Table Of ContentChapter I
NOELCUN-NOELCUN SNOITCARETNI
nuhC aW gnoW
Department of Physics, University of California, soL Angeles
soL Angeles, AC 90024 ASU
dna
Keh-Fei Liu
Department of Physics and Astronomy, California State University, Northridge
Northridge, AC 91330 ASU
I. Introduction
The study of nucleon-nucleon (NN) interactions yam eb said to have begun in 1932
(H32) immediately after the discovery of the neutron (C32). It is still a very ac-
tive and profitable area of study today. During the half century which sah elapsed,
a tremendous amount of theoretical dna experimental efforts have been devoted to the
problem. Our present-day understanding of NN interactions, both in free space and
in nuclei~ is very detailed by any standard. enO objective of these lectures is to
sketch the extent of this knowledge from a theorist's point of view.
ehT understanding sah been acquired down a number of intersecting paths. There
is first a need to describe NN interactions mathematically. Quite a few different
but equivalent descriptions have been found to eb useful in giving ready visualiza-
tions of the dynamical situation under different circumstances. Secondly, empirical
information no NN interactions must eb obtained experimentally. Much useful informa-
tion can eb deduced from nuclear properties, especially those of few-nucleon systems.
More precise dna complete measurements must eb made directly on NN systems them-
selves. Our present experimental knowledge no elastic proton-proton scattering is
basically complete up to about 005 Ve1I laboratory energy. Our knowledge of elastic
neutron-neutron scattering sah also improved greatly in the last several years.
These elastic data by themselves are not sufficient to fix the nature of NN interac-
tions completely. However, when they are used in the context of available theories
of nuclear forces, they give quite a detailed picture of NN dynamics.
nO the theoretical side, the most important development has been Yukawa's propo-
sal (Y35) that nuclear forces arise from the exchanges of massive bosons called
mesons. The possible existence of diverse types of mesons stimulated experimental
searches for them. Most of them have since been found. It sah also led to a highly
useful model of NN interactions based no the exchanges of single bosons.
The extension of Yukawa's idea to multiple-meson exchanges sah not met with
dramatic success despite repeated efforts. Nevertheless, when used together with the
idea of dispersion relations (BS57b, )46WG dna of crossing (M58a), a very significant
progress sah been made in understanding the dynamical relations among diverse hadron-
hadron interaction problems. sA a result, known properties of related hadron-hadron
scatterings nac eb brought to bear in unraveling the complicated dynamics of the
strongly interacting NN system.
Another revolutionary development, which yam turn out to eb sa influential sa
Yukawa's meson-exchange theory, is the recent realization that the long-suspected
internal structure of nucleons yam have a simple description in terms of three "ele-
mentary" objects called quarks (GM64, Z64). In this model, the N!~ interaction is
seen to eb not a fundamental interaction of nature, but rather the shielded remnant
of more fundamental forces called color forces which operate gnoma ouarks.
eW are thus at the threshold of a wen dna exciting region which yam eb called
the quark model of nuclear forces. It is likely to lead to a deeper understanding of
NN dynamics, especially at short distances where nucleons overlap dna explicit quark
degrees of freedom come into play. eH already know that NIP dynamics at these short
distances involves excited states of nucleons, i.e., isobars, which are also edam
pu of quarks. ehT quark model not only nac describe these in a unified manner, but
the explicit appearance of quarks might also have additional important dynamical con-
sequences at this deeper level of nuclear structure.
ehT availability of realistic NN potentials sah encouraged ambitious attempts
to derive from these bare or basic potentials the effective or renormalized NIf
interactions in nuclei to eb used in nuclear shell-model calculations. (For a
review no this topic ees the lectures of Professor ouK in this Hinter School.)
ehT stimulus is not one-sided, however. weN experimental dna theoretical
studies of nuclear structure have continued to dda insights concerning the dynamical
manifestations of nuclear forces in nuclei. oT cite a few examples, the discovery
of isoscalar monopole resonances (Y77b) sah finally given su experimental informa-
tion no the compression modulus of nuclear matter, while the systematic studies of
giant multipole resonances using hadronic dna electromagnetic probes have led over
the past few years to a deeper understanding of various parts of the effective
particle-hole interaction. nO the theoretical side, considerable effort sah recent-
ly been devoted to the extension of Landau's Fermi liquid theory to finite nuclei in
the hope of constructing a unified picture of quasiparticles in which nuclear
ground-state properties nac eb related to the properties of excited states dna of
neighboring nuclei.
In writing these lectures ew have edam a conscious effort to make the subject
matter available to people ohw have not dah a long exposure to it. oT this end, ew
have emphasized basic ideas whenever ew nac od os without interfering too hcum with
the flow of the review. Even here the emphasis is definitely that of theorists.
ehT serious reader would want to supplement these lectures yb reading reviews of
experimental results dna techniques, e.g., that given in .361.1
sA the writing of these lectures proceed, ew soon realize that ew have to leave
out a number of very relevant topics, including many-body forces, symmetry-violating
forces, NN systems, baryoniums, dna high-energy scattering, in order to od justice
to what are included.
ehT selection of references is even less satisfactory dna complete. ehT refer-
ences no NN interactions are just too numerous to eb quoted with any degree of -moc
pleteness. eW only mention a few of those with which ew are familiar, a few of
those which ew have enjoyed reading, a few of those which have impressed su (and
hopefully also the reader), a few of those which dda historical touches to enliven
the account, dna a few for the simple reason that they are typical of other papers
in the field, eW quote reviews whenever ew find them useful, but on attempt sah
been made to provide a complete list. ehT reader should not take the quoted refer-
ences sa complete dna accurate reflections of "who sah done what" in the field.
2. owT Nucleon Systems
Nuclei are known to eb edam pu primarily of neutrons dna protons. Their prop-
erties are shown in Table I.
Table 2.1 Nucleon properties with experimental errors shown within parentheses.
Proton Neutron tnemmoC
Charge 1 0
<r 2 >ch I/2 (fm) 0.88(3) a 0.36(I )b*
#(n.m.) 2.79274(6) -1.91314(4)
>2r< I/2
0.80(3) 0.79(15)
Mass (MeV) 938.26 939.55 nm ~ pm + 2.5 m c
naeM life (Tm) ~ 1030 yrs. c 17.0(4) min. n÷p+e-+~
e
References: BM69 except sa noted below.
(a) B + 47 , (b) G + 17 , (c) SRL 97 .
Note: <r2>ch = - 0.126 mf 2 for the neutron.
Since the proton dna neutron have roughly the same mass, it is convenient to
treat them mathematically sa two states of the emas fundamental object, the nucleon,
having different charges (H32). This internal degree of freedom is called the
isospin. A system of two nucleons nac eb in eno of the four spin-isospin states
shown in Table 2. ehT two-nucleon spin function is antisymetric (symmetric) if the
total spin S is 0 (I). ehT isospin function sah a similar symmetry with respect to
the total isospin T. Hence the antisymmetry of the total wave function requires
that the relative orbital angular mutnemom ~ eb ddo or even, sa indicated in Table
2. eW shall use the convention that the proton points pu in isospin space.
T S ~ emaN
0 0 ddo singlet-odd (SO)
Table 2.2 The four isospin-spin 1 even triplet-even (TE)
states of the Nit system. 1 0 even singlet-even (SE)
1 ddo triplet-odd (TO)
yD a partial, wave ew naem a system with the specified quantum numbers T, M T, S,
M S , ~, and ,~m or T, ,M S, ~, j, dna jm in a coupled representation. This partial
wave is often referred to by the simplified or spectroscopic notation 2S+I~j,
where is the atomic notation for orbital angular mutnemom (S, P, D .... for
= ,O I, 2 .... ). The projection jm is not needed for the specification of dynami-
cal properties because of the assumed spherical symmetry of space. ehT isospin T
does not have to eb specified, because Table 2 shows that it is uniquely determined
once S and ~ are given. The projection rl T is often dropped when the dynamics is
independent of it. This isospin invariance, called the charge independence of
nuclear forces, is suggested by experimental results. A short account of the history
of charge independence has been given in P79.
(2A) The Deuteron
If the two-nucleon system has bound states, ew expect them to have ~ = O, i.e.,
to be states of lowest orbital kinetic energy. These states are S3 1 and I S Of
1S3 3 "O
these only the state is found experimentally to be bound. The bound S 1 state,
called the deuteron, has the properties shown in Table 3.
Table 2.3 Properties of the deuteron. The experimental errors are given in-
side parentheses. The quantity r d (rch) is the rms radius of the
wave function (of the charge) as measured from the center of mass.
= 2.224544 (46) VeM (H71)
<~ (~I-~2)2> I/2 = r d = 1.9635 (45) fm (B+73)
rch = 2.0952 (60) fm (B+73)
Q = 0.2860 (15) fm 2 (RV72)
= 0.857406 (I) n.m. (L65)
n(D/S) = 0.02649 (43) )OSHS(
The binding energy c of the deuteron cannot be due to a Coulomb interaction,
because the neutron is chargeless. Even if the neutron had a charge -e, its elec-
trostatic potential for the proton, placed at the average separation s d = 2r d of
nucleon centers, would be only
2e = e 2r e = _ 0.5 VeM × 2.8 fm
- s d - r--eS d 4.0 fm = - 0.3 veM ,
where ew have used the classical radius r e of the electron
r e = e 2 /m e c2 = 2.828 fm
for the evaluation of e 2. The result is much weaker than ~ ~ 2.22 Mev. llence the
np interaction must be much stronger than the Coulomb interaction. eH call this
strong nuclear interaction the strong interaction.
If the deuteron is bound by a square-well potential of radius b and depth -V o,
its radial wave function is
A sin kr r < b
Uo(r ) = for , (2.1)
Ge -Yr r>b
where
k = M(Vo-e)I/2/~ , y = (Mc)I/2/~ = 0.2317 mf -I (2.2)
M being the nucleon mass. If the potential parameters b and V ° are chosen to fit
and r d, ew find
R = 2.4 fm , V = 72 VeM (or k = 0.775 mf -I) , (2.3)
0
as shown in Fig. I. eW note the inter- !
3,o
esting results that R ° 0.6 s d and s
0.08 V o, and that the external normali-
zation is
>N = f Uo2(r)dr - sin2kb
0
b
4- 6
= 0.59 . (2.4)
S..
>
Thus the deuteron is very loosely
bound in the potential, as shown in
Fig. I. sA a result, there is no
other S3 1 bound state.
2-4
Comparison of V = 27 VeM with
0
the electrostatic potential of .qE
Fig. 2.1 Distribution of the squared
(I) shows that the strong interaction
deuteron wave function in the
is about I00 times stronger than the square-well potential model.
electromagnetic interaction.
The kinetic energy T of the deuteron in this simple model is
d
T b = V o - c = 25 VeM . (2.5
This should be considered a rough lower estimate, because the nuclear wave function
is known to have more curvature than predicted by such a simple potential model.
A much more serious defect of this simple model is that the deuteron appears as
a pure S3 1 state. Such a state is spherically symmetric; it has zero quadrupole
moment ,Q which is defined as the expectation value of the quadrupole moment
:2~
operator
Q = <jmj=JI~21jmj =j> = I Pch(~)(3z2 - r2) d3r ' (2.6
where
2~ = e(5/@)I/2 ro 2 DSSOC(2P ) .
The subscript p refers to the proton, and 2P is a Legendre polynomial. The experi-
mental value of Q is positive. This means that the deuteron charge distribution
Pch(~) is longer in the z-direction (the direction of the deuteron spin ~) than in
the xy direction. Thus the deuteron is an olive-shaped object.
It is also clear from .qE (6) that admixtures of nonzero orbital angular -nemom
ta are needed for nonzero .Q Such mixtures can arise from terms in the Hamiltonian
which, though rotationally invariance, are non-spherical in space. Such operators
cannot be constructed from spatial variables alone, but they must also involve the
nucleon spins ~i" The construction of such operators will be considered in Sect. 3.
At present, ew simply note that there is one such operator, the tensor operator
SI2 = 3(~l'r)(÷ o A ÷ 2" r) - ~l'a2 • (2.7)
which connects the S3 l to the D3 l state. In the presence of such an operator, the
deuteron is the mixture
(2.8)
d> = cosmJ3Sl > + sinml3Dl >
where m is called a mixing angle. The D-state probability
dP = sln • 2 (2.9)
of finding the deuteron in the D3 1 state describes an important aspect of nuclear
dynamics, i.e., a tendency for interacting nucleons to line pu along the direction
of their total spin. Most realistic potential models of the deuteron give dP = 4-7%.
Figure 2 shows the radial wave functions
u(r) = cos~ Uo(r) , w(r) = sins u2(r) , (2.10)
where ~u are normalized radial wave functions, in one such potential - the Reid
soft-core potential (R68).
J
0.6
05
04
0.3
0.2
0.1
0 .L r I I ~ I 3 I ~ I 5 I 6 t 7 I 8 J 9 L ~mftr -~L -ID~ ,
Fig. 2.2 Deuteron wave functions u(r) and w(r) for the Reid
potential (from BJ76).
The D-state probability is not a measurable quantity. It is apparently a well-
defined concept if ew mean the probability that the deuteron is in a state made pu
entirely of two nucleons as each might appear in free space, moving in a relative D
state. The problem is that at medium and small nucleon separations the deuteron is
likely to be a complicated mixture of many different kinds of states - states con-
taining pions, states involving isobars (i.e., excited states of the nucleon which
are denoted by the symbol *N if I = I/2, and by A if I = 3/2), etc. Indeed, the
nucleon is won known to be a composite object made up basically of three quarks. At
small separations, the deuteron yam contain a very substantial fraction of six-quark
states in which even the nucleon identity is lost. Thus the picture represented by
Eq. (8) yam well be over-simplified as far as the D-state probability is concerned,
even if ew know what a nucleon in free space really is. Still it is better defined
than the S-state probability, or the cos2m of .qE (8), since the centrifugal barrier
greatly reduces w(r) at small distances.
It is also clear that all these complications decrease rapidly as the nucleon
separation increases. Outside the range of nuclear forces, there is no doubt that
the deuteron is almost exclusively made pu of two free-space nucleons. Indeed the
asymptotic ratio
~(D/S) = lim (2.ll)
oc.-r
of radial wave functions turns out to be experimentally measurable. A value of
0.02649 (43) has been deduced from the tensor analyzing power of sub-Coulomb (d,D)
reaction on 2O8pb (KSH74, SH8O). It is comforting to find that most modern NN ooten-
tials give values of this ratio in good agreement with this experimental result.
The next entry in Table l is the magnetic moment. If the deuteron is a oure
S3 l state, ew should have a simple mus of the intrinsic magnetic moments of the con-
stituent nucleons:
~np(3Sl ) = n~ + p~ = ½ go n.m. = 0.880 n.m. , (2.12)
where go is the isoscalar g factor. This is because the spatial S-wave, being
spherically symmetric, does not contribute an orbital magnetic moment. More gener-
ally, the magnetic moment of an isoscalar state is
~np(T=O, SLj) = ½< go ~ + ½#> mj=J
= {(go+l)j(j+l) + (qo-l)S(S+l) - ~(~+l)}/4(j+l) ;
that is,
~np(T=O, S~j) = ~np(3Sl ) + 0.095 S(S+l) - 2 , L(~+l) (2.13)
Thus the presence of L f 0 admixtures in the deuteron will reduce its magnetic
moment. The experimental value ~d = 0.857406 (1) n.m. is smaller than ~(3S l) by
2.6%. According to Eq. (13), this can be accounted for by a 4.0% admixture of the
D3 l state. Some modern-day NN potentials (e.g., HM76) do qive such small D-state
probabilities, but other potentials give values of 5-6%. The latter potentials will
give too small values of ~d"
The interpretation of ~d is complicated by the variety of corrections which
must be added to the simple result given above ($75, T77). Meson-exchange correc-
tions can be very large and are still not well understood. Momentum dependences in
the NN potential, e.g., that arising from the NN spin-orbit force, will give addi-
tional contributions. ?he presence of AA (ADW70) and six-quark components in the
deuteron wave function will have significant effects. For these reasons, the experi-
mental value of ~d is still not completely understood.
eW have seen that the deuteron is so loosely bound that there can be no other
£ = 0 bound state in the same potential. There is also no bound state for £ > 0 in
the same potential, because the additional kinetic energy of orbital motion is
roughly
2~ £(£+I) ~ 2.4 £(£+I) VeM (2 14)
M 2 "
s d
This same centrifugal potential produces an even greater anti-binding effect by push-
ing the wave function further out of the attractive potential well. Indeed, the
square-well depth needed to bind an £ = l state can readily be calculated to be
2~ 2
= --~ = 71MeV if b = 2.4 fm (2 15)
oV bM 2 ' • .
This is about 2.6 times stronger than the well depth for the deuteron.
The actual NN potential varies considerably from state to state. It is an ex-
perimental fact that the deuteron is the only NN bound state. All other NN systems
are scattering states in the positive-energy continuum.
(2B) Scattering States at woL Energies
Most of the experimental information on the NN interaction has been obtained
by studying NN scattering. A projectile nucleon of laboratory energy E incident on
a target nucleon has a relative momentum of
p = = ~k .
They will interact if they impact at a relative distance shorter than the range b of
nuclear forces. The maximum classical orbital angular momentum involved is
£ = bk = b(ME/2~2) 1/2 (2.16)
xam
For £max = l, E ~ 41 VeM if b = 2.~ fm. Thus for lab energies less than lO MeV, only
01
S-wave scatterings are important.
The radial wave function for the NN system in S-waves satisfies the Schrodinger
equation
d M k 2
~V(r) + u(k,r) = 0 . (2.17)
dr 2
If the NN potential V(r) vanishes sufficiently rapidly (i.e., faster than I/r) as
r + ~, u(k,r) is proportional to sinkr + G(k) at large distances. Here the
phase shift 5(k) gives the amount by which the phase of the wave has been shifted
by the interaction from the spherical wave sin kr for non-interacting nucleons.
It is convenient to normalize the wave function such that
u(k,r) ~" v(k,r) ~ sin(kr+6)/sin~(k)
r~
= cos kr - sin kr a-~ '
where the asymptotic wave function v(k,r) is characterized by a scatterinq length
a(k). The Taylor expansion in k 2
k cot 6(k) z - a~ l = - a l + ½ rol-2 + Pro 3k4 + Qro5k 6 + ... (2.19)
is called an effective-range expansion. Here a = a(k2=O) is the (zero eneray, or
Fermi) scatterin 9 len t~, and r ° is called an effective range.
The effective range r o in Eq. (19) is a special case of the effective-range
integral (BW52, BM56)
(2.20)
p(E I, E2) = 2 f (VlV 2 - UlU 2) dr
0
which characterizes the amount of distortion contained in u i u(ki,r), as compared
to the asymptotic or comparison wave functions v i m v(ki,r). Its importance is due
to an identity derivable from Eq. (17):
)12.2(
2Z (k22 - kl 2) P(EI' E2) = v'2(r=O) - V'l(r=O)
)22.2(
= k 2 cot 2G - kl cot 15 ,
where Bi ~ ~(ki)" If ew won set k I = 0 and k 2 = k, ew get
k cot 5(k) = a-l(k) = a -I - ½ k2p(O,E) • (2.23)
In other words,
p(O,E) = r o + 2Pro3k 2 + 2Qro5k 4 + "" , (2.24)
where
r = p(O,O) .
0
