On Mathematical Foundations of Quantum Collisions and Nuclear Reactions and Outcome of Certain Physical Phenomena from Them

V.S. Olkhovsky^1,

¹Institute for Nuclear Research of NASU, prospekt Nauki, Kiev, Ukraine

Cite this paper:
V.S. Olkhovsky. On Mathematical Foundations of Quantum Collisions and Nuclear Reactions and Outcome of Certain Physical Phenomena from Them. International Journal of Physics. 2014; 2(2):41-75. doi: 10.12691/ijp-2-2-4

Abstract

I. Analytical structure of the non-relativistic unitary and non-unitary S-matrix is reviewed for the cases of any interactions with any motion equations inside a sphere of radius a, enclosed by centrifugal and rapidly decreasing (exponentially or by the Yukawian law or by the more rapidly decreasing) potentials. Some kinds of the symmetry conditions are imposed. The Schroedinger equation for the particle motion in the external region where r > a and the completeness of the correspondent wave functions are assumed. The connection of the obtained results with the causality is examined. Partially some analytical properties for the multi-channel S-matrix are reviewed and the sum rules for mean compound-nucleus time delays and the density of compound-nucleus levels. Sometimes (as physical manifestations of the profound general methodic and in very good consistent accordance with the experiment) observable physical effects, such as parity violation enhancement and time resonances or explosions, are appeared. Finally a scientific program of future search is presented as a clear continuation and extension of the obtained results. II. It is already known the appearance of time advance (due to distortion by the non-resonant background) instead of the expected time delay in the region of a compound-nucleus resonance in the center-of-mass (C-) system. Here at the same conditions we study cross sections and durations of the neutron-nucleus scattering in the laboratory (L-) system. Here it is shown that such time advance is a virtual paradox but in the L-system the time-advance phenomenon does not occur and only the trivial time delay is observed. At the same time the transformations from C-system into the L-system appeared to be different from the standard kinematical transformations because in the C-system the motion of a compound nucleus is absent but it is present in the L-system. We analyze the initial wave-packet motion (after the collision origin) and the cross section in the laboratory (L-) system. Also here (as physical revelations of profound general methodic and in very good consistent accordance with the experiment) several results of the calculated cross sections for the neutron-nucleus in comparison with the experimental data in the L-system at the range of one or two overlapped compound resonances are presented. It is shown in the space-time approach that the standard kinematical transformations of cross sections from the C-system to the L-system are not valid because it is necessary to consider the center-of-mass motion in the L-system. Finally on a correct self-consistent base of the space-time description of the nuclear processes in the laboratory system with 3 particles in the final channel, it is shown the validity of the former approach, obtained for the space-time description of the nuclear processes with 2-particle channels earlier.

Keywords:
(I) S-matrix condition of completeness of external wave functions external centrifugal and rapidly decreasing potentials causality time resonances or explosions parity violation enhancement; (II) space-time approach to nuclear collision time delay time advance transformations of cross sections from the C-system to the L-system interference phenomena

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	Bogolyubov N.N., private communication: mathematics now become partly the range of the theoretical physics (for instance, the quantum collision theory).
[2]	Landau L.D., private communication: a new good method in physics is better than any effect because the good method can bring us to some or even many new effects which can in a new way explain the experimental data.
[3]	V.S.Olkhovsky, Time as a quantum observable, canonically conjugate to energy. Time analysis of quantum processes of tunneling and collisions (nuclear reactions) [LAP LAMBERT Academic publishing, 177 pages, 2012].
[4]	Heisenberg,W., Die “beobachtbaren Groessen” in der Theorie der Elementarteilchen, Zs.f. Phys. A Hadrons and Nuclei, 120(1943)513, 673-702.
[5]	Heisenberg,W., Der mathamatische Rahman der Quantentheorie der Wellnfelder, Zs.Naturforsch. Teil A, 1(1946) 608.
[6]	Hu, Ning, Application of the Heisenberg’s S-matrix theory for the resonance scattering and nuclear reactions, Phys.Rev., 74(1948)131-140.
[7]	Van Kampen,N., The S-matrix and the causality condition. I. The electromagnetic field., Phys.Rev., 89(1953)1072-1079.
[8]	Van Kampen,N., The S-matrix and the causality condition. II. Non-relativistic particles., 91 (1953)1267-1276.
[9]	Olkhovsky, V.S. and Tsekhmistrenko,Yu.V., On the analytical structure of the S-function of elastic scattering in the case of infinite potentials, Ukrainian Physical Journal, 6(1961)149-156 (in Ukrainian, abstract in English; partially in English); see also Theor.and Math.Physics, 147(2006)539-551, Appendix 1, pp.547-549.
[10]	Olkhovsky,V.S., To the study of properties of the unitary and non-unitary S-matrix on the base of caysality and the completeness conditions for wave functions, Teor.Matem.Fiz, 20(1974) 211-222.
[11]	Nikolayev,M.V., and Olkhovsky,V.S., On the analytical Structure of the S-matrix for some classes of potentials, Teor.Mat.Fiz., 31(1977)214.
[12]	Nikolayev,M.V., and Olkhovsky,V.S., On Completeness of Radial Equation Solutions and S-matrix Singularities in case of Complex Potentials, Lett.al Nuovo Cimento, 8(1973)703-707.
[13]	Olkhovsky,V.S., and Zaichenko,A.K., Analytical properties and resonance structure of the S-matrix in case of non-central and parity-violating interactions, Nuovo Cim., A63(1981)155-170.
[14]	Baldo,M., and Olkhovsky,V.S., Analytical properties of the S-matrix for interactions with Yukawa-potential tails, Theor. Mathem.Phys., 147 (2006)539-551
[15]	Olkhovsky,V.S., On the analytic properties of the S-matrix for the unknown interactions surrounded by centrifugal and rapidly decreasing potentials, Central Eur. J. Phys., 2011,v.9, N1,13-44.
[16]	Saavedra,I., Dispersion relations, “causal” inequalities and the completeness of the set of eigen-states in non-relativistic potential scattering, Nucl.Phys., 29(1962)137-152; Completeness, unitarity, and dispersion relations in potential scattering, Nucl.Phys., 87(1966) 267-272.
[17]	R.Newton, Scattering Theory of Waves and Particles, McGRAW-HILL BOOK COMPANY, New Yorl–San Francisco–St.Louis–Toronto–London–Sydney, 1969.
[18]	Gasymov, M.G., On the expansion over eigen functions og non-self-adjoint boundary-value problem for the differential equation with a singularity in zero, Doklady AN SSSR, 165(1965)261-264.
[19]	Bertero, M., and Dillon,G., An outline of scattering theory for absorptive potentials, Nuovo Cim., A2(1971)1024-1038.
[20]	V.De Alfaro, and T.Redge, Potential scattering, North-Holland Publish.Company, Amsterdam, 1965.
[21]	E.C. Titchmarsh, The Theory of Functions, Oxford University Press, 1939.
[22]	Jost, R., Uber die falschen Nullstellen der Eigenwerte der S-matrix, Helv.Phys.Acta, XX (1947)256-266.
[23]	Ma, S., Phys.Rev., On a General Condition of Heisenberg foe the S-matrix, 71(1947) 195-200.
[24]	I.S.Gradstein and I.M.Ryzhik: Tables of Integrals, Sums, Series, and Products (Moscow, Fizmatgiz,1963) [in Russian]; English translat.: Tables of Integrals, Sums, Series, and Products, Acad.Press, New York(1965).
[25]	A.S.Davydov, Theory of atomic nucleus, Fizmatgiz, Moscow (in Russian), 1958.
[26]	Simonius,M., Overlapping resonances and unitarity of the scattering S-matrix, Nucl.Phys., A218(1974)53-60.
[27]	Olkhovsky,V.S., Nucl.Phys., A425 (1984)445-457 D’Arrigo,A., Giardina,G., Olkhovsky,V.S., et al, Internat. J. Mod. Phys., E2(1993)809-820.
[28]	Smith, F.T., Phys.Rev., 118 (1960) 349; Phys.Rev.,119 (1960) 2098.
[29]	Lyuboshitz, V.L., Phys.Lett.,B, 72N1(1977)41; Lyuboshitz, V.L., Sov.J.Nucl.Phys., 27(1978)502; Lyuboshitz, V.L., JETP Lett.,28:1(1978)32;
[30]	Baz’, A.I.,JETP Lett.,27:2(1978)132-135}.
[31]	V.S.Olkhovsky, Time as a quantum observable, canonically conjugate to energy. Time analysis of quantum processes of tunneling and collisions (nuclear reactions) [LAP LAMBERT Academic publishing, 177 pages, 2012]
[32]	Gosset, J., Gutbrod, H., Meyer, W. et al, Phys.Rev.,C16(1977)629.
[33]	Westfall, G., Sexstro, R., Poskanzer, A. et al, Phys.Rev.,C17 (1978)1368.
[34]	Cumming, J., Haustein, P.,Ruth, T. et al, Phys.Rev., C17( 1978) 1632.
[35]	Kaufman, S., Steinberg, E., Phys.Rev.,C17(1980)167.
[36]	Fortney, D., Porile, N., Phys.Rev.,C21(1980)2511.
[37]	Sandoval, A.,Gutbrod, H., Meyer, W. et al, Phys.Rev., C21(1980)1321.
[38]	Danielewicz, P., Pan, O., Phys.Rev.,C46(1992)2002.
[39]	Alba, R., Coniglione, R., Del Zoppo, A. et al,Phys. Lett., B322 (1994)38.
[40]	Chen, K., Fraenkel, Z., Friedlander, G., et al, Phys.Rev., 166 (1968) 949.
[41]	Amsden, Antony A., Goldhaber, Alfred S., Harlow, Francis H., and Rayford, Nix J., Phys.Rev., C17 (1978)2080-2096.
[42]	Izumo, Ko, Araseki, H., Progr. Theor. Phys., 69 (1983)158-170.
[43]	Goto, T., Yamaguchi, K. and Sudo, N., Progr. Theor. Phys., 64 (1980)1-17.
[44]	Olkhovsky,V.S., Dolinska,M.D., Omelchenko,S.A., Central Eur. J. Phys., 4(2006)1-18;
[45]	Olkhovsky,V.S., Dolinska,M.D., Omelchenko, S.A., Central Europ. J.Phys.,9:N4(2011) 1131-1133.
[46]	Westfall, G.D., Gosset, J., Johansen, J. et al, Phys.Rev. Lett., 18: 37(1976) 1202.
[47]	Gosset, J., Kapusta, J.I.,Westfall, G.D., Phys.Rev.,C18(1978)844.
[48]	Prilepsky, Yu., Shmonin, V.,Acta Phys.Polon.B 18(1987)729.
[49]	Olkhovsky, V.S., Sov. J. Part. Nucl.,151(1984)30-148.
[50]	Olkhovsky, V.S., Doroshko, N.L., Europhys. Lett., 18 (1992) 483.
[51]	D’ Arrigo, A., Doroshko, N.L., Eremin, N.V., Olkhovsky, V.S. et al, Nucl. hys., A, 549 (1992)375.
[52]	D’Arrigo, A., Doroshko, N.L., Eremin, N.V., Olkhovsky, V.S. et al, Nucl. Phys., A, 564 (1993)217.
[53]	Kelkar, N.G., J. Phys. G: Nucl. Par. Phys., 29 (2003) L 1-L 8.
[54]	Kelkar, N.G., Nowakowski, M. and Khemchandani, K.P., Nucl. Phys., A 724 (2003) 357; Kelkar, N.G., Nowakowski, M. and Khemchandani, K.P. and Jain, B.K., Nucl. Phys., A730 (2004) 121.
[55]	Kelkar, N.G., Khemchandani, K.P. and Jain, B.K., J. Phys. G: Nucl. Part. Phys., 32: 3 (2006) L 19.
[56]	Prokopets, G.A., Nuclear Physics (Russia), volume 74 (2011), p. 740-746.
[57]	Eremin, N.V., Giardina, G., Olkhovsky, V.S., Omelchenko, S.A., Mod. Phys. Lett., A, 9 (1994) 2849.
[58]	Olkhovsky, V.S., Dolinska, M.E., Omelchenko, S.A. and Romanyuk, M.V., Internat. J. Mod. Phys. E, 19: 5-6 (2010), рр. 1212-1219.
[59]	Olkhovsky, V.S., Dolinska, M.E., Omelchenko, S.A., arXiv. 1101. 5541v1 [nucl-th], Jan. 2011; Appl. Phys. Let., 99 (2011) 244103 (1-3).
[60]	Olkhovsky, V.S., Dolinska, M.E., Omelchenko, S.A., On the cross section and duration of the neutron- nucleus scattering with a resonance, distorted by a non-resonant background, in the center-of-mass system and laboratory system, Proc. of the 4-th Internat.Conf. Current Problems in Nuclear Physics and Atomic Energy (NPAE-2012, sept. 3-7, 2012, Kyiv, Ukraine), Kyiv 2013, pp. 198-201.
[61]	Olkhovsky, V.S., Sov. J. Particles Nucl.(Engl. Transl.);(United States), 1984, v. 15, (2), p. 130-148; Olkhovsky, V.S., Time as a Quantum Observable, Canonically Conjugated to Energy, and Foundations of Self-Consistent Time Analysis of Quantum Processes, Advances in Math. Phys., vol. 2009 (2009), аrticle ID 859710, 83 pages.
[62]	Olkhovsky, V.S., Doroshko, N.L., Lokotko, T.I., On the cross section and duration of the neutron-nucleus scattering with two overlapped resonances in the center-of-mass system and laboratory system, Proc. of the 4-th Internat. Conf. Current Problems in Nuclear Physics and Atomic Energy (NPAE-2012, sept. 3-7, 2012, Kyiv, Ukraine), Kyiv 2013, pp. 192-197.
[63]	The JEFF – 3.1.1. Nuclear Data Library, OECD NEA/NEA 2009; EXFOR 13759.002, J.A. Harvey, D.C. Larson, ORNL, 1974.
[64]	Brusegan, A., Rohr, G., Shelley, R., Macavero, E., Van Der Vorst, C., Poortmans, F., Mewissen, I., Vanpraet, G., Very high resolution transmission measurements and resonance parameters of Ni 58 and Ni60, 1994.
[65]	Olkhovsky, V.S., Zaichenko, A.K., Phys. Lett. B 272 (1991) 183-185.
[66]	D’ Arrigo, A., Fazio, G., Giardina, G. et al. Progress of Theor. Phys. 87 (1992) 1359-1365.
[67]	Kopylov, G.I., Podgoretskij, M.I., Sov. J. Nucl. Phys. 15 (1972) 219-223.
[68]	M.L. Goldberger, K.M. Watson, Collision Theory. Wiley J. & Son Inc., 1964.
[69]	A.I. Baz, Ya.B. Zel’ dovich, A.M. Perelomov, Scattering, Reaction in Non-Relativistic Quantum Mechanics. Jerusalem, 1969.
[70]	Rosenfeld, L., Nucl. Phys. 70 (1965) 1-27.