotion of k *  We have here s l i g h t l y generalized Hu's c a l c u l a t i o n , i n that he assumes b = 1 from the s t a r t .  9  Now i f k s K i s a zero of S ( k ) , then By substitution of (29) into the expression loTE ( rWi^) (  / i ^ j i ) )k r K  L  d 0/dkdr) 2  k  =  K  / i  / iK^(dp/dk)  k  =  K  one e a s i l y shows that -2iKb|(dS/dk) k  = ^(d ^/dkdr) 2  E  / i p |  k  =  K  - (d^g/dr) ( d p / d k ) ^  . .  (30)  Now from consideration of the Schroedinger Wave Equation i t follows that i b f (dS/dk) k  K  a fj^  T  ~ L. 0^ ) a  • •  <) 31  2K  where r = a i s the range of the p o t e n t i a l , V ( r ) . If  i s very nearly real at r = a and i n a  neighbourhood thereof, and i f the major contribution to the integral i n (31) ocours i n t h i s region, then (dS/dk) » k  cannot equal zero. of the f i r s t order.  E  Thus the pole of S(k) at k s -K must be Prom equations (24), (25), and the  conclusion drawn from the above, i t follows that indeed S(k) has the form given by (27). I t i s worth while to point out that the assumption under whioh Ning Hu's resonance scattering formula holds i s i d e n t i c a l with that under whioh Siegert's holds.  (Compare  above discussion with comments connected with equations (16) and (17) i n Siegert's derivation.)  10  I I - Proof of the Equivalence of Siegert s and Ning Hu'3 1  Derivations For the sake of s i m p l i c i t y we confine our attention to the case 1 « 0, where 1 i s the angular momentum of the inoident p a r t i c l e .  I f we denote byrf^the scattering cross  section given by Siegert and (T^ the scattering cross section given by Hu, we have, as stated i n the previous section rfT* = s  4H \R/I \  (32-a)  4<U S(k) - 1 \ \ 2ik \  z  (32_b)  In order to find the resonance maxima of the cross section, both authors look, as we have seen, for s i n g u l a r i t i e s of the right hand side of (32) as a funotion of complex k . Evidently the resonance part of the cross section w i l l be, i n both cases, exactly the same i f the moduli of the residues of R(k)/I(k) and of S ( k ) / 2 i k are equal, provided the s i n g u l a r i t i e s of these two expressions are poles of the f i r s t order* Now both authors assume that the behaviour of the incident p a r t i c l e "inside" the nucleus i s described by the equation  |^Pn  /  fW - V) J2) = 0 n  n  (33)  2m where 0 (r) n  i s the wave function corresponding to an energy  of the compound nucleus, characterized by k£ = 2m¥n/E 2  (34)  11  and they both assume that p (r)  -  n  b e n i k  for r s a  r  n  (35)  where & i s the "nuclear radius", and b i s a complex n  amplitude. I t i s thus almost obvious that the residues are Indeed equal.  The formal proof of t h i s equality i s as follows;  Prom the r e l a t i o n S(k)S(-k) = 1, one derives, on the assumption that the pole of S(k) at k = k^ i s of the f i r s t order, the r e l a t i o n Residue S ( k )  s -l/fas/dk)^  ^  k B  s  ^  • . (36)  That t h i s equation i s v a l i d i s easily seen from the following arguement:  If k = k  i s a pole of S(k), we may  D  expand S(k) i n a Laurent series about k s i  , v a l i d at least  n  i n the neighbourhood of the pole S(k)  =  a_i  (k -  a  f  j  a  i  f  ]  £  „  k  n  )  /  (  3  7  )  k ) n  Now i f k s k. i s a zero fo S(k), then we may expand S(k) i n a Taylor series about k = k , , v a l i d i n the neighbourhood of  K S(k)  =  a{(k ~ k . )  /  By virtue of (24), i f k * k certainly k = - k  n  a£(k - k . ) n  2  /  (38)  i s a singularity of S(k), then  i s a zero of S ( k ) .  Thus we may rewrite  (38) as S(k)  =  a{(k / k ) n  /  al(k / k ^ )  with k i n (39) equal to - k i n (37).  2  /  Thus from (24), (37),  (39) , and the remark just made, we have  (39)  IS  SOt)S(-k)  = -euiai  /(-aai  /  a_ aj)(k-k ) 1  n  /  (40)  low since t h i s holds i n a neighbourhood of k = S^, i t must also hold at the point 1% i t s e l f . a^a*  We have,  = -1  But a£ = (dS/dk) r-tjj »  (41)  s i n c e  a  k  S(k)  k  therefore,  - l  i  s  e  9  *° residue  u a l  . j , , equation (36) follows. Thus Residue S(k)/2ik  =  -  1  ]_  On the other hand, from equation (13) of our outline of Siegert s oalculation and equation (35) of t h i s section 1  i t follows, on the same assumption Residue R / I s -  bj  £j&r  . _1 /  i  flg(a)  2  . . . .  (43)  *  21% Wow since the Schroedinger  operator  i s even i n k,  i t i s obvious from equation (31) that ib*(dS/dk) „ ^ k  *  jZ^dr  / i ^ 0 2k  . . .  (44)  n  Thus by substituting (44) into (43) and comparing the resulting equation with (42), one sees that the two residues are indeed equal. We have thus shown that whatever the f i n a l expression for the resonance scattering oross section may be, i t must be the same i n the case of Siegert s calculation and that of 1  King Hu, provided, of course, that no more or less arbitrary approximations are introduced into the calculations.  13 Actually, as was stated i n Section I , both Siegert and King Hu do introduce certain approximations, but the approximations are i d e n t i c a l i n both oases, so that the f i n a l result i s the same, i f we correot the t r i v i a l error i n King Hu's calculation, which we have mentioned i n the introduction. Ill  A Further Discussion of the Relation between Siegert's Formulae and those of King Hu In view of the arguement presented i n Section I I i t i s  c e r t a i n l y evident that Siegert's derivation and Hu's derivation are fully equivalent.  However, we can make the  interdependence of these two derivations even more e x p l i o i t i n the following simple manner. We may rewrite the singular part of equation (18) of our summary of Siegert's paper as |2 i^*»  k  2 n  - k  2 * (-o. I  kr>(a)j1 e  12  J Ipnl dr  /  e  12 2 i ( k a  i |p (a) 1 e D  2  n  nr  (45) z  *n  How making an approximation i d e n t i o a l to that made i n obtaining (19) from (18) - i.e. that 0 i s very nearly r e a l n  i n the region of major contribution to the integral i n the denominator of (45) - we see that the second term i n the denominator i s very much less that the f i r s t .  Thus oombining  (45) with (14), one obtains R/I  r  2m/n  2  (4  -  ._ e i l l k ) j. k£) 2  (  k  n  . . . .  (46)  14 and from (15) one sees that k *  -  #  n  k*  =  iY„2m/n  (47)  (kn - kn)  (48)  2  Thus (46) beoomes R/I  =  - i e  1  ,2 fen - k .2  Now  2t|R/il and  =  lat(fc - ift) i I (k •» kg}) I n  ( 4 9 )  sinoe  2*0% - k ) *  (k/knHk-kn)  n  (k / *£)(k - k„) (50)  -  and | (k - k ) ( k / k ) | n  =  n  |(k - k ) ( k / k n ) |  (51)  n  then 2k[R/l(  =  | (k / k ) (k - k ) n  (k  I  = U I  (k / kp) (k - k ) I  -  D  D  - k ) (k / n  *  kj)  *  i  ( B 8 )  I  (k - k n ) ( k / kj)  Now i t f o l l o w s from (32) that 2kJR/l|  =  Js(k)  - l(  (53)  We see, t h e r e f o r e , that Ning Hu's expression S(k)  =  (27) f o r S(k)  (k - k ) (k / k ) (k / k j ) ( k - k ) n  D n  i s indeed compatible w i t h equation  (52) which was here derived  fro£ S i e g e r t s theory without any a r b i t r a r y assumption. 1  '1 BIBLIOGRAPHY (1) B r e i t , G.  The Interpretation of Resonances i n Nuclear Reactions Physioal Review, 58, 506, 1940  (2) B r e i t , G.  Scattering Matrix of Radioactive States Physical Review, 58, 1068, 1940  (3) Peshbaoh, H . , Peaslee, D. C . , & Weisskopf, V. F,  On the Scattering and Absorption of P a r t i c l e s by Atomic Nuclei Physical Review, 71, 145, 1947  (4) Hu, Ning  On the Application of Heisenberg's Theory of S Matrix to the Problems of Resonance Scattering and Reactions i n Nuclear Physios Physical Review, 74, 131, 1948  t 5 ) t J o s t , R.  Uber die falsohen N u l l s t e l l e n der Eigenwerte der S Matrix Helvetioa Physica Acta, 20, 256, 1947 On the Derivation of the Dispersion Formula for Nuclear Reactions Physical Review, 56, 750, 1939  (6) Siegert, A . J . P . (7) Wheeler, J . A .  On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure Physical Review, 52, 1107, 1937  (8) Wigner, E . P .  Resonance Reactions and Anomalous Scattering Physical Review, 70, 16, 1946  

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