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1 ELEMENTARY CONCEPTS OF NUCLEAR PHYSICS 1.1 The force between two nucleons Nucleons interact with each other mainly through the strong interaction which obeys fundamental symmetries. Discarding at first a possible dependence on the nucleon’s spins and momenta the Hamiltonian for two assumedly point-like particles can be written as ˆ H = ˆ p 2 1 2m 1 + ˆ p 2 2 2m 2 + ˆ V (r)= ˆ p 2 2μ + ˆ V (r)+ ˆ P 2 2M (1.1) with M = m 1 + m 2 and μ = m 1 m 2 /M being the total and the reduced mass, respectively. The second form on the right follows directly from the definition of the vectors of relative motion for coordinates and momenta, r = r 1 r 2 p =(m 2 p 1 m 1 p 2 )/M , (1.2) and the corresponding ones for the center of mass, R =(m 1 r 1 + m 2 r 2 )/M P = p 1 + p 2 . (1.3) Translational invariance is reflected by the property of ˆ H to commute with the total momentum P. As the latter is a conserved quantity, the center of mass motion of the two particles can be split off such that only the Hamiltonian of relative motion ˆ H rel = ˆ p 2 /2μ + ˆ V needs to be considered. It turns out that the strong interaction ˆ V is practically the same no matter which combinations of the two nucleons one takes. In accord with this feature, neutrons and protons may be treated as two realizations of one object, the nu- cleon. Formally, this can be achieved by introducing the concept of isospin. It is described by a vector operator t whose components fulfill the same commutation rules as angular momentum or spin. Hence, one may apply the same represen- tation as for a spin 1/2 particle: The states |p and |n are the two components of one spinor and represent a proton or a neutron, respectively. By convention we will choose the |p to belong to the eigenvalue 1/2 of the z-component ˆ t z and |n to 1/2. Either state can be changed into the other by introducing the com- mon ladder operators: ˆ t + |n = |p and ˆ t |p = |n . For a two-particle system the total isospin can be calculated according to the rules of combining angular momenta, namely T = t 1 + t 2 . The two-particle state may then be symmetric or antisymmetric with respect to an exchange of the partners, depending on the quantum number T, formal details details are found in Section 26.7. The fact

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Page 1: ELEMENTARY CONCEPTS OF NUCLEAR PHYSICS ...einrichtungen.ph.tum.de/T36/HH-WaNuc-chap1.pdfELEMENTARY CONCEPTS OF NUCLEAR PHYSICS 1.1 The force between two nucleons Nucleons interact

1

ELEMENTARY CONCEPTS OF NUCLEAR PHYSICS

1.1 The force between two nucleons

Nucleons interact with each other mainly through the strong interaction whichobeys fundamental symmetries. Discarding at first a possible dependence onthe nucleon’s spins and momenta the Hamiltonian for two assumedly point-likeparticles can be written as

H =p2

1

2m1+

p22

2m2+ V (r) =

p2

2µ+ V (r) +

P2

2M(1.1)

with M = m1 + m2 and µ = m1m2/M being the total and the reduced mass,respectively. The second form on the right follows directly from the definition ofthe vectors of relative motion for coordinates and momenta,

r = r1 − r2 p = (m2p1 −m1p2)/M , (1.2)

and the corresponding ones for the center of mass,

R = (m1r1 +m2r2)/M P = p1 + p2 . (1.3)

Translational invariance is reflected by the property of H to commute with thetotal momentum P. As the latter is a conserved quantity, the center of massmotion of the two particles can be split off such that only the Hamiltonian ofrelative motion Hrel = p2/2µ+ V needs to be considered.

It turns out that the strong interaction V is practically the same no matterwhich combinations of the two nucleons one takes. In accord with this feature,neutrons and protons may be treated as two realizations of one object, the nu-cleon. Formally, this can be achieved by introducing the concept of isospin. It isdescribed by a vector operator t whose components fulfill the same commutationrules as angular momentum or spin. Hence, one may apply the same represen-tation as for a spin 1/2 particle: The states |p

⟩and |n

⟩are the two components

of one spinor and represent a proton or a neutron, respectively. By conventionwe will choose the |p

⟩to belong to the eigenvalue 1/2 of the z-component tz and

|n⟩

to −1/2. Either state can be changed into the other by introducing the com-

mon ladder operators: t+|n⟩

= |p⟩

and t−|p⟩

= |n⟩. For a two-particle system

the total isospin can be calculated according to the rules of combining angularmomenta, namely T = t1 + t2. The two-particle state may then be symmetricor antisymmetric with respect to an exchange of the partners, depending on thequantum number T, formal details details are found in Section 26.7. The fact

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4 Elementary concepts of nuclear physics

of the NN interaction being charge independent may simply be written in theform [ H , T ] = [ V , T ] = 0. For a more elaborate discussion of these featureswe refer to Section I.6 of (de Shalit and Feshbach, 1974).

Besides conservation of isospin and total momentum, which reflect symme-tries with respect to charge exchange and translation in space, the strong in-teraction obeys additional conservation laws. First of all there is the rotationalinvariance. Again, this symmetry goes along with a vanishing commutator of theinteraction V with the corresponding generator for rotations, which here is thetotal angular momentum J = L + S, involving both orbital part as well as spin,[ H , J ] = [ V , J ] = 0. In the center of mass system the total orbital angularmomentum is given by

L = l1 + l2 = l where Lc.m. = 0 (c.m.-system) , (1.4)

with l being the angular momentum of relative motion, l = [r× p]. For thecombination of spin and angular momenta see Section 26.7.

In addition there are the symmetries for time reversal (t→ −t) and parity orspace reflection (r→ −r). With respect to the latter it must be noted that parityis not conserved in weak interactions. The latter may play some role in nuclearsystems, both in the two-body problem (see e.g. (Ericson and Weise, 1988)) aswell as in heavier nuclei, for instance in β-decay. Such problems will not betouched on in this book. Likewise, there is no room to discuss nuclear propertieswhich involve antiparticles. For the latter a further invariance of the stronginteraction comes into play, that with respect to charge conjugation turningparticles into antiparticles.

1.1.1 Possible forms of the interaction

The fundamental symmetry properties enable one to constrain possible formsof phenomenogical NN interactions . There are good reasons to assume that analmost static approximation will do, meaning that the interaction should involveonly low powers in the relative momentum of the NN system. This is suggestedboth from experimental evidence as well as from considerations which engagesub-nuclear degrees of freedom. Since one expects both spin and angular mo-mentum to play an important role an ansatz like the following one is suggested:

V = Vc(r) + VT (r)S12 + VLS(r)L · S + VLL(r)L12 (1.5)

with the central interaction being given by

Vc = V0(r) + Vσ(r)(σ1 · σ2

)+ Vτ (r)

(τ 1 · τ 2

)+ Vστ (r)

(σ1 · σ2

) (τ 1 · τ 2

)(1.6)

and with similar expressions for VT , VLS and VLL. In (1.5) the so-called tensoroperator S12 appears:

S12 =12

r2(s1 · r) (s2 · r)− 4 (s1 · s2) =

6

r2

(S · r

)2

− 2S2 . (1.7)

Clearly, this tensor force is not spherically symmetric in coordinate space. Sucha component is necessary, however, to explain the measured asymmetry of the

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The force between two nucleons 5

charge distribution of the deuteron. In (1.5) the first two terms represent purelystatic (velocity independent) forces. The third term is a typical spin-orbit inter-action; it is linear in the relative velocity or momentum. Finally, the L12 of thelast term of (1.5) is given by

L12 =(δLJ +

(σ1 · σ2

))L2 −

(L · S

)2(1.8)

and in a sense acts as a spin-orbit force of second order, see also sections 1-4 and2-5 of (Bohr and Mottelson, 1969). It is not difficult to prove that (1.5) fulfillsthe basic symmetry relations mentioned previously.

1.1.2 The radial dependence of the interaction

In the general expressions for the interaction as in (1.5) there appear functionsof r. A moment’s thought tells one that these forces cannot only have attractiveparts. Otherwise, it would be very hard to imagine that nuclei could exist asstable objects. The same would hold true for nuclear matter or neutron matterwhich supposedly exists in one way or another in stars. One needs some mecha-nism to prevent the nuclear matter from collapsing. The most obvious possibility,at least within the non-relativistic approach we are looking at, is offered by at-tributing to the interaction a repulsive core. The latter will not allow the nucleonsto come too close. How in the end this leads to stable configurations at valuesof the density close to the observed one is the subject of quite involved theories;some of which will be described in Chapter 2. The existence of a repulsive com-ponent is confirmed in scattering experiments of nucleons on nucleons at higherenergies, above about 200 MeV. This may be attributed to a more or less “hard”core with a radius of rc ≃ 0.4− 0.5 fm. Such a core does not influence the crosssection very much at lower energies.

Knowing such elementary features the following scheme may then be set tofind acceptable functions for all the form factors Vi(r) necessary for an ansatz likethat given by (1.5). These forms need to have both attractive as well as repulsiveparts like the one shown in Fig. 1.1, adapted from the so-called ”Reid soft corepotential” (Reid, 1968) in the L = S = 0 channel. Once these functions arespecified by a set of parameters the latter may be fixed by calculating differentialcross sections and comparing them with experimental results. In this way onemay hope to determine the interaction V . Unfortunately, for several reasons thismethod breaks down at energies above several hundred MeV, one being that inthis regime meson production starts to play an important role. Nevertheless, itallows one to obtain interactions which may be applied to procedures of many-body physics such that in the end one may deduce properties of nuclear matteror even that of finite nuclei.

To build on this more conventional way of picturing the NN interaction is inline with the general scheme of the book, in which we like to concentrate on thetypical many-body effects which have been developed to understand the physicsof heavier nuclei. For this reason we want to touch upon sub-nuclear degrees offreedom only marginally.

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6 Elementary concepts of nuclear physics

0

100

-100

0 1.0 2.0

RADIUS (fm)

PO

TE

NT

IAL

(MeV

)

Fig. 1.1. A typical form of the radial dependence of the nucleon–nucleon inter-action, adapted from (Reid, 1968).

1.1.3 The role of sub-nuclear degrees of freedom

In the past decades much efforts have been undertaken to understand the physicsof the nucleon in terms of quantum chromodynamics (QCD), which suggests thatthe nucleon is built out of quarks and gluons. There is no doubt that considerableprogress has been achieved. Unfortunately, this is less so for the NN interaction,at least on the more quantitative level. It is probably fair to say that to date stillno satisfactory answer to this problem has been found which could be used forthe description of heavier nuclei or even for the idealized case of nuclear matterat zero temperature.

This situation is very different as soon as one keeps the nucleon as an ele-mentary particle and takes into account in explicit fashion mesonic degrees offreedom. As has been well known since the pioneering work of H. Yukawa (1935),the exchange of mesons between nucleons leads to a mutual force, by analogywith the Coulomb force which in a field theoretical approach comes about asan exchange of photons. Again, we would not like to go into any details. Theinterested reader is referred to (Siemens and Jensen, 1987) for an introductorydiscussion on the construction of an interaction from meson exchange. An exhaus-tive presentation is found in (Ericson and Weise, 1988). On the very qualitativelevel, the essential elements can be summarized as follows.

The attractive, long-range part of the interaction can be explained by theexchange of π-mesons, even on a quantitative basis. In this way one also under-stands the appearance of the tensor force. Important information on the role of

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The model of the Fermi gas 7

the pion in the NN interaction can be deduced from studies of typical propertiesof the deuteron (Ericson and Weise, 1988). Furthermore, the exchange of ρ and ωmesons allows one to understand on a qualitative level the existence of a repulsivepart. Typically, the range of the forces are given by the Compton wavelength ofthe associated meson, which for a pion is ~/(mπc) ≃ 1.4 fm and for the ρ meson≃ 0.3 fm (or the ω which has about the same mass as the ρ meson). It must besaid, however, that for the shorter distances the exchange model does not workwell. Partly this is due to the fact that the smaller the distances being probed inthe reaction, the higher must be the wave number and thus the energy. But athigher energies one needs to consider multiple meson exchange, which is difficultto handle in detail. Moreover, at the smaller inter-nuclear distances it becomessomewhat difficult to visualize the two nucleons as independent particles, to-gether with the “existence of a meson in between”. Considering such features itappears astonishing that by meson exchange one may nevertheless explain manyproperties at intermediate distances correctly (Ericson and Weise, 1988). It maythus be no surprise that this picture can be used for parameterizations of theNN interactions. The problem at short distances is coped with by introducingsome free parameters again. These interactions have a much more complicatedstructure than the Reid potential mentioned previously. Amongst other things,they do not only depend on spin, angular momentum and the distance of relativemotion, but on other combinations of the particle’s momenta as well (in some lowpower of it). Moreover, the excitation of the ∆ resonance of the nucleon can beseen to deliver an effective three-body force in the nuclear medium. As we shallsee later, such three-body forces turn out to be necessary for obtaining correctvalues for nuclear binding, both for nuclear matter as well as for finite nuclei. Forrecent reviews on these features we recommend (Baldo, 1999), (Brockmann andMachleidt, 1999), (Bombaci, 1999),(Brockmann and Machleidt, 1999), (Heisel-berg and Pandharipande, 2000). Likewise we should like to mention the newerdevelopment applying effective field theories; for a review see e.g. (Beane et al.,2001). In the past few years great success has been achieved to connect suchmodels to low-energy QCD. One is guided by the chiral symmetry and accountsfor the manifestation of its spontaneous breaking by the very fact of a finite pionmass. This opens the way for systematic and consistent perturbation expansionswhich constrain possible forms of effective Lagrangians, for a recent and concisereport see (Weise, 2005). In Section 2.1.3 applications of this approach to nuclearmatter will briefly be described.

1.2 The model of the Fermi gas

The reader may already be familiar with this simple model from quantum me-chanics or quantum statistics. We will at first restrict ourselves to zero temper-ature postponing the generalization to later sections. One assumes the fermionsto be free particles the motion of which is governed by the Schrodinger equation

−~2∆

2mψk(x) = ǫ(k)ψk(x) . (1.9)

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8 Elementary concepts of nuclear physics

For a homogeneous system with translational invariance the single particle wavefunction ψk(x) in space will be a plane wave. Discarding for the moment anyadditional degrees of freedom and assuming motion in three dimensions, it isthus given by

ψk(x) = V−1/2 exp(ikx) ≡⟨x|k⟩. (1.10)

It is convenient to assume periodic boundary conditions,

ψk(x, y, z) = ψk(x+ Lx, y + Ly, z + Lz) , (1.11)

with the Li being the lengths of a box of volume V = LxLyLz. They pose thefollowing restrictions on the wave numbers

ki =2π

Lini ni = 0,±1,±2 . . . for i = x, y, z . (1.12)

The normalization of the wave function may thus be written as

∫ Lx

0

dx

∫ Ly

0

dy

∫ Lz

0

dzψ∗k(x, y, z)ψk ′(x, y, z) = δk1k′1δk2k′

2δk3k′

3≡ δkk ′ (1.13)

where on the right-hand side Kronecker symbols appear, rather than δ-functionsas would be given for plane waves in an infinitely large box, for which the ki

would be continuous variables. Indeed, it is often of advantage to work with sucha continuum of states. Then the summation over the discrete indices appearingin (1.12) can be replaced by the following integration in (three-dimensional) kspace

k

f(k) ≡∑

nx,ny,nz

f(nx, ny, nz) −→ V∫

d3k

(2π)3f(k), (1.14)

which is easier to handle. Such an approximation should not matter too much aslong as one is interested only in average quantities, or to get a first orientation onthe order of magnitudes of various quantities. If spin and (and isospin) degreesof freedom are to be included, represented by spinors like χ(ms,mτ ), the form(1.10) generalizes to

ϕα(x) = ψk(x)χ(ms,mτ ). (1.15)

For the level density this implies including a spin-isospin degeneracy factor gs.i.,which equals 4 if one counts neutron and proton as the same particle. In mostcases only the absolute value of the wave vector matters. Calling dn(k) thenumber of levels in the intervals dni (i = 1− 3) one gets

dn(k) = gs.i.dkxdkydkz

(2π)3 V ≡ gs.i.

V(2π)

3 4πk2 dk . (1.16)

The density of levels g(e) = dne/de in energy space can be obtained wheneverthe relation e = e(k) is known. For the Fermi gas this is simply given by thekinetic energy, which is to say

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The model of the Fermi gas 9

e(k) =~2

2mk2 ⇐⇒ 2k dk =

2m

~2de . (1.17)

Notice that this is still correct even if the particles move in a medium–providedthe effects of the latter can be accounted for by an effective mass. Often this isa good approximation for the conduction electrons in metals or semiconductors.Also for the motion of nucleons in a nuclear medium, like for the model of infinitenuclear matter, there may exist such a relation e = e(k), although in the generalcase it will not be given just by the kinetic energy. From (1.16) one gets for dne

dne(e) = gs.i.V

(2π)2

(2m

~2

)3/2 √e de . (1.18)

Here a small detour is in order to systems of dimensions d = 2 or 1, whichmay be realized for electrons in semiconductors, for instance. The equations frombefore are easily generalized to any d. Eq. (1.16) turns into

dn = gs.i.

(1

)d

Vdddk, (1.19)

if Vd and ddk represent the volumes in d-dimensional space and k-space, respec-tively. For the electron gas the gs.i. would have to be chosen equal to 2. For thelevel density dne/de one gets the following relations:

dne(e) = gs.i.S

m

~2de, for d = 2, (1.20)

with Vd = S = LxLy and d2k = 2πkdk = 2π(m/~2)de, and

dne(e) = gs.i.L

√m

2~2

de√e

for d = 1 , (1.21)

with V = L and dk for the volume element in k-space. Notice that for d = 2 thedne is independent of energy.

1.2.1 Many-body properties in the ground state

For the many-body ground state all single particle states are filled up to the so-called Fermi energy, or the Fermi momentum kF. Suppose there are A particlesin the box of volume V .3 This number must be given by

A ≡ n(kF) =

∫ kF

0

dn(k) =gs.i.6π2VkF

3 (1.22)

3Similar results may also be derived for a system in which the particles are put inside acubic box of length a with infinitely deep walls. As in this case the density of nucleons vanishesoutside the box one may in addition study surface effects both on the density of states as wellas on the total kinetic energy, for details see e.g. (de Shalit and Feshbach, 1974)

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10 Elementary concepts of nuclear physics

such that the density of particles becomes

ρ0 =A

V = gs.i.1

6π2k3F . (1.23)

For the total (kinetic) energy per particle one gets

Ekin

A=

∫ kF

0e(k)dn(k)

dk dk∫ kF

0 dn=

~2

2m

∫ kF

0k4dk

∫ kF

0 k2dk=

3

5

~2kF2

2m≡ 3

5eF . (1.24)

On the very right the Fermi energy eF has been introduced with the help ofwhich the level density g(e) can be evaluated to (recalling (1.18))

g(e) =dne

de=

3

2

√e

eF

A

eF, (1.25)

a form which is independent of gs.i. and thus valid for any Fermi gas of A particlesin three dimensions.

Let us apply these expressions to a simple picture of a heavy nucleus . Itturns out that the density in its interior takes on the value ρ0 ≃ 0.172 fm−3.Using a degeneracy factor of gs.i. = 4 this implies the following values kF ≃1.36 fm−1, ǫF ≃ 38 MeV and Ekin/A ≃ 23 MeV for Fermi momentum, Fermienergy and the total energy per particle, respectively; for the natural units ofnuclear physics see Chapter 27. As it should be, the kinetic energy per particle isa positive quantity. The total energy E/A, on the other hand, must be negative;otherwise the system would not be bound. As we shall see later the E/A is knownto be about −16 MeV. The difference has to come from the particle’s mutualinteraction which in the present formulation of the Fermi gas model was discardedentirely. Suppose this interaction may be treated on average by way of a meanfield (or potential) U . The latter then has to be negative and might be accountedfor by simply adding U to the kinetic energy Ekin (subtracting the absolute value|U |) to get E = Ekin + U ≃ −16A MeV. Thus the potential energy per particlemust be quite large, namely of the order of U/A = −Ekin/A − 16 ≃ −39 MeV.Later on we will address the problem of calculating U in more quantitativefashion.

Symmetry energy For most nuclei the neutron number differs from the protonnumber. On the level of the Fermi gas model this may be accounted for bystudying two independent gases, in which case the total energy would be givenby

Ekin =3

5eF

(e(n)F N + e

(p)F Z

), (1.26)

as follows directly from (1.24). The two different Fermi energies can be expressedby the one associated with the total particle number A = N + Z

e(n)F =

(2N

A

)2/3

eF and e(p)F =

(2Z

A

)2/3

eF . (1.27)

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The model of the Fermi gas 11

One only needs to calculate the corresponding Fermi momenta from (1.22) usinggs.i. = 4 for particle number A and gs.i. = 2 for the N neutrons and the Zprotons. Thus the total energy may be written as

Ekin =3

5

((2N

A

)2/3

N +

(2Z

A

)2/3

Z

). (1.28)

This form allows for an expansion into a power series with respect to the neutronexcess λ = (N −Z)/A. A short calculation shows that to order two this becomes

Ekin ≃3

5eFA+

1

3eF

(N − Z)2

A. (1.29)

Taking the value given above for the Fermi energy the coefficient of the symmetryterm takes on the value 13, which, as we shall see later in Section 1.3.3.2, is abouthalf the realistic one.

1.2.2 Two-body correlations in a homogeneous system

Suppose we are given a system of 2n neutrons and protons, such that the totalnumber of nucleons becomes A = 4n = 2N = 2Z, with n being an integer.The ground state is given by the Slater determinant |ΦSL

αl...αA(1, 2, . . .A))

⟩built

out of single particle states like those given in (1.15). Let us enumerate themby associating the index l = 1 . . . A with the set of quantum numbers αl. Theone-body density operator

ρ(1)FG = A tr |2...A |ΦSL

α1...αA(1, · · · , A)

⟩⟨ΦSL

α1...αA(1, · · · , A)| (1.30)

can then be seen to become

ρ(1)FG =

A∑

l=1

|αl

⟩⟨αl| =

n∑

i=1

|ki

⟩⟨ki|

4∑

µ=1

|χµ

⟩⟨χµ| . (1.31)

For formal details of the derivation we may refer to Chapter 19 (see Section 19.2.1in particular), together with Section 26.7 for the treatment of the spin-isospindegrees of freedom. In (1.31) the summation over the (4)-spinors just leads tothe unit 14 matrix in four dimensions. Therefore the density matrix becomes

ρ(1)FG(x; x ′) =

⟨x|ρ(1)

FG|x ′⟩

=14

Vn∑

i=1

eiki(x−x ′) (1.32)

This is still an operator in spin and isospin space. Averaging over these degrees

one gets ρ(1)FG(x; x ′) = tr |στ ρ

(1)FG(x; x ′) = (4/V)

∑ni=1 e

iki(x−x ′), the diagonal

matrix elements of which reduce to the space density ρ(1)FG(x; x) = A/V = ρ0.

The two-body density gives an interesting insight into those space whichare generated by the Pauli principle. Let us just sketch the calculation for the

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12 Elementary concepts of nuclear physics

simple case when spin and isospin degrees of freedoms are discarded. The two-body density matrix is given by (19.42). Inserting there the space part of theone-body density of (1.32), the diagonal elements become

ρ(2)FG(x1,x2; x1,x2) =

1

V2

n∑

k,l=1

(1− cos (2kklr)) , (1.33)

with the vectors r = x1 − x2 of relative distance and kkl = (k1 − k2)/2 ofrelative momentum (in units of ~). Obviously, the magnitude of kkl is bound bythe Fermi momentum, |kkl| = |(kk − kl)| /2 ≤ kF. To proceed further we againreplace summations over discrete indices by integrals over wave numbers makinguse of (1.14). Writing cos (kkr− klr) = coskkr cosklr + sinkkr sin klr integralsof the type

∫d3k

(2π)3cos(kr) =

1

(2π)2

∫k2dkd(cos θ) cos(kr cos θ)

=1

2π2r2

[sin(kFr)

r− kF cos(kFr)

] (1.34)

appear. Since the ones over sin kkr vanish, the two-body density becomes

ρ(2)FG(x1,x2; x1,x2) =

(ρ20

)g−(kFr) for P x

12 = −1 . (1.35)

The function g−(u) is defined as

g−(u) = 1− (C(u))2

with C(u) =3

u2

(sinu

u− cosu

). (1.36)

The P x12 introduced in (1.35) represents the permutation operator in coordinate

space. As the two particle state must be totally antisymmetric its eigenvaluemust be −1 if spin and isospin are discarded.

In the general case one must distinguish between the three different possibil-ities of permutations given by the space part, the spin and the isospin degreesof freedom. Calling the corresponding operators for permutations P x

12, Pσ12 and

P τ12, respectively one needs to have

P x12P

σ12P

τ12 = −1 . (1.37)

For a given symmetry of the space part the product of the two other permutationsmay thus be expressed as

P σ12P

τ12 =

−1 if P x

12 = +1

+1 if P x12 = −1 .

(1.38)

The second case has just been treated. A totally symmetric space part can behandled analogously. One may convince oneself by direct calculation that the

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The model of the Fermi gas 13

diagonal element of the density matrix is given by an expression like (1.33)but where the signs in front of the cosines turn into a plus. Likewise, the finalresult can be expressed like before in (1.35) with the g+(u) replaced by g+(u) =1 + (C(u))2. After averaging over spin and isospin one obtains

ρ(2)FG = ρ2

0

(1− 1

4(C(kFr))

2

), (1.39)

with the ρ0 being the actual nuclear density A/V . The factor 1/4 comes about inthe following way. The average over spin-isospin implies that the functions g− andg+ get weights of 10/16 and 6/16, respectively. There are ten possibilities withP σ

12Pτ12 = 1 and six with P σ

12Pτ12 = −1: In the former case they consist of combi-

nations of the three triplets for both spin and isospin plus the one of the spin andisospin singlets; a spin-isospin antisymmetric state requires the combination ofsinglets with triplets. It is then easy to see that the appropriate synthesis of bothcases leads to (1.39), indeed. As indicated earlier, the functional form (1.39) rep-resents those two-body correlations in the system of independent nucleons whichare there simply because of the Pauli principle. They have nothing to do withcorrelations which may come from the two-body interaction. As the C(kFr)) ap-proaches 1 for r → 0 the correlation reduces to the value of 1/4 at zero (relative)distance (it vanishes for the totally antisymmetric space part). This implies thaton average the nucleons repel each other leaving a dip in the correlation functionthe extension of which is of order r ≃ (2 − 3)/kF (where the contribution fromC2 becomes negligibly small). It is instructive to calculate the average deviation

1/ρ0

∫d3x(ρ

(2)FG − ρ2

0) of the correlation in the two-body density from the uncor-related one, which is easily seen to become −(ρ0/4)

∫d3x(C(kFr))

2 = −1. Thisresult may be interpreted by saying that in the neighborhood of each nucleonone nucleon less is found than in the uncorrelated system. For further discussionswe like to recommend the textbooks by A. de Shalit and H. Feshbach (1974) andA. Bohr and B.R. Mottelson (1969).

In momentum space the two particles fill two Fermi spheres to the Fermimomentum kF each. Of interest is the absolute value |kkl| of their relative mo-mentum, which is bound by kF. The corresponding probability distribution P (k)may be defined as

P (k) =

∫d3k1 d

3k2 δ(1

2 (kk − kl)− k

)∫d3k1 d3k2

. (1.40)

By a straightforward calculation it is readily shown to become

P (k)dk =6

πkF3

[1− 3

2

k

kF+

1

2

(k

kF

)3]

4πk2 dk for k ≤ kF . (1.41)

The P (k) tends to zero both for k → 0 as well as for k → kF and it reachesa maximal value at k ≃ 0.6kF. The most probable kinetic energy of relative

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14 Elementary concepts of nuclear physics

motion therefore is (with the reduced mass µ = mN/2) er = 2 · (0.6)2 · eF ≃27 MeV. This is the energy the two particles have in their center of mass system(defined such that their total momentum is zero). In the lab frame this wouldcorrespond to a scattering of a nucleon of twice the energy of a nucleon at rest.The maximal relative kinetic energy, which corresponds to the maximal relativemomentum kF, is twice the Fermi energy of one particle, viz of the order of80 MeV. For a nucleon–nucleon reaction in the lab this corresponds to an energyof about 160 MeV. Suppose nucleonic dynamics inside a nucleus were to showno correlations beyond those of the Fermi gas. Then it would suffice to performnucleon–nucleon scattering experiments in the lab up to projectile energies ofthat order to get the required information on the (bare) two-body interaction.However, it turns out that correlations of shorter range are important implyingthat the nucleon–nucleon interaction must also be tested at higher energies.

1.3 Basic properties of finite nuclei

An important means of measuring properties of nuclei is in reactions with otherparticles. Such measurements are undertaken in accelerator experiments, whichmay involve elementary particles or nuclear ions, but one may also simply exposenuclei to the gamma radiation and neutrons. Over the years one has been ableto accelerate heavier and heavier ions. In the present chapter we will look atthe interaction of nucleons with nuclei, mainly to get a first orientation of hownucleons propagate in a nuclear medium. Later in Chapter 13 we will returnto heavy-ion collisions to learn about essential features of such more complexreactions.

1.3.1 The interaction of nucleons with nuclei

One way of treating the collision of a nucleon N with a nucleus N is to applypotential scattering. One only has to find appropriate forms of the potentialU(r).

1.3.1.1 The approximation of a real mean field for nucleonic motion It is anestablished fact that in N-N reactions at low energies events occur with highprobability which may indeed be described simply as a propagation of the nu-cleon in the mean field of all the others–although historically it did not at alllook like that for quite some time. Indeed, already in the 1930s description ofthe interaction of a nucleon with a nucleus had been attempted by means of areal potential U(r). The problem was that in this way it was impossible to getthe narrow resonances on the scale of electronvolts seen in neutron scattering,to which we will come to shortly. One may convince oneself (see exercise 1) thatthe widths of resonances in a potential of realistic size are in the range of MeV.For this and other reasons, the model of independent particles moving in a meanfield U(r) had almost be given up. However, in 1949 and thereafter, new ev-idence about nuclear structure appeared in favor of this picture. In that yearM. Goeppert-Mayer and J.H.D. Jensen (see (1955)) suggested the inclusion of

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Basic properties of finite nuclei 15

σpot

4π/k2

Γ

ER

Cro

ss

sect

ion

Energy

Fig. 1.2. The behavior of the cross section σ(E) of eqn(1.43) in the neighbor-hood of the resonance. The cross section σpot(E) for pure potential scatteringis obtained for Γ = 0.

a strong spin-orbit interaction Uls(r). With such an interaction it was possibleto explain the so-called magic numbers. These refer to the particle number forwhich nuclei were known to be particularly stable. The breakthrough came bychoosing the sign of the spin-orbit interaction opposite to that which one knewfrom atomic physics. This observation led to the shell model. In its simplestversion the nucleons move independently of each other, which is to say that nointeraction is taken into account beyond that of the mean field U(r) + Uls(r).However, as there are interactions Vres residual to the mean field, this modelmust be considered a first approximation only.

1.3.1.2 Compound reactions The presence of the Vres implies that in realitythe reaction of N with N is a more complex process. The physical reason forthis behavior is not difficult to understand. When a N hits a N the substructureof the latter becomes relevant at practically all energies. This is an essential,qualitative difference from the NN scattering where the sub-N degrees of freedom,which actually consist of different types of particles, manifest themselves onlyat larger energies. For N-N reactions, on the other hand, the impinging nucleon“sees” partners of its own type even at low energies, with which it may performindividual NN collisions, before it will leave the nucleus again.

Typically, in such experiments one uses slow neutrons (or protons) hitting anucleus of neutron number N and proton number Z. Here ”slow” means thattheir initial kinetic energy is not larger than a few eV. When a neutron enters thenucleus the total available excitation energy is given by this kinetic energy plusits binding energy. The latter is determined by the separation energy Sn(N+1, Z)of the least bound neutron of the compound nucleus with mass number A + 1,

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16 Elementary concepts of nuclear physics

and may thus be calculated from the difference of the corresponding bindingenergies as Sn(N + 1, Z) = B(N + 1, Z) − B(N,Z). This difference typicallyranges between 5 and 10 MeV.

A picture of the angle integrated cross section as function of the incidentkinetic energy is shown in Fig. 1.2 for one resonance. It can be said to be genericfor resonances produces in reactions with slow neutron, see e.g. Fig. 2-8 of (Bohrand Mottelson, 1969). The widths of these resonances are smaller by ordersof magnitude when compared to the several MeV wide resonances of potentialscattering. Qualitatively and physically correct, this can be understood from thetime-energy uncertainty relation

δE τdelay ≃ Γ τdelay ≃ ~ . (1.42)

Estimating the δE as the width of the resonance, it is seen that the nucleon isre-emitted only after the time delay τdelay. The latter is much larger than thetime ∆t it takes (see exercise 1) for a slow neutron to cross a potential welltypical for the mean field of the nucleus. Actually, this ∆t is in the same relationto the width ∆E of the shape resonances as τdelay is to δE. As the δE is in therange of a few eV, the corresponding time delay τdelay is larger by several ordersof magnitude. This feature, together with the fact of the interaction between thenucleons being strong, led N. Bohr to describe the N-N reaction as a sequenceof individual N–N collisions rather than a motion of the incoming nucleon in themean field of the others. He compared the situation to that of an ensemble ofbilliard balls put into a shallow basin. The additional ball brought into motionwould initially collide with all, or many, of the other balls (nucleons) before oneis again leaving the ensemble with some finite probability. Put in pictorial termsthis is the essence of the model of the compound nucleus presented in a talkgiven at the Copenhagen Academy by N. Bohr (1936). As the basic hypothesis,N. Bohr assumed that the available energy is equally shared among all nucleons,as it would be in a microcanonical ensemble of statistical mechanics. As thesystem passes through such a state before it again decays the latter processshould be independent of the way the intermediate state was produced. In theliterature this statement is referred to as N. Bohr’s independence hypothesis: The“exit channel” is independent of the “entrance channel”, save for those quantitieswhich are conserved because of underlying symmetries, like total energy, totalangular momentum, parity etc. N. Bohr’s independence hypothesis is criticallyexamined in Section 18.4.4 after the cross section has been derived thoroughlywithin nuclear reaction theory in Chapter 18. In Section 18.2.4.1 the followingformula for the cross section is derived

σ(E) =4π

k2

∣∣∣ sin δ − eiδ Γ/2

E − ER + iΓ/2

∣∣∣2

, (1.43)

with k being the wave number for relative motion of energy E = ~2k2/2m.This formula determines the Breit–Wigner type resonance at E = ER of width

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Basic properties of finite nuclei 17

Γ, together with its interference with the “background”. The latter is due topotential scattering, which itself would occur with cross section (4π/k2) sin2 δ.In Fig. 1.2 the implication of the interference is clearly visible: the cross sectionfirst decreases below that of the background before it starts to rise sharply.

A closer look exhibits that for resonance scattering the Bohr hypothesis is in asense naturally fulfilled without having to rely on statistical concepts; for detailssee the discussion in Section 18.4. However, with increasing energies the reso-nances lie closer and closer, reflecting the fact that the density of levels increasesexponentially. It soon becomes infeasible to treat resonances in explicit fash-ion, as it is impossible to describe the scattering process by solving Schrodingerequations for the correct many-body wave functions. This feature is intimatelyconnected to the picture of the compound nucleus introduced above. The latternecessarily must include quantum states which are much more complex thanthose of the independent particle model. Whereas the latter are Slater deter-minants, the more complicated states involve correlations beyond those impliedby the Pauli principle. Like the example of the billiard balls, these correlationscome up in individual nucleon–nucleon collisions but manifest themselves onlyafter times much longer than the transit time through the nucleus. Within sucha picture one may distinguish between prompt events and the delayed ones of thecompound reactions. In the statistical model of nuclear reactions, developed ingreater detail in Section 18.4, one then argues that no phase correlations in theamplitudes (or T-matrices) exist which are associated with the prompt or delayedprocesses. As a consequence the cross section is just a sum of probabilities

σ = σ(P) + σ(FL), (18.133)

without interferences between the two types of reactions. Here, “P” stands for“prompt” and the suffix “FL” for “fluctuating” is to indicate that the crosssection of compound reactions varies on a much shorter energy scale than theσ(P) of the shape resonances. The bar put over the σs in (18.133) is to reflect thatwe are dealing with averaged quantities. For the statistical model this averageis to be understood as an ensemble average carried out in practice as an energyaverage. In the spirit of statistical mechanics this implies an ergodic hypothesis.The necessity of averaging over energies is already required by experimentaldemands: With increasing energy it becomes less and less feasible to measurethe structure of individual resonances in detail. This is possible only aroundand slightly above neutron threshold, i.e. when the excitation energy becomesequal to or a bit larger than the separation energy Sn. As the average spacingD of levels decreases exponentially with E one soon reaches the regime wherethe widths Γ become larger or even much larger than D, which is to say theresonances begin to overlap more or less strongly.

As indicated earlier the independence hypothesis of N. Bohr implies that thecompound reaction takes place in two steps, the production of the compoundnucleus and its subsequent decay, and that during the first stage a statisticalequilibrium is reached in the sense of a microcanonical ensemble. Of course, this

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18 Elementary concepts of nuclear physics

may happen only for sufficiently fast relaxation to equilibrium. Indeed, there isa whole class of reactions for which this condition is not fulfilled. They are calledpre-compound or pre-equilibrium processes and will be discussed in Chapter 9.

1.3.1.3 Gross and intermediate structure As indicated previously, energy av-erages may be advantageous for experimental reasons, eventually even at lowerenergies. Interesting features arise when such averages are performed on ampli-tudes, which in the end imply loss of probability flux. The total cross sectionis no longer determined by that of elastic scattering, called σel in the sequel.Rather one has to account for absorption to get

σtot = σabs + σel . (1.44)

One source of absorption is found in inelastic processes where part of the incom-ing flux may flow into other open channels. After the reaction, the two fragmentsmay differ from the initial ones in respect to e.g. particle number or excitationenergy. However, loss of probability may not only come from genuine reactions.If compound states are excited they take away flux from the prompt processes,simply because of the time delay involved for re-emission. This part of σabs is ob-tained by energy averaging the wave functions for compound elastic scattering.Altogether the absorption cross section is then given by the sum

σoptabs = σrea + σ

(C)el . (1.45)

Here, the averaging interval ∆ must be large enough to totally wipe out the de-tailed structure of the compound resonances. This ensures that no delayed pro-cesses are counted anymore. For elastic scattering, which is to say for σrea = 0,averaging the (compound) wave function delivers the only source of absorption.Due to the energy average, the Schrodinger equation for the prompt channelacquires an imaginary part for the potential, which is shown explicitly in Sec-tion 18.3.1 and demonstrated further in Section 18.4.3. Due to analogies withpropagation of waves in dispersive media this approximation goes along withthe notion of the optical model, details of which will be discussed below. Forthe inclusion of inelastic processes we refer to the literature, like e.g. (Feshbach,1992) or (Hodgson, 1994). By analogy with (1.44) the total cross section is thenwritten as σopt

tot = σoptrea +σopt

el . In Section 18.4.3.1 a simple but instructive modelis used to demonstrate the connection of the optical to the compound model.

So far variations in the cross section have been discussed which may be as-sociated with widths of resonances of very different size, the ΓGS for the grossstructure seen in the optical model, and the very small ΓCN for compound scat-tering. Structure of an intermediate nature is also seen where the typical widthsΓd lie in between the former two,

ΓGS ≫ Γd ≫ ΓCN . (1.46)

This happens when the so-called doorway mechanism is involved (for which rea-son the index “d” is used). In this case the entrance channel couples to the

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Basic properties of finite nuclei 19

“doorway state” but not directly to the more complicated compound states.The latter are only reached through doorway states. Both couplings, the onefrom the entrance- to the doorway state and from there to the compound states,may be parameterized in terms of a Γ↑d and a Γ↓d, respectively. For the sake ofsimplicity let us assume that just one open channel exists, the general case beingdiscussed in Section 18.3.2. For the elastic cross section the generalization of(1.43) then reads

σ(E) =4π

k2

∣∣∣ sin δ − eiδ Γ↑d/2

E − Ed + i2

(Γ↑d + Γ↓d

)∣∣∣2

, (1.47)

The denominator in the resonance term is still determined by the full widthΓd = Γ↑d + Γ↓d. In the numerator, however, only the width Γ↑d for the coupling tothe exit channel appears. This feature implies loss of probability, as may be seenfrom the (in this case) diagonal element of the S-matrix, which is given by

S = e2iδE − Ed + i

2

(Γ↑d − Γ↓d

)

E − Ed + i2

(Γ↑d + Γ↓d

) . (1.48)

This is easily verified from (1.47) knowing the general relation S = 1− 2πiT tothe T-matrix and that (for appropriate normalization of the wave functions) the

cross section is given by σ(E) = (4π2/k2) |T |2. Evidently, the absolute value ofS is less than unity, and, hence, flux is not conserved; for a detailed discussionof these properties see Section 18.1. A more direct and useful way of exhibitingthis “violation of unitarity” is given by looking at the so-called transmission

coefficient T = 1−S2

, for which (1.48) leads to

T (E) =Γ↑dΓ↓d

(E − Ed)2

+ 14

(Γ↑d + Γ↓d

)2 . (1.49)

1.3.2 The optical model

Above it has been argued that the optical model is introduced to describe theprompt part of the scattering of single particles at nuclei. In a microscopic sensethis requires one to perform energy averages over an interval ∆ which is largerthan both the widths ΓCN of the compound resonances (of the order of fractionsof an eV to a few hundred of keV), as well as their average spacing D. On theother hand, the ∆ must be small enough to preserve the typical structure in thesingle particle motion, which varies over ranges of several MeV. As mentionedearlier, averaging implies that the single particle potential gains an imaginarypart. In general the effective, non-Hermitian Hamiltonian will be of non-localform. In practical applications this is approximated by a local part of the type(see e.g. (Feshbach, 1992), (Hodgson, 1994) or (Mahaux and Sartor, 1991))

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20 Elementary concepts of nuclear physics

hOM =p2

2mN+ U − iW (1.50)

(with both U = U † and W = W †). Such a model was introduced into nuclearphysics by H. Bethe in (1940). Important progress in the microscopic derivationof the general optical model was achieved by H. Feshbach, C.E. Porter and V.F.Weisskopf (1954) in the early 1950s. In Section 18.3 a possible formal derivationis described using Feshbach’s projection technique. This leaves open the questionof how the microscopic input for the Hamiltonian is to be obtained upon whichthe projection technique is based. This latter question is closely related to thefundamental problem of how the nuclear many-body system is to be treatedstarting from the nucleon–nucleon interacting. This topic will be addressed inthe next chapter.

1.3.2.1 The optical model and the loss of probability The imaginary part ofthe potential is associated with loss of probability, as measured by the densityρ(r, t) = ψ∗(r, t)ψ(r, t) together with the continuity equation. For a H of theform of (1.50) the latter reads

∇ · j(r, t) +∂

∂tρ(r, t) = −2

~W (r)ρ(r, t) , (1.51)

where j(r, t) = (~/2imN)(ψ∗(r, t)∇ψ(r, t)− ψ(r, t)∇ψ∗(r, t)) is the current den-sity. The ψ(r, t) is the solution of the time-dependent Schrodinger equation incoordinate representation,

~i

∂tψ(r, t) = hOMψ(r, t) , (1.52)

with the adjoint equation for ψ∗(r, t) obtained by complex conjugation, in which

way the h†OM appears.It is helpful to look at a schematic one-dimensional case for which the U and

W are assumed constant. The stationary Schrodinger equation is then

hOMφ(x) ≡(− ~2

2mN

∂2

∂x2+ U − iW

)φ(x) = Eφ(x) (with E = E∗)

(1.53)and is satisfied by the ansatz φ ∼ exp(iKx) ≡ exp((ik−κ)x), where k is supposedto be positive, to ensure a wave propagating from left to right. Its value, aswell as that of κ, the real and imaginary parts of K, can be determined fromthe two equations ~2(k2 − κ2) = 2mN(E − U) and ~2kκ = mNW . It is seenthat for positive W the κ is positive, too. This means that a wave travelling inthe x direction experiences an attenuation according to |φ(x)|2 ∼ exp(−2κx):The imaginary part of the potential describes a decrease in intensity due to the”absorption” of particles of given energy. One may thus define a mean free path

λ =1

2κ=

~2

2mN

k

W. (1.54)

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Basic properties of finite nuclei 21

Notice that the eigenvalues E of the Hamiltonian hOM are real reflecting the factthat we are dealing with elastic scattering.

At this stage it is worthwhile to briefly mention the close analogy with elec-tromagnetic waves. Optical properties of dielectric materials are specified by thedielectric function ε(ω) which may be split into its real and imaginary parts bywriting ε(ω) = ε′(ω) + iε′′(ω). Any finite ε′′(ω) implies loss of energy of the elec-tromagnetic wave to the medium. In such a case the index of refraction n =

√ε

becomes complex (assuming the magnetic susceptibility µ to be close to unity).As in the case discussed before, the attenuation of the electromagnetic wave canbe described by introducing a complex wave vector.

1.3.2.2 The realistic three-dimensional case As indicated above, in the pastone has been able to develop methods that allow one to understand nucleardynamics on a microscopic basis. Unfortunately, this knowledge does not alwayssuffice to actually calculate basic quantities on the same level. Rather, one has torely on phenomenological approaches. One needs to invent a plausible, physicallyreasonable form for the average potential. This involves a set of parameters whichmay then be found by optimizing the evaluated cross section to fit experimentalresults best. In this way one gets important information about details of suchpotentials, which one may then try to understand from a “microscopic” point ofview. Equally important is the prediction of the outcome of other experiments,which is possible because the number of parameters is much smaller than theavailable data. To calculate the cross sections for a given potential one needsto solve numerically the Schrodinger equation to find the phase shifts and fromthem the scattering amplitude and thus the cross section. As explained in Section18.1.3, a partial wave analysis can also be applied to absorptive processes. Foran instructive example the reader is referred to the case of scattering at a blacksphere discussed in an exercise in Section 18.1.3.

Because of the importance of the spin-orbit interaction a corresponding termhas to be added to the Hamiltonian (1.50) to get

hOM =p2

2mN+ U(r) − iW (r) + Uls(r) l · s . (1.55)

The r = |~r| measures the distance of the nucleon whose motion we study to the

center of the heavy nucleus. Here, l = [r × p] and s are the operators for itsorbital angular momentum and spin, respectively. For the three functions of rappearing in (1.55) the following forms have turned out to be useful (see e.g.(Mahaux and Sartor, 1991), (Gadioli and Hodgson, 1992), (Hodgson, 1994))

U(r) = Uu(E)f

(r −Ru

au

)(1.56)

W (r) = Wv(E)f

(r −Rv

av

)− 4Ws(E)as

∂rf

(r −Rs

as

)(1.57)

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22 Elementary concepts of nuclear physics

Uls(r) = Uls1

r

∂rf

(r −Rls

als

)

If a proton interacts with the nucleus one needs to add to U(r) its Coulombpotential UC(r) in the field of the other Z protons, which are usually assumedto be distributed homogeneously. The f (x) is the Fermi function

f (xi) =1

1 + exp(xi)with xi = (r −Ri)/ai . (1.58)

The values for the radii orient themselves at the space distribution of the densityto which we will come to below. For the real part of the mean field of A particlesone has, for instance,

Ru(A) = ruA1/3 with ru = 1.25 fm (1.59)

This value of ru is slightly larger than the one for the density given below in(1.61). In this difference the finite range of the two-body interaction is hidden.Like the density, the potential has a diffuse surface determined by the parameterau, whose value is about au = 0.65 fm. The potential is uniform in the interiorand has a surface of thickness 4.4a (measuring the range where the potentialfalls from 0.9 to 0.1 of its value in the interior). In nuclear physics a potential ofthis shape is called a Woods–Saxon potential. Its depth parameter Uu and thestrength of the spin-orbit interaction are of order −50 MeV and 30 MeV fm2/~2,respectively, but they may vary with A in a different manner. Moreover, theydepend on isospin, which is to say on N −Z, and on energy. It must be said thatthese features only describe global properties. The determination of more detailsof the optical potential is indeed a subtle problem in the way microscopic aspectsare to be combined with phenomenological trial and error. The interested readermay wish to look into the review article by C. Mahaux and R. Sartor (1991).For a recent exhaustive study of this type we may refer to (Bauge et al., 1998).

For the present purpose it may suffice to consider briefly the following twoproperties of the imaginary part. As seen from (1.57) theW has a volume partWv

and a surface part Ws. Whereas the Wv is larger in the interior of the nucleus, theWs is peaked at the surface: The derivative of the Fermi function is maximal atRs

and drops on both sides over a distance determined by the as. A further remarkmay be made on the energy dependence of the functions Wv(E) and Ws(E): Atlow energies the Wv(E) is known to increase roughly linearly with E whereasthe Ws(E) decreases; a reasonable estimate for the Wv(E) is Wv(E) ≃ 0.22E forE v 10 MeV. This reflects the general trend that at larger energies the volumepart becomes more important. Later on we will discuss on a more microscopiclevel why the single particle motion becomes increasingly damped the more itsenergy deviates from the Fermi energy. In fact, along with this behavior we willencounter a similar variation with the temperature of heated systems.

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Basic properties of finite nuclei 23

1.3.3 The liquid drop model

An important characteristic for the nature of nuclear dynamics is the mean freepath λ of nucleons moving in the nuclear medium. If the λ were much smallerthan the nuclear diameter one would be inclined to conclude that the nucleusas a whole behaves as a drop of liquid. Indeed, this was the conjecture of N.Bohr and F. Kalckar (1937) drawn in analogy with the compound model. Wewill return to this question below after having examined whether or not such ahypothesis still holds true in the light of the experience one has gained in themeantime.

1.3.3.1 The puzzle of the mean free path In (Siemens and Jensen, 1987) thefollowing example is used to obtain some reasonable orientation for a typical caseat zero thermal excitations. For the interior of the nucleus the mean free pathmay be estimated on the basis of our considerations in Section 1.3.2.1. Using theestimates for Wv one finds the nucleonic mean free path to be of the order ofthe size of a heavier nucleus. For the example of a neutron of 40 MeV one getsλ ≃ 5 fm ≃ R. This is an astonishingly large number on account of the largevalue of the angle integrated cross section σ for the NN scattering. For the samevalue for the relative kinetic energy one would get σ ≃ 17 fm2. Estimating themean free path in the way one would do if the nucleons were to behave like asystem of classical particles one would find

λclass ≈1

ρσ≈ 0.4 fm ≪ λ . (1.60)

In such a case the nucleons would behave like molecules or atoms of a classicalliquid–or like the ensemble of billiard balls mentioned above. The form (1.60)assumes the subsequent collisions of the nucleons to be independent of each other,which is actually not the case as we shall argue below. Within such a picture,formula (1.60) is immediately understood if one realizes that λclassσ defines thatvolume in which on average a moving nucleon meets one scattering partner, forwhich reason one must have ρσλclass ≈ 1. To get the number shown in (1.60) thedensity has been taken to be 0.17 nucleons per fm3, see below.

In fact, even in the book by J.M. Blatt and V.F. Weisskopf (1952) it wasbelieved that the mean free path could be estimated as in (1.60). As just stated,now we know better. However, that does not mean that the concept of thecompound nucleus must be totally ruled out. It only has to be modified and putinto a more correct or appropriate quantum basis. After all, sharp resonances areseen in experiments. As indicated in the previous section, they can be explainedonly by accounting for a strong interplay of the wave function of the projectilewith an adequate many-body function of the target nucleus.

1.3.3.2 The picture of a liquid drop Many properties of finite nuclei exhibit asmooth behavior when looked at as a function of particle number or the totalenergy. In this sense the nucleus is said to exhibit “macroscopic” behavior. The

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24 Elementary concepts of nuclear physics

truth is that many of these features show similarities to those of a liquid drop.Let us discuss a few of them, beginning with the density distribution in a three-dimensional space.

The density distribution: It can be measured in scattering experiments, in par-ticular using electrons which do not underlie the strong interaction. This is ofsome advantage, as for example there will be no absorption. Using electrons ofsufficiently high energies, their scattering can be described in Born approxima-tion. This allows for an immediate deduction of the form factor, which in turn isa direct measure of the distribution of the protons. Such charge densities followlargely the functional form ρ(r) = ρ0[1 + exp(r−R/a)]−1 with the three param-eters, the densityρ0 in the interior, the radius R and the surface diffuseness a.One finds

R(A) = r0A1/3 , with r0 = 1.1− 1.15 fm and a ≃ 0.55 fm. (1.61)

From the radius the average density can be estimated to be

ρ0 =A

V =3

4πr30≃ 0.16− 0.17 nucleons fm−3 . (1.62)

Its value corresponds quite closely to the central density of a heavy nucleus. Com-bining (1.23) with (1.62), namely ρ0 = 3/(4πr30), one gets for the dimensionlessquantity r0kF the following simple relation:

r0kF =

(9π

8

)1/3

≃ 1.523 (1.63)

The fact that ρ0 is largely independent of the size of the nucleus is related tothe short range nature of the strong interaction. Inside the nuclear medium theindividual nucleon encounters other nucleons only in its immediate neighbor-hood. In this sense one speaks of ”nuclear saturation”. Unlike the electrons inan atom, on average the nucleons do not come closer when additional particlesare added. It is only that the radius changes according to (1.61). This behavioris analogous to the one of molecules in a liquid, which may not be too much ofa surprise looking at the form of the average two-body interaction. The lattershows similarities to the van der Waals force, being repulsive at short distancesand attractive at larger ones.

The static energy as function of particle number: Let us next look at the bindingenergy B(N,Z) of a nucleus of N neutrons and Z protons. It is determined bythe mass decrement of the N + Z nucleons to the M(N,Z) of the combinedsystem,

B(N,Z) = Nmnc2 + Zm2

p −M(N,Z)c2 ≡ −E(N,Z) . (1.64)

By definition the B(N,Z) is a positive quantity whose negative value we associatewith the energy E(N,Z). As a most striking feature, the variation of B(N,Z)

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Basic properties of finite nuclei 25

(or E(N,Z)) with mass number A = N+Z shows oscillations around an averagetrend of very small amplitude. Their magnitude is of the order of less than 1%.The average trend is well represented by formulas of the type

ELD(N,Z) = EvolLD + Esurf

LD + ECoulLD (1.65)

with

EvolLD = −avol

(1− κvol

(N − Z)2

A2

)A (1.66)

EsurfLD = asurf

(1− κsurf

(N − Z)2

A2

)A2/3 (1.67)

ECoulLD =

3

5

Z2e2

Rc(1.68)

and the following values of the parameters involved: avol ≃ 16 MeV, asurf ≃21 MeV, κvol ≃ 1.9, κsurf ≃ 2.3 and Rc ≃ 1.16 A1/3 fm. In the literaturesuch formulas are commonly associated with the names of H.A. Bethe and C.F.von Weizsacker. Here, we confine ourselves to a simple version just to exhibitthe essential features. In the literature much more involved forms are in use,in particular by the Berkeley–Los Alamos collaboration (see e.g. (Moller et al.,1995)). Such refinements are necessary for truly quantitative descriptions of nu-clear masses, which can be obtained after considering contributions from thequantal single particle motion; these problems will be addressed in Chapter 5.Whereas the liquid drop model treats nuclear matter as incompressible, this as-sumption is eased in the “droplet model” developed by W.D. Myers and W.J.Swiatecki (1969). Moreover, neutrons and protons are no longer assumed to havea common surface. Altogether this leads to terms additional to those shown in(1.65)–(1.68) by which the series in terms of A−1/3 continues down to 1/A (fordetails and further references see (Moller et al., 1995)). Another improved for-mulation of the traditional liquid drop formula can be found in (Pomorski andDudek, 2003).

Whereas the B itself takes on values of the order of up to about 1800 MeVfor the heaviest nuclei, the deviations from the form (1.65) are less than about10 MeV! (Commonly one plots the quantity B/A which is known to increasefrom about 5 MeV for the lightest nuclei (not counting the special cases of thedeuteron and the α particle) to about 8.8 MeV in the iron region, to drop almostlinearly afterwards down to a value of about 7.4 MeV.

The expansion (1.65) is typical for a so-called leptodermous system, domi-nated by a large volume and a relatively thin skin. Often one approximates thesmooth space density by a sharp one with a vanishing diffuseness parameter a,but the more detailed forms of (Moller et al., 1995) take into account a diffusesurface. The leading term in (1.65) is given by the so-called “volume energy”which would be there also for an infinitely large system. The main correction isproportional to the size of the surface, hence the name “surface energy”. Both

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26 Elementary concepts of nuclear physics

terms contain (comparatively small) isospin corrections. Finally, there is theenergy corresponding to a homogeneously charged sphere. The whole pictureresembles that of a charged liquid drop with a more or less pronounced sharpsurface, for which reason we have put the index LD on the binding energy BLD

of formula (1.65). The expansion (1.65) may be considered a prime example ofmacroscopic behavior which can be seen in many nuclear properties. Often suchquantities behave smoothly in respect to their variation in particle number, or inother quantities by which such properties may be parameterized such as the en-ergy or deformation of the nuclear shape. Quantum effects which accompany thesmallness of a system only show up in the fluctuations around the average trend.Although such deviations may be small, they may be decisive in understandingessential nuclear properties.

The deformed liquid drop model: According to (1.65) the liquid drop energy isgoverned by the volume energy, the surface energy and the Coulomb energy, cal-culated for spherical shapes, discarding again the other terms mentioned before.It is an established fact that the density distribution of nuclei may be deformed.The degree of deformation may be characterized by some parameter Qµ. Oneis mostly concerned with deformations which maintain the volume constant. Itis, therefore, only the surface energy and the Coulomb energy which vary withthe Qµ. The coordinate dependence of the total energy may thus be simply ex-pressed by multiplying the corresponding terms of the Bethe–Weizsacker formulaby appropriate factors, namely

EsurfLD = asurfA

2/3S0 EcoulLD =

3

5

Z2e2

RcC0 . (1.69)

The one for the surface is simply given by S0(Q) = S(Q)/(4πR2(A)). One as-sumes that the density which is constant in the interior, has the value given in(1.62) and drops to zero sharply at the radius R(A) given by (1.61). Evidently,the S0(Q) is larger than 1 for any shape Q which differs from that of a sphere.Conversely, the C0 for the Coulomb energy must be smaller than unity, becausefor a deformed system the charges lie farther apart from one another.

For small deviations from a sphere the deformation parameters may be intro-duced as follows. One parameterizes the shape as ρ(r, ϑ, φ) = ρ0 Θ(R(ϑ, φ)− r),where R(ϑ, φ) is the absolute value of the radius vector which starts at the centerof mass and which points in the direction (ϑ, φ). The function R(ϑ, φ) may beexpanded in spherical harmonics in the form

R(ϑ, φ) = R0

(1 +

λµ

αλµY∗λµ(ϑ, φ)

)(1.70)

with

αλµ = (−1)µα∗λ−µ = R−10

∫dΩR(ϑ, φ)Yλµ(ϑ, φ) . (1.71)

The R0 is the radius of the sphere (of identical volume) which in the nuclearcase is related to the particle number A by R0 = r0A

1/3; as indicated in (1.68),

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Basic properties of finite nuclei 27

for the charge density r0 ≈ 1.24. Volume conservation requires the conditionα0 = − 1

∑λµ |αλµ|2. Another one is to be put for the α1µ to ensure that the

center of mass remains at rest; proofs of these relations can be found in Appendix6A of (Bohr and Mottelson, 1975).

Denoting by X either the surface or the Coulomb energy, to order |αλµ|2 thelatter can be written in the form

EXLD(αλµ) = EX

LD +1

2

λµ

(CLD

λ

)X|αλµ|2 , (1.72)

with the spherical parts EXLD being given by (1.67) and (1.68). The stiffness for

the surface energy turns out to be

(CLD

λ

)surf

=1

4π(λ− 1)(λ+ 2)A2/3asurf , (1.73)

and the one for the Coulomb energy

(CLD

λ

)Coul

= − 3

λ− 1

2λ+ 1

e2

r0Z2A−1/3 . (1.74)

Summing up both, it is seen that for quadrupole distortions (λ = 2) and for

Z2

A>

(Z2

A

)

crit

=10

3

(r0asurf

e2

)≈ 49 (1.75)

the overall stiffness becomes negative, such that the nucleus is unstable: it maybe driven apart by the Coulomb force. This feature may be expressed throughthe fissility parameter x

x =Z2

A

(Z2

A

)−1

crit

=ECoul

LD

2EsurfLD

≈ 0.0205Z2

A. (1.76)

The instability occurs when x becomes larger than unity. Its expression in termsof the Coulomb and surface energy of the spherical drop is easily recognized from(1.67) and (1.68). These estimates go back to the famous paper by N. Bohr andJ.A. Wheeler (1939). Later in Section 8.2.2 we are will study refinements of theseproperties due to the possibility of nuclear rotations.

For quadrupole deformations (at λ = 2) it is customary to transform to thebody fixed frame by writing

α2µ =∑

ν

a2µD2µν(φ, θ, ψ) . (1.77)

Here, φ, θ and ψ are the three Euler angles which specify the body fixed framewith respect to the lab frame and D2

µν(φ, θ, ψ) is the transformation matrix.The a2µ are subject to the conditions a2−1 = a21 = 0 and a2−2 = a22. For

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28 Elementary concepts of nuclear physics

the two parameters which besides the (φ, θ, ψ) are independent it is convenientto introduce the β and γ defined through a20 = β cos γ and a2−2 = a22 =√

1/2β sin γ. The β specifies the size of the deformation and γ may be viewedas a kind of polar angle. To linear order in β the shape represents an ellipsoidwith the increment of the three principal axes given by

δRk =

(5

)1/2

β R0 cos

(γ − k 2π

3

)k = 1, 2, 3 . (1.78)

This relation allows one to understand the geometrical meaning of the parame-ters; for a detailed description, also of the underlying symmetries we may referto App.6B-1 of (Bohr and Mottelson, 1975).

1.3.3.3 The liquid drop model and the compound nucleus Already in their pi-oneering paper (1937) N. Bohr and F. Kalckar had suggested generalizing theliquid drop model to include nuclear dynamics. Drawing on analogies to the dy-namics of a drop of water it became possible to envisage that a nucleus mayperform collective motion, such as vibrating about a spherical configuration.Another and less trivial example is nuclear fission, which was discovered in thefollowing year. The liquid drop model delivers the simplest way to understandthe most basic physics of this process. The timely sequence to the advent ofthe compound model in 1936 is no accidence. Indeed, in this paper the authorsspeak of the LDM as being the classical analog to the compound picture. Ithas been said above that the compound model assumes the existence of strongcorrelations among the nucleons. Moreover, in 1937 the mean free path of nu-cleons inside the nucleus was still meant to be strictly smaller than the nucleardiameter. As mentioned before, in these days the λ was estimated on the basisof (1.60). In a classical interpretation such features are characteristic of a liquid.

We shall see later that nuclear matter behaves like a quantum liquid, whichat T = 0 is essentially non-viscous. (It is interesting to note that Bohr andKalckar also left out the complications of viscosity, although they were awarethat this might be too drastic an assumption). For higher temperatures, wherethe impact of the Pauli principle diminishes, one may expect the quantum liquidto resemble a classical one more and more. One way to see this feature is bystudying the temperature dependence of nuclear viscosity. Unfortunately, this issuch a hard task that to date there is still no convincing answer; we will returnto this problem later.

For the static interpretation of the liquid drop model, on the other hand, thereis evidence that its validity is warranted at higher temperatures. The smooth partof the total energy, as represented by the Bethe–Weizsacker formula, correspondsto the free energy when calculated on a quantal basis for temperatures of T ≈2 MeV. As we shall see later, at such excitations those quantum effects whichare responsible for the fluctuations of nuclear masses or their binding energieshave already disappeared.

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Basic properties of finite nuclei 29

1.3.3.4 First observations about nuclear transport It has been mentioned thatthe compound model of nuclear reactions assumes that in an intermediate stepthe excitation brought into the nuclear system is equally shared among all de-grees of freedom, very much in the spirit of a statistical equilibrium. It has alsobeen said that these conditions are not always met. There is a whole class of typesof reactions which go along with the concept of pre-equilibrium. This notion in-dicates that one may wish to describe nuclear dynamics as a transport process.This turns out necessary if processes are involved in which many nucleons partic-ipate ”collectively”, and which already for this reason occur on long timescales.One may then introduce the concept of partial equilibria for the “fast” degreesof freedom. That nuclei exhibit collective motion has known been at least sincethe discovery of fission and has become manifest in the 1950s for other degrees offreedom as well. Notwithstanding, it was not before the late 1970s that transportmodels became fashionable, again one might add. Indeed, after the liquid dropmodel of Bohr and Kalckar, it was H.A. Kramers who already in (1940) had setup a transport model to describe the dynamics of nuclear fission. He implied thatcollective dynamics was irreversible, tracing dissipation back to viscosity of thenuclear fluid. This hypothesis were fully justified if the mean free path λ wouldindeed be short compared to the overall size of the nucleus. It was only after thedevelopment of the shell model by M. Goeppert-Mayer and J.H.D. Jensen thatone realized the opposite were true, at least at zero or small excitations.

This short discussion shows the relevance of the microscopic understandingof the damping of single particle motion–as reflected in the imaginary part ofthe mean field, for instance–for the understanding of the nature of damping ofcollective motion. It has already been indicated that λ can be expected to becomeshorter when the nucleus gets heated up. For the moment let us take this forgranted. Evidently, when λ becomes shorter than the dimension of the nucleus,one may expect that the nuclear fluid becomes genuinely viscous, in the sense ofbeing collision dominated. Even before that happens, one may anticipate that theparticle dynamics follows transport equations similar to that of the Boltzmannequation, which is known to describe the dynamics of gases (and of liquids tosome extent) away from equilibrium. Indeed, in nuclear physics such kind oftransport equations find application as well, not only the one of Kramers.

Exercise

1. Potential scattering of s-waves: (i) Calculate the width of a resonance ap-plying time independent scattering theory.(ii) Discuss the extension to the scattering of wave packets and the associ-ation of the resonance width to a time-delay.