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 i J THE UNIVERSITY OF BRITISH COLUMBI AUGUST, 1949 ABSTRACT In this 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 ina l expression for the resonance scattering cross section may be, i t must be the same in 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 appreciation to Professor W. Opeohowski of the Department of Physios, 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 oarried on. Thanks are also due to the National Research Council of Canada for a Bursary, during the tenure of which, t h i s work has been done. TABLE OF CONTENTS Introduction . . . . . . . . • I I - Preliminary Considerations . . . . . . 1 1. Siegert 's Calculation . . . . . . . . . . . . . 1 8. Some General Properties of the Scattering Matrix 5 3. Hu's Calculation 8. II *- Proof of the Equivalence of Siegert 's and Ning Hu's Derivations • 10 I I I - A Further Discussion of the Relation between Siegert 's Formulae and those of Ning Hu 13 I IHTRODUCTIOH There are several derivations of the Breit-Wigner dispersion formula for nuclear resonance scattering. These derivations can be divided into two groups: those using time-dependent wave functions; and those using time-independent wave functions. Among the derivations 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 Si e g e r t * 1 * , B r e i t * 2 * , Wigner^ 3*, Feshbaoh and c o l l a b o r a t o r s f 4 ) ang King H u ^ . Although the interdependence of the above derivations has been p a r t l y cleared up by the authors themselves, there are certain points whioh seem to require further i n v e s t i g a t i o n . In t h i s thesis detailed calculations are given of a o r i t i o a l comparison of King Hu's and Siegert's work. A o r l t i c a l i nvestigation of King Hu's r e s u l t s seemed p a r t i c u l a r l y desirable beoause h i s resonanoe scattering (1) - Siegert - Physioal Review, 56, 760, 1939 (2) - Breit - Physical Review, 58, 506, 1940 58,1068, 1940 (3) - Wigner - Physioal Review, 70, 15, 1946 (4) - Feshbaoh, Peaslee, and Welsskopf - Physioal Review, 71, 145, 1947 (5) - King Hu - Physioal Review, 74, 131, 1948 II formula differed from the usual one, and beoause the author himself attributed this difference to the fact that h is "formulae rest on a much more so l id 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 calculat ion. 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 br ief "Erratum" was published by King Hu in Physical Review^6) i n which he concedes more or less statement fa) and retracts his above quoted sentence. However, he does not make any new statement about the re la t ion of his derivation to the previously published ones. In Section I of th is thesis we sha l l give a br ie 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 calculat ion. 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 relat ion of Hu's expression for the "scattering matrix" to Siegert 's formulae. * - By stating that Hu's derivation i s exactly equivalent to Siegert 's , we do not say that Hu's paper does not go beyond Siegert 's results in 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 section we confine our attention to those aspects of Siegert's and Hu's calculations which are of di r e c t i n t erest i n Sections I I and I I I , For convenience we have also added a short note on some general properties of the S c a t t e r i n g matrix". (1) Siegert's Calculation I f i s the solu t i o n of the r a d i a l part of the Sohroedinger Wave Equation f o r the case 1=0, where 1 i s the angular momentum of the inoident p a r t i o l e , then pe s r if must be a solution of the equation / 'OS - = 0 (1) where the primes on p e denote d i f f e r e n t i a t i o n with respeot to r , E i s the energy of the inoident p a r t i c l e and V = V(r) for r < a 2 V • 0 for r > a where a may be regarded as the "nuclear radius". The asymptotic solution of the wave equation may be written as Pe s I s i n kr / R e i l E r (2) k" where R and I are funotions of k and k 2 = 2mE/n2 (3) The scattering cross section i s then given by / PnPe s 0 (9) Integrating (9) from 0 to a and using the boundary conditions (7), the author obtains fl'(a) - ifcCL(a) s (Wn - E) ( ) How assuming the eigenvalue Wn i s not degenerate, then i n the l i m i t as E Wn, 0e — ^ pn P — • / 1 « W • . . d l ) = * » ( a , l **n 1 For the numerator of R/I we have i n the l i m i t as E —>-Wn P e(a) cos ka - P e(a) s i n ka /k P n(a) (oos kjja - i s i n k^a) n pn(&) e - l k n a ( 1 Z ) Thus i n the l i m i t , R/I becomes R/I e 1 n8/2m ft2(a) e ~ 2 1 k n a / f (B) , . . . (13) ^ 2k n where f(E) i s a regular funotion i n the surrounding of Wn and gives r i s e to tha "potential scattering". In order to express the scattering oross section i n more f a m i l i a r terms, Siegert derives, from the Sohroedinger Wave 4 Equation, (1), and i t s complex conjugate, the re la t ion 8 2m JUa> Y r*i ,* i2 J« I I L I 4 R ( 1 4 ) *n 79 A H where Wn = - ijfn (15) 2 with E n and >« both r e a l . How for suff ic ient ly small values of tf„we can multiply p n by a suitable oonstant, A, of modulus one, so as to make Apn real near r a a. Then i f we assume that^ihe major contribution to the integral in the denominator of (13) occurs for r very nearly equal to a, we may write / l 2 0 2dr = jfyafar (16) W*) « |fa(aj| e ^ W V8> ( 1 7 ) where S « i s a phase determined entirely 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 va 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 H2/2m | p n ( a f e 1 *" / f(E) (18) 2 *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 2 4"i» e / f(E) (19) 2 That the eigenvalues and eigenfunotions given by the boundary oondition characterize a long l ived oompound nucleus may be shown by the following arguement: I f the levels are narrow, and therefore the escape of a par t ic le 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 par t ic le altogether. It i s obvious that (7) i s equivalent to preventing the escape of the pa r t i c le , since i n the l i m i t as k k n , I 0, and hence we have no stream of incident pa r t i c les . Therefore, there can be no scattered beam, so that there actually are no par t ic les escaping from the nuoleus. (2) Some General Properties of the Scattering Matrix The "scattering matrix", o r ig ina l ly introduced by (rt \ Wheeler' has been used by several authors - Wheeler, Wigner, Bre 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 par t ic les . According to his 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 levels of the system from the "scattering matrix", one must proceed with the utmost caution, since for a long range potential we are led , i n some cases, to redundant energy values. The re la t ion 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 non-rela t ivis t io par t ic le i n a central f i e l d of force. The Sohroedinger Wave Equation i s , i n this oase, 1**^) / k V - X(A j 1 U / V W - 0 . . (20) r 2 where X i s the angular momentum of the pajrblole. I f we set p a r ^ , then the asymptotio solution of the wave equation i s given by p = s in (kr { \ t ( l L ) ) (21) where (a) 1 e (45) 2 2 * (-o. I 12 j 12 2 i (k n a k n - k J e Ipnl dr / i |pD(a) 1 e n r z 2 * n How making an approximation identioal to that made i n obtaining (19) from (18) - i.e. that 0n i s very nearly real 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/n2 . _ e i l l . . . . (46) (4 - k 2 ) ( k n j. k£) 14 and from (15) one sees that k#n* - k* = iY„2m/n2 (47) Thus (46) beoomes R/I = - i e 1 (kn - kn) (48) .2 ,2 fen - k Now 2t|R/il = lat(fcn- ift) i ( 4 9 ) I (k •» kg}) I and sinoe 2 * 0 % - k n) * ( k / k n H k - k n ) - (k / *£)(k - k„) (50) and | (k - k n ) ( k / k n ) | = |(k - k n ) ( k / kn)| (51) then 2k[R/l( = | (k / k n) (k - k D) - (k / kp) (k - k D) I I (k - k n) (k / kj) * = U * i ( B 8 ) I (k - kn)(k / kj) I Now i t follows from (32) that 2kJR/l| = Js(k) - l ( (53) We see, therefore, that Ning Hu's expression (27) for S(k) S(k) = (k - kn) (k / kD) (k / k j )(k - kn) i s indeed compatible with equation (52) which was here derived fro£ S i e g e r t 1 s theory without any a r b i t r a r y assumption. '1 BIBLIOGRAPHY (1) Bre i t , G. (2) Bre i t , G. (3) Peshbaoh, H . , Peaslee, D. C. , & Weisskopf, V. F, (4) Hu, Ning t5)tJost , R. (6) Siegert, A . J . P . -(7) Wheeler, J . A. (8) Wigner, E. P . The Interpretation of Resonances i n Nuclear Reactions Physioal Review, 58, 506, 1940 Scattering Matrix of Radioactive States Physical Review, 58, 1068, 1940 On the Scattering and Absorption of Par t ic les by Atomic Nuclei Physical Review, 71, 145, 1947 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 Uber die falsohen Nul ls te l len 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 On the Mathematical Description of Light Nuclei by the Method of Resonating Group Structure Physical Review, 52, 1107, 1937 Resonance Reactions and Anomalous Scattering Physical Review, 70, 16, 1946