Index_1983_Direct-Nuclear-Reactions

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Index 
A 
Addition theorem 
solid spherical harmonies, 282 
spherical Bessel function and spherical 
harmonic, 282 
spherical harmonic, 351 
Alpha scattering 33, 159, 231-340 
Analog reactions, 310 
Angular distribution, 7, 22, 30 
and conservation laws, 7, 10-12, 95, 145 
Angular momentum, 9, 95, see also 
Conservation laws 
coupling, 351-353, see also 
Clebsch-Gordan coefficients 
critical, 316 
grazing collision, 313 
recoupling, 353-356 
Antisymmetrization, 54-56, 201-206 
Asymptotic behavior, 67, 73, 87, see also 
Boundary conditions 
complete wave function, 20-21 
Coulomb functions, 28, 30 
radial wave function of distorted wave, 29 
Β 
Bessel function, spherical, 28, 362 
addition theorem, 282 
Bethe-Goldstone equation, 182, 184 
Born approximation, see also Coupled channel 
Born approximation; Distorted-wave Born 
approximation 
optical potential, 24 
plane wave, 7, 24 
Born series, 65 
Boundary conditions, 47, 298, see also 
Waves, incoming and outgoing 
channel, 74, 136-137 
Lippman-Schwinger equation, 63-64 
partial wave, 303-304 
Breuckner G-matrix, see G-matrix 
C 
Central field, 14, 16, see also Optical potential 
Channel 
boundary conditions, 74, 136-137 
coordinate, 57, 275, see also Relative 
coordinate 
coupled, 163-166, see also Coupled 
equations 
defined, 15 
entrance, 16-17 
function, 134, 303 
open, 17 
projection operator, 128 
truncation, 128-130 
373 
374 Index 
Charge exchange reactions, 2, 119-120, 310 
Clebsch-Gordan coefficients, 351-353 
angular momentum conservation, 58, 93, 
145 
Coherence, 126, 135, 160-163, 293, see also 
Correlations 
Collective transitions, 160-163, 168, 173 
Coulomb excitation, 123-124 
surface vibrations, inelastic excitation, 
106-111 
Compound nuclear reactions, 2-3 
competition with direct, 5-6 
Conservation laws, 10-12, 145, see also 
Selection rules 
angular momentum, 7, 10, 58 
Coordinate space representation, 62-64, 67 
Core polarization, 143, 168-174, 185 
Correlations, 2, 163 
two-nucleon transfer reactions, 251, 
261-263 
Coulomb 
barrier energy, 41, 314 
excitation, 123-124, 245 
functions, 28, 31 
phase shift, 28 
potential, 30, 33, 43 
scattering amplitude, 30 
Coupled-channel Born approximation, 
293-301 
amplitudes, 295 
source term method, 297-299 
Coupled equations, 242, 297 
deformed rotational nuclei, scattering, 
227-228 
inelastic scattering, 125-128 
partial wave expansion, 133-137, 303 
Coupled reaction channels, 293, 301-302 
Creation operators, 98-100, 150 
Cross section, 21, 30, 59, 138-139, 260 
differential, see Angular distribution 
partial-wave expansions and derivation, 
303-305 
plane-wave approximation, 215-216 
D 
Deflection function, 317-326 
Deformed nuclei, see Collective transitions; 
Rotational levels 
Direct integrals, 55, 90, 252, 275 
Direct interaction, 111, 113, 119, 156-160, 
see also Effective interaction 
Direct nuclear reactions, 2 - 3 , 13, 45-46 
competition with compound, 5-6 
Distorted wave, 27, 72-73 
convention for incoming, 74-75 
Distorted-wave Born approximation, 45-60, 
77-78, 160, 252, 260 
antisymmetrization, 54-56 
charge-exchange reactions, 119-120 
Coulomb excitation, 123-124 
coupled equations solution, 137 
discussion of approximations, 53-54, 292 
distorted-wave impulse approximation, 
120-123 
(d, p) stripping reaction, 90-93 
Gell-Mann-Goldberger transformation, 
48-49 
generalized, see Coupled channel Born 
approximation 
Green's function, 46-48, 76-77 
impulse approximation, 120-122 
nuclear rotational levels, inelastic excitation, 
110 
one-nucleon transfer, 97-104 
pairing theory, 104-106 
post and prior forms, 78 
second, 79-80 
single-nucleon states, inelastic excitation, 
111-119 
surface vibrations, inelastic excitation, 
106-110 
time reversal, 49, 78 
transition amplitude, 50-53, 78 
calculation, 89-124 
coordinate dependences, 275-277 
multipole expansion, 56-60 
two-potential formula, 49-50 
zero-range approximation, 94-95 
Distortion function, 214-215 
(d,p) stripping reaction, 90-93 
Ε 
Effective interaction, 127, 131, 141-145, 
156-158 
energy dependence, 175-189 
for exchange scattering, 201 
and free interaction, 175-189 
multiple scattering series, 131-133, 176 
Index 375 
multipole expansion, 113-115, 144, 154, 
206-212 
non-locality, 131 
Elastic scattering, see also Optical potential; 
Scattering amplitude 
alpha particles, 33 
defined, 1-2 
heavy ions, 37, 326-330 
nucléon, 1-2, 33 
Exchange integrals, 55, 112, 201-206 
Exchange operators, 113, 119 
F 
Finite-range interaction, transfer reactions, 
273-291 
distorted-wave Born approximation 
amplitude, coordinate dependences, 
275-277 
parentage expansions, 277-278 
selection rules contrasted with zero-range, 
285-286 
(t,p) reaction, 278-288 
two-nucleon transfer, heavy ions, 288-291 
Flux, 26, 139 
Form factor, 307, see also Transition density 
collective transitions, 160-163 
nuclear, 146-149 
one-nucleon transfer, 322-323 
single-particle, 117 
two-nucleon transfer, 256-259, 322-323 
Fourier transform, 203 
Fractional parentage coefficients, 102-103, 
270, see also Parentage expansion 
Free nucleon-nucleon interaction, 177, see also 
T-matrix, free two-body 
G 
Gaussian wave function, 254 
Gell-Mann -Goldberger transformation, 
48-49 
generalization, 71-75 
G-matrix, 133, 143, 176, 178-185, 190-191, 
221-222, see also Watson's multiple 
scattering series 
local coordinate-space representation, 
193-199 
Grazing peak, 317 
Green's function, 18 
coordinate space representation, 63-64 
distorted wave, 46, 76-77 
expansion, 79 
free, 19, 67, 73, 76 
outgoing wave, 19, 47, 73, 243 
Green's theorem, 85-87 
H 
Hamiltonian, 14-15, 71 
Hermitian and Green's theorem, 85-87 
non-Hermitian, 74-75 
partitions, 17, 71 
shell model, 14-15 
Harmonic oscillator, see Oscillator wave 
function 
Heavy ion reactions, 313-349 
elastic scattering, 37-39, 318, 326-330 
finite-range interaction, 274, 288 
inelastic scattering, 326-330 
recoil effects, 290-291 
S-matrix, 318, 324-325, 334-336 
special features, 313-314 
two-nucleon transfer between, 288-291, 
330-333 
Higher-order processes, transfer reactions, 
292-312 
I 
Imaginary potential, 26, 33, see also Optical 
potential 
mean free path, 40, 322 
Impulse approximation, 120-122 
Indirect transitions, 135, 232, 271, 310, 330 
higher-order processes, 292-312 
Inelastic scattering, 1, 110-119 
alpha, 231-240 
deformed rotational nuclei, see Collective 
transitions; Rotational levels 
heavy ions, 326-330 
microscopic theory, 140-167, 190-222 
applications, 160-167, 216-222 
coordinate-space representation, 193 -199 
coupled equations, interactions, 141-143 
cross sections, 215-216 
direct interaction, 156-158 
376 Index 
effective interaction, high-energy domain, 
199-201 
exchange contribution, 201-206 
matrix elements, 143-145 
multipole expansion, 206-212 
nuclear form factor, 146-147 
nuclear structure, 140-141, 149 
particle creation and destruction operators, 
150-151 
particle-hole configurations, 151 -152 
quasi-particle configurations, 152-154 
selection rules, 145-146, 158-160 
shell-model configurations, 149-150 
single-particle form factors, 147-149 
transition densities, 212-215 
two-body amplitude, 192-193 
rotational states, 110, 223-240 
single-particle states, 111 -120 
surface vibrations, 106 
transition amplitude, 118 
Integral equation of scattering theory, 70, 
72-73, 75, 84-85 
formal solution, 65-66 
Lippmann-Schwinger equation, 62-65 
scattering amplitude, 17-20 
Interference, 310, 340, see also Coherence 
Internal coordinates, 18, 57, 252, 275-276 
Intrinsic coordinates, 18, 57, 252, 275-276 
J 
3 - y' symbol, see Clebsch-Gordan coefficient 
6 - j symbol, 134, 145, 353-354 
9 - j symbol, 145, 355-356 
Jacobian, 57, 276 
Κ 
Kinetic energy operator, 14, 16, 27 
L 
Laguerre polynomial, 362 
Laplace operator, 350 
Legendre expansion, 113-114, 144, 281 
Legendre function, 351 
Lippmann-Schwinger equation, 62-65 
Long-range absorption,245 
M 
Mol 1er operator, 66 
Momentum space representation, 63-64 
Multiple scattering series, 131-133, 176, see 
also Watson's multiple scattering series 
Multipole expansion, see also Partial wave 
expansion 
deformed optical potential, 224-226 
transition amplitude, 56-60 
two-nucleon interaction, 113-115, 144, 
154, 206-212 
Ν 
Nuclear form factor, 275-277 
Nuclear overlap function, 90, 253, 278, see 
also Form factor 
Nuclear potential, 43, see also Optical 
potential 
Nuclear structure amplitudes, 146, 149-152, 
206, 212 
Nucleon-nucleon interaction, 177, see also 
T-matrix, free two-body 
Ο 
One-nucleon transfer reaction, 2, 8, 90-106 
contrast with two-nucleon transfer, 250-251 
distorted-wave Born approximation 
amplitude, 90-92 
(d,p) stripping reaction, 90-93 
improvements in framework of 
distorted-wave Born approximation, 97 
spectroscopic factors, 97-106 
transition amplitude, 92 
zero-range approximation, 94-95 
Optical potential, 45-46, 130, 142, 187-189, 
326 
alpha particles, 33 
calculation of specific components, 241-249 
Coulomb potential, 30-32 
deformed nuclei, 225-226 
distorted wave, 27, 48-49, 72-75 
elastic scattering amplitude, 29-30 
generalized, 71, 75, 294-296 
Green's function, 46-48, 76-77 
heavy ion scattering, 37-39 
imaginary potential, 40-41 
I n d e x 377 
non-locality, 42, 131, 243 
nucléon elastic scattering, 33-37 
nuclear potential, 30-32 
parametrization, 32-33 
partial wave expansion, 26-29 
partial wave sum convergence, 43-44 
radial wave function, 26-29 
rationale, 24-26 
relation between spherical and deformed 
nuclei, 228-231 
spin-orbit interaction, 33-37 
systematics, 41 
trivially equivalent local potential, 244 
Oscillator wave function, 254, 361-362 
Ρ 
Pairing theory, 104-106, 152-153, 270 
Parentage expansion 
one-nucleon, 91, 97-106 
two-nucleon, 253, 267-271, 277 
Partial wave expansion, 26-27, 31-32, 
48 
convergence, 43-44 
Coulomb, 31 
coupled equations, 133-135, 303-305 
for particles with spin, 35, 39 
transition amplitude, 56-59 
Particle-hole configurations, 151-152 
Partition, 15, 21, 74, see also Channel 
Permutation operator, 54 
Phase shift, 29 
Coulomb, 28 
Pick-up reactions, 2, 8, see also One-nucleon 
transfer reaction 
Plane wave, see also T-matrix 
Born approximation, 7, 24 
expansion, 9, 364 
normalization, 63 
theory, 7-12 
Polarizability, nuclear wave functions, 
338-349 
adiabatic estimate, 345-347 
dynamical description, 347-349 
Potential, see also Coulomb potential; Optical 
potential 
energy of two nuclei, 16 
Projection operators, 127, 156, 179, 279 
Propagator, see Green's function 
(p,t) reaction, 262-266, 278, 287-288 
Q 
Quasi-particles, 152-154, 271 
R 
Racah coefficient, 134, see also 6 - j symbol 
Rearrangement 
amplitude, 67, see also Scattering 
amplitude; T-matrix 
collisions, 71, 73, 89, see also 
One-nucleon transfer reactions; 
Two-nucleon transfer reactions 
Reaction amplitude, see T-matrix 
Reduced matrix element, 357-360 
Relative coordinate, 15, 56 
Relative motion, 16 
optical potential, 25, 71, 296 
Rotation functions, 224, 227, 361 
Rotational levels, 110, 307-310 
alpha scattering, 231-240 
coupled equations, 227-228 
multipole expansion, 224-226 
optical potential, 228-231 
wave functions, 223-224 
S 
Scattering amplitude, see also T-matrix 
Coulomb, 30 
defined, 17-20 
elastic, 19, 29 
inelastic or reaction, 20, 129, 304-306 
Schrödinger equation, 10, 62, 299-300, 
362 
Coulomb potential, 31 
entire system, 17 
optical potential, 25, 72, 75, 294 
shell model, 15 
Selection rules, 93, 119, 145, 147, 285-286, 
290 
direct interaction, 158-160 
spin-isospin, 256 
Separation method, 183-184 
Shell model, 14-15, 149, see also 
Single-nucleon states 
Single-nucleon states 
harmonic oscillation wave function, 254, 
257, 361 
378 Index 
inelastic excitation, 111-119 
wave function, 91, 147 
Single-nucleon transfer, see One-nucleon 
transfer 
S-matrix, 31, 136-137 
amplitude, 318, 324-325 
structure, 334-336 
Source term method, 297-299 
Spectroscopic factors, one-nucleon transfer, 
97-106 
sum rules, 98-100 
Spherical Bessel function, see Bessel function, 
spherical 
Spherical harmonics, 350-351 
addition theorem, 351 
closure, 47 
plane-wave expansion, 9, 364 
Spherical tensors, 356 
scalar product, 358 
vector product, 359 
Spin-orbit interaction, 33-36 
Stripping reaction, see One-nucleon transfer 
reaction 
Structure amplitude, see Nuclear structure 
amplitudes 
Sum rules, 98-100 
Surface vibrations, excitation, 106-110, 
168-170 
Τ 
Talmi-Moshinski transformation, 257, 362 
Time reversal, 49, 78 
T-matrix, 24, 66-70, 74, 129, see also 
specific reactions 
coupled-channel Born approximation, 295 
defined, 20 
distorted-wave Born approximation, see 
Distorted-wave Born approximation 
free two-body, 120-122, 132, 175, 
191-199 
Gell-Mann-Goldberger transformation, see 
Gell-Mann -Goldberger transformation 
one-nucleon transfer, 92 
plane wave, 8, 12-13, 21 
post and prior form, 70 
scattering amplitude, see Scattering 
amplitude 
series expansion, 70, 79-85 
two-nucleon transfer reaction, 259 
Transition amplitude, see Scattering amplitude; 
T-matrix 
Transition densities, 212-215, see also Form 
factors 
Transition operator, 69-71, see also Watson's 
multiple scattering series 
Triton wave function, 280 
Truncation, 127-130 
Turning point, classical, 43, 315-316 
Two-nucleon interaction, 16, 113, see also 
Effective interaction 
repulsion, 83-84, 125, 178 
Two-nucleon transfer reactions, 2, 250-272 
between heavy ions, 288, 330-332 
between rotational nuclei, 307-310 
between vibrational nuclei, 305-307 
correlations, 251, 261-263 
form factor, 259 
parentage amplitudes, 267-271 
selection rules, 256, 261 
selectivity of, 251, 254, 261-262 
structure amplitudes, 258 
transition amplitude, 259-260 
Two potential formula, 49 
Two-step processes, 79, see also Higher-order 
processes; Indirect transitions 
V 
Vibrational nuclei 
indirect transitions, 330-333 
inelastic excitation, 106-110 
two-nucleon transfer, 305-307 
W 
Watson's multiple-scattering series, 81-85, 
175 
Waves, see also Distorted wave 
incoming or outgoing, 28, 31, 49, 63 
non-Hermitian Hamiltonian, 74 
Wigner-Eckart theorem, 357 
Woods-Saxon potential, 14, 32, see also 
Optical potential 
Wronskian, 47-48 
Ζ 
Zero-range interaction, 57, 94, 255, 261, 274, 
285