Scattering Theory of Waves and Particles pp 514-587 | Cite as

# Inelastic Scattering and Reactions (Multichannel Theory), II

## Abstract

We now want to approach the manychannel problem from the point of view of the coupled radial Schrödinger equations (16.67) or (16.67a). Their structure resembling that of (15.92), the procedure of solving them is very much like that of Sec. 15.2.2. The main complication there arose from the coupling between different orbital angular momenta. Because it is generally possible to excite internal energy levels of different angular momenta (i.e., different fragment spins), that complication is present here also. In many practical cases, however, when the fragment spin is really an internal *oribital*angular momentum, it is somewhat less serious.

## Keywords

Riemann Surface Inelastic Scattering Faddeev Equation Fredholm Determinant Physical Sheet## Preview

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## Notes and References

- 17.1.1The argument leading to (17.5) was first given by R. G. Newton (1963a).Google Scholar
- 17.1.1aA coupled-Schrödinger-equation approach to reactions was employed first by G. Breit (1946).Google Scholar
- 17.1.1bThe treatment here follows mostly R. G. Newton (1958). See also T. Sasakawa (1964) and Sasakawa and Tsukamoto (1965).Google Scholar
- 17.1.2The final-state interaction theory is due to K. M. Watson (1952). See also sec. 9.3 of Goldberger and Watson (1964b). For a generalization to three-particle final states, see R. D. Amado (1967 and 1975), Adhikari and Amado (1974), and to four-body final states, Adhikari and Amado (1977). See also Aitchison and Kacser (1966); R.T.Cahill(1974).Google Scholar
- 17.1.3The existence of a function from which all S-matrix elements can be obtained as in (17.42) and (17.51) was first demonstrated in a more special case by K. J. LeCouteur (1960). That the function is the Fredholm determinant of the scattering integral equation was shown by R. G. Newton (1961a). Section 17.1.3 essentially follows this reference. See also Chan Hong-Mo (1961, 1963); R. Blankenbecler (1964); Sugar and Blanken-becler (1965); W. Glöckle (1966); Dahmen and Reiner (1967); R. G. Newton (1967c); Benayoun and Leruste (1969 and 1970); W. P. Reinhardt (1973).Google Scholar
- 17.1.5For mathematical treatments of uniformization, see, for example, G. Springer (1957), pp. 7–11. The specific uniformization for the two-channel problem given in (17.69) is due to J. R. Cox (1962). That in (17.69) was given by M. Kato (1965). The argument connected with Fig. 17.3, concerning the impossibility of mapping the three-channel Riemann surface onto the plane (without use of doubly periodic functions), is due to C. Goebel (private communication). See also H. A. Weidenmüller (1964). The fact that, as the forces change, the poles which are near the upper rim of the cut and thus cause the resonances change places as they pass a threshold was independently pointed out by M. Ross (1963); Eden and Taylor (1963); D. Amati (1963); Dalitz and Rajasekaran (1963); Nauenberg and Nearing (1964). See also R. L. Warnock (1964). For generalizations of the double dispersion relation to many-channel processes, see Fonda, Radicati, and Regge (1961); J. Underhill (1962 and 1963). Generalizations of the
*N/D*method to many-channel problems and discussions of them can be found in D. J. Bjorken (1960); Chew and Mandelstam (1960); M. Froissart (1961); Frye and Warnock (1963); E. J. Squires (1964); M. Luming (1964); H. Munczek (1964); Bander, Coulter, and Shaw (1965); Nath and Shaw (1965b); Bali, Munczek, and Pignotti (1965).;. Munczek and Pignotti (1965); K. Kikawa (1965); R. L. Warnok (1967). See also C. van Winter (1966) and F. Riahi (1969).Google Scholar - 17.2The arguments of Sec. 17.2.1 are due to R. G. Newton (1963b). Among papers concerned with the analytic behavior of the S matrix near a threshold we may mention R. J. Eden (1949 and 1952). The threshold anomaly as an observable phenomenon was first pointed out by E. P. Wigner (1948) and is therefore sometimes referred to as a
*Wigner cusp;*however, see also E. Fermi (1936). Applications can be found in V. Mamasakhlisov (1953); Capps and Holladay (1955), appendix B. General treatments of the anomaly are to be found in Guier and Hart (1957); G. Breit (1957 and 1959), p. 274; A. I. Baz (1957 and 1959); R. G. Newton (1958 and 1959); Baz and Okun (1958); L. M. Delves (1958–1959); L. Fonda (1959, 1961a and b); Jackson and Wyld (1959); Y. Yamaguchi (1959); Nauenberg and Pais (1961); W. E. Meyerhof (1962 and 1963);T. F. Tuan (1963); S. K. Adhikari (1973); Bardsley and Nesbet (1973); I. Aronson et al. (1975). The usefulness of the effect for parity information was emphasized by R. K. Adair (1958). The influence of the Coulomb potential was investigated by Fonda and Newton (1959) and applied to X-ray scattering in Newton and Fonda (1960); see also F. H. M. Faisal (1968); Shakeshaft and Spruch (1979). A similar threshold effect in three-body channels was treated in Fonda and Newton (1960b); I. G. Halliday (1966 and 1967a). See also Dragt and Karplus (1962); N. S. Kronfli (1963); R. Roskies (1966); Amado and Rubin (1970). If the particles in the threshold channel are unstable, then the cusp is of course “washed out.” This effect was emphasized in A. I. Baz (1961); Nauenberg and Pais (1962); Fonda and Ghirardi (1964a). In the graphic terminology of the second reference it is called a “woolly” cusp. For references concerning Eq. (17.78) see the Notes and References to Sec. 12.2 referring to Eq. (12.165). For threshold influences on resonances, see J. B. Ehrman (1951); R. G. Thomas (1952); A. I. Baz (1959); D. R. Inglis (1962); Barker and Traecy (1962); F. C. Barker (1964); H. A. Weidenmüller (1965).Google Scholar - 17.3The content of Sec. 17.3.1 follows mostly Fonda and Newton (1960c); R. G. Newton (1961); that of Sec. 17.3.2 the last reference. See also Fonda, Radicati, and Regge(1961). The Wigner-Weisskopf model was introduced by Wigner and Weisskopf (1930) and studied also by M. Moshinsky (1951); M. Wellner (1960). The
*Bargmann potentials*have been generalized to the coupled-channel case by J. R. Cox (1964); Cox and Garcia (1975); E. B. Plekhanov et al. (1981). The*effective-range theory*has been generalized to multichannel problems in the following articles: L. M. Delves (1958); Ross and Shaw (1960 and 1961); Shaw and Ross (1962); Nath and Shaw (1965a). Other solvable models are discussed by J. B. McGuire (1964); Brezin and Zinn-Justin (1966); C. N. Yang (1968); F. Calogero (1969 and 1971a); J. R. Dodd (1970); C. Marchioro (1970); Kamal and Kreutzer (1970); C. K. Majumdar (1972); McGuire and Hurst (1972); M. G. Fuda (1973); Calogero and Marchioro (1974); Gerber and Rosenbach (1974); E. A. Rohlfing et al. (1980). See also R. F. Dashen et al. (1976).Google Scholar - 17.4.2The Faddeev equations are due to L. D. Faddeev (1960, 1961, 1962 and 1963). The slight modification of the Faddeev equations given in this section corresponds to that of C. Lovelace (1964a and b). The procedure is related to the older method of K. M. Watson (1953 and 1957). The modification necessary in the presence of three-body forces was first given by R. G. Newton (1966).Google Scholar
- 17.4.3The Weinberg-van Winter equations are due to S. Weinberg (1964a), C. van Winter (1964, 1965, and 1970). The Rosenberg method was invented by L. Rosenberg (1964b); see also R. G. Newton (1967a). The coupled-channel array (CCA) method originated with Baer and Kouri (1973); Kouri and Levin (1974d); W. Tobocman (1974).Google Scholar
- 17.4.4The use of Fredholm methods in the three-particle problem began with M. Rubin et al. (1966 and 1967). The discussion in this section is based on R. G. Newton (1971 and 1974d). The issue of
*spurious solutions*was first raised by P. Federbush (1966). For its further discussion see J. V. Noble (1966); R. G. Newton (1967d); Y. Hahn et al. (1974b); Adhikari and Glöckle (1978, 1979, 1980b): C. Chandler (1978); K. L. Kowalski (1978b); V. Vanzani (1978b and c); S. K. Adhikari (1979c and 1982); Chandler and Sloan (1980). Analyticity properties are discussed by M. M. Islam (1965); I. Sh. Vashakidze et al. (1965); M. Rubin et al. (1966 and 1967); D. D. Brayshaw (1968b); Hartle and Sugar (1968); R. D. Amado et al. (1971); Y. Avishai (1971b).Google Scholar - 17.4.5The method and results of this section are based on R. G. Newton (1972a) and S. P. Merkuriev (1971). See also F. A. Berezin (1965); J. Nuttall (1967a and 1971); Nuttall and Webb (1969); Doolen and Nuttall (1971); S. P. Merkuriev (1973); Osborn and Bollé (1973); C. Gignoux et al. (1974); W. Glöckle (1974); V. Vanzani (1978); Cattapan and Vanzani (1979a); S. Servadio (1981). The recognition of the “double scattering difficulties” is originally due to E. Gerjuoy (1970 and 1971) and J. Nuttall (1971). That the resulting three-particle cross section is nevertheless finite was shown by Newton and Shtokhamer (1976) and Potapov and Taylor (1977). For an application, see R. G. Newton (1976b).Google Scholar
- 17.4.6The introduction of the variables
*ψ, θ*,*φ*,*E*_{1}*,E*_{2}*, E*_{3}into the Faddeev equations is due to Omnes, and so is the subsequent angular-momentum analysis. The present procedure essentially follows Omnes in this, except for the correction of some minor errors. The following papers deal with angle-functions and angular-momentum analyses of 3-particle or*n*-particle systems: E. Fabri (1954); R. H. Dalitz (1954); V. I. Ritus (1961); A. J. Macfarlane (1962); W. Zickendraht (1964 and 1965); Badalyan and Simonov (1966); Lévy-Leblond and Lévy-Nahas (1965); Pustovalov and Simonov (1966and 1968); Y. A. Simonov (1966); E. L. Surkov (1967); Raynal and Revai (1970); M. Fabre de la Ripelle (1971); Shukre and Winternitz (1972b); A. Lindner (1973); Bollé and Osborn (1974); A. Kupperman (1975); K. K. Fang (1977 and 1979); D. Eppel (1978); B. R. Johnson (1980); J. L. Friar et al. (1981). For relativistic generalizations see M. I. Shirokov (1961); G. C. Wick (1962); J. A. Lock (1975).Google Scholar - 17.4.7The results of this section are mostly due to R. G. Newton (1974c and 1975). See also J. A. Wright (1965), D. B. Pearson (1966); E. A. Remler (1966), D. R. Yafaev (1978).Google Scholar
- 17.4.8The discovery of the Efimov effect in three-particle systems is due to V. Efimov (1970, 1973, 1976). A more rigorous proof of its existence was given by Amado and Noble (1971 and 1972). Amada and Greenwood (1973) showed that there is no analogous effect in
*n*-particle systems*(n >*3) at “zero-energy bound states” of its*m*-fragment subsystems for*m >*2. There is, nevertheless, such an effect in an*n*-particle system if among one of its three-fragment states at least two have simultaneous zero-energy half-bound states; I. M. Sigal (unpublished); Kröger and Perne (1980). A rigorous proof of the Efimov effect was given by Ovchinnikov and Sigal (1979). See also B. Simon (1970); D. R. Yafaev (1974); A. C. Fonseca et al. (1979); Fonseca and Shanley (1979); Adhikari and Fonseca (1981). Numerical calculations were done by Stelbovics and Dodd (1972); A. G. Antunes et al. (1976).Google Scholar - 17.4General. Further references on three-particle scattering are: Feshbach and Rarita(1949); R. Clapp (1949 and 1961); M. Verde (1949); Troesch and Verde (1951); Massey and Moiseiwitch (1951); Borowitz and Friedman (1953); Boyet and Borowitz (1954); Skorniakov and Ter-Martirosian (1956); G. V. Skorniakov (1956); Derrick and Blatt (1958 and 1960); L. Eyges (1959); G. H. Derrick (1960 and 1962); Fonda and Newton (1960b); Cohen, Judd, and Riddell (1960); L. M. Delves (1960); G. S. Danilov (1961); Gallina, Nata, Bianchi, and Viano (1962); Barsella and Fabri (1962); F. T. Smith (1962); R. D. Amado (1963); Mitra and Bhasin (1963); J. R. Higgins (1963); A. I. Baz (1964); A. Chakrabarti (1964); Aaron, Amado, and Yam (1964 and 1965); Bhatia and Temkin (1964); M. McMillan (1964); M. H. Choudhury (1964); C. Zemach (1964b); R. L. Omnes (1964); Omnes and Alessandrini (1964); Bianchi and Favella (1964a); W. Hunziker (1964); Ghirardi and Rimini (1965b); J. T. Cushing (1965); Amadzadeh and Tjon (1965); L. Basdevant (1965); N. Mishima (1965); Mishima and Yamasaki (1965); Block and Gillet (1965); M. Bander (1965); Levy-Leblond and Levy-Nahas (1965); Tadic and Tuan (1965); G. Immirzi (1965); S. Mandelstam (1965); Berman and Jacob (1965); Alessandrini and Omnes (1965); C. Zupancic (1965); Hetherington and Schick (1965); A. Tucciarone (1966); Aaron and Shanley (1966); I. G. Halliday (1966); R. Roskies (1966); E. O. Alt et al. (1967); MacDonald and Mekjian (1967); Malfliet and Ruijgrok (1967); N. Mishina (1967); Tobocman and Nagarajan (1967); van Nieuwenhuizen and Ruijgrok (1967); W. Zickendraht (1967); Ball and Wong (1968); G. Doolen (1968); J. R. Jasperse (1968); D. D. Brayshaw (1969); M. G. Fuda (1969); Grassberger and Sandhas (1969); V. Vanzani (1969, 1973, 1974); R. J. Yaes (1969); W. W. Zachary (1969); Kazaks and Greider (1970); McKee and Rolph (1970); J. C. Y. Chen et al. (1971); Chen and Hambro (1971); Chen and Joachain (1971); W. Glöckle (1971); Osborn and Kowalski (1971); S. C. Pieper et al. (1971); T. Sasakawa (1971, 1972, 1973, 1977a); E. O. Alt (1972); Baer and Kouri (1972); Bahethi and Fuda (1972); K. L. Kowalski (1972a and 1973); W. Sandhas (1972); K. Takeuchi (1972 and 1973); Kloet and Tjon (1973); R. D. Amado (1974); M. Fuda (1974); Hirooka and Sunakawa (1974); Klink and Johnson (1974); Kouri and Levin (1974a and 1975c); Larson and Hetherington (1974); Schmid and Ziegelman (1974); Adhikari and Slaon (1975); Karlsson and Zeiger (1975); Kim and Tubis (1975); D. Bolle (1976); H. Hannover (1976); D. J. Kouri et al. (1976); S. P. Merkuriev et al. (1976); J. Ginibre (1977); V. B. Mandelzweig (1977); Osborn and Wilk (1977); Sasakawa and Sawada (1977b); O. A. Yakubovskii (1977); V. I. Kukulin et al. (1978); Kukulin and Pomerantsev (1978a, b, c); V. E. Kuzmichev (1978); T. Preist (1978); A. G. Sitenko (1978); K. Yajima (1978); Adhikari and Glöckle (1980a); N. Austern (1980); Dodd and Nieukerke (1980); V. V. Komarov et al. (1980); M. Kawai et al. (1981); G. Payne (1981); I. H. Sloan (1981); Kröger and Fenske (1981); Fonseca and Shanley (1981); Faddeev and Yakubovskii (1981). B. C. Eu (1971a) treated the classical three-body problem in analogy with Faddeev. The following papers deal with electrically charged three or
*n*-particle systems: J. V. Noble (1967); L. Schulman (1967); van Winter and Brascamp (1968); Hamza and Edwards (1969); A. M. Brodskii et al. (1970); A. M. Veselova (1970b, 1972, 1978); Gy. Bencze (1972); L. Rosenberg (1973); Gibson and Chandler (1974); E. O. Alt (1976 and 1978); E. O. Alt et al. (1976 and 1978); S. P. Merkuriev (1976, 1977, 1978a, b, 1979a, b, 1980); Alt and Sandhas (1978 and 1980); Z. Bajzer (1978); Chandler and Gibson (1978c); Kharchenko and Shadchin (1978); P. V. Sauer (1978); W. Thirring (1978); Bencze and Zankel (1979); Kuzmichev and Kharchenko (1979); V. Enss (1979); Sasakawa and Sawada (1979); M. L. Zepalova (1979); J. Zorbas (1979); C. Chandler (1980); L. P. Kok (1980); L. P. Kok et al. (1980); Kok and van Haeringen (1980); Evans and Hoffman (1981). For bounds on the number of bound states see D. R. Yafaev (1972 and 1976); G. M. Zhislin (1972); see also Buslaev and Merkuriev (1969); Bollé and Chadan (1982). Relativistic treatments of three-particle systems are given by J. Nuttall (1967b); Bhasin and Mitra (1970); P. du T. van der Merwe (1972). For approximation methods, see R. Aaron et al. (1961); Mitra and Bhasin (1963); Sitenko and Karchenko (1963); Dodd and Greider (1966); Greider and Dodd (1966); Wong and Zambotti (1967); M. G. Fuda (1968); I. H. Sloan (1968); K. L. Kowalski (1969); Amado and Rubin (1970); Brodskii and Potapov (1970); J. A. Tjon (1970); Becker and Harrington (1971); R. L. Kelly (1971); I. Manning (1972); J. M. Namyslowski (1972); Sinfailam and Chen (1972); H. Ziegelmann (1972); S. K. Adhikari (1973); Bakker and Sandhas (1975); T. Sasakawa (1977b); Cattapan and Vanzani (1978); J. A. Lock (1978); V. S. Potapov (1980); Tomio and Adhikari (1980 and 1981); D. Eyre et al. (1981); G. A. Hagedorn (1981). Treatments of four-particle equation are to be found in the following papers: A. N. Mitra et al. (1965); Takahashi and Mishima (1965); V. A. Allessandrini (1966); Mishima and Takahashi (1966); Popova and Komarov (1967); Yakubovskii (1967); E. O. Alt et al. (1970); I. Sloan (1972); D. Bessis (1974); W. Sandhas (1975); Fonseca and Shanley (1976); T. Sasakawa (1976 and 1978); J. A. Tjon (1976 and 1980); Adhikari and Amado (1977); Kharchenko and Levachev (1977); Kholyavin and Yakubovskii (1977); S. K. Adhikari (1979a); Haberzettl and Sandhas (1981). For reviews of «-particle scattering theory see L. D. Faddeev (1970); V. Vanzani (1976, 1978a, b); K. L. Kowalski (1978a); M. J. Moravcsik (1978); E. F. Redish (1978); W. Sandhas (1978); Gy. Bencze (1980); F. S. Levin (1981b). See also the conference proceedings B. Pisent et al. (1978); H. Zingl et al. (1978); and F. S. Levin (1981a). The following papers treat general*n*-particle scattering equations in detail: W. Hunziker (1965, 1968, 1977); J. Weyers (1966); Grassberger and Sandhas (1967); T. Sasakawa (1967, 1977, 1978); R.Omnes(1968);W. Bierter (1969); J.-M. Combes(1969and 1974); K. Hepp (1969); Komarov and Popova (1969); W. Glöckle (1970); Y. Avishai (1971a); Narodetskii and Yakubovskii (1971); L. Rosenberg (1971); V. Vanzani (1971, 1976, 1978b); Y. Hahn (1972d, 1976a, 1977); Hahn and Watson (1972); Gy. Bencze (1973, 1976, 1977); Y. Hahn et al. (1974a); Karlson and Zeiger (1974); Kouri and Levin (1974d, 1975b, 1977); E. F. Redish (1974); W. Sandhas (1974, 1976); W. Tobocman (1974b, 1975b); P. Benoist-Gueutal (1975); Cotanch and Vincent (1976); E. F. Redish et al. (1976); J. Schwager (1976); Gy. Bencze et al. (1977); Bencze and Tandy (1977); Cattapan and Vanzani (1977, 1979b, 1981); M. L’Huillier et al. (1977); D. J. Kouri et al. (1977); K. L. Kowalski (1977); P. Benoist-Gueutal et al. (1978); K. L. Kowalski et al. (1978); Polyzou and Redish (1978 and 1979); I. M. Sigal (1978); Greben and Levin (1979); Hahn and Retter (1979); Lovitch and Vanzani (1979); Ludeking and Vary (1979); Goldflam and Kowalski (1980a); B. E. Grinyuk (1980); F. S. Levin (1980 and 1981c); W.N. Polyzou (1980 and 1981); G. Cattapan et al. (1981); K. L. Kowalski et al. (1981); Levin and Li (1981); S. F. J. Wilk et al. (1981); see also B. R. Karlsson (1982). A relativistic version of*n*-particle scattering theory is discussed by L. Rosenberg (1966). N-particle scattering with*dilation analytic*potentials is treated by Balslev and Combes (1971); I. M. Sigal (1978a); C. van Winter (1978).*Optical-model*potentials are derived from*n*-particle scattering theory by K. L. Kowalski (1979) and by Goldflam and Kowalski (1980a).*Numerical tests*of*n*-particle scattering formalisms were performed by Lewanski and Tobocman (1978) and by W. Tobocman (1981a, b).Google Scholar