Povh B., Rith K., Scholz Ch., Zetsche F. Particles and Nuclei: An Introduction to the Physical Concepts

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282 17 The Structure of Nuclei

One likes to compare the (2ν)andthe(0ν) life-times for m

= 1 eV and

= 1 MeV. For these values of the neutrino mass we have

0ν

≈ 3 · 10

years, (17.62)

much longer than τ

2ν

. Of course as was said earlier this is just a crude estimate

and the real value could be factors of 10 diﬀerent and 1 eV/c

is certainly

an upper bound on the Majorana mass of the electron neutrino. This long

lifetime shows how diﬃcult the search for neutrinoless double β-decay will

be.

One setup used in experiments is the well-shielded Germanium counter.

One of the Germanium isotopes,

Ge, undergoes the double β-decay into

Se. In Fig. 17.24, the energy spectrum of the charged particles in the dou-

ble β-decay is shown. Neutrinoless decay would demonstrate itself in the

monoenergetic line with the full energy of transition. Recently a new analysis

of the results collected in the span of ten years by the Heidelberg-Moscow

Collaboration on the double β-decay of

Ge were published ([Kl02]). The

measurements have been performed in the Gran Sasso underground labora-

tory (1500 meters under ground) using Germanium counters enriched in

content to 86%. At the energy of the expected total energy peak 16 events

were observed. At the same time in the continuum of the (2ν) double β-decay

113764 events were counted. From these numbers the following lifetimes of

Ge can be deduced. For the (2ν) decay

2ν



1.74 ± 0.01(stat)

+0.18

−0.16

(syst)



· 10

years, (17.63)

and for the (0ν) decay

0ν



1.5

+1.68

−0.7



· 10

years. (17.64)

In this experiment the estimate of the neutrino mass, in fact the Majorana-

neutrino mass, is

=0.39

+0.45

−0.34

eV. (17.65)

Fig. 17.24. The continuum is

due to the sum energy of the

two charged leptons in the (2ν)

decay, the monoenergetical line

with the full energy of the tran-

sition comes from the (0ν)de-

cay.

Problems 283

In view of the large uncertainties in the background subtraction and very low

statistical signiﬁcance of the published results of [Kl02], measurements with

statistics that are at least an order of magnitude higher would be desirable as

to conclude beyond doubt that the neutrinos are in fact Majorana particles.

All possibilities that allow neutrinoless double β-decay require non-trivial

extensions of the standard model: massive Majorana neutrinos, right-handed

coupling of the weak interaction, or doubly charged Higgs particle. Any of

these possibilities would illuminate the path to the goal for a grand uniﬁed

theory of particle physics.

Problems

1. Fermi gas model

Calculate the dependence of the Fermi pressure upon the nuclear density. How

large is this pressure for a density 

= 0.17 nucleons/fm

? What is this in

macroscopic units (bar)?

2. Shell model

a) In the following table we present the experimentally determined spins and

parities of the ground states and ﬁrst excited states of some nuclei:

3/2

−

3/2

7/2

−

9/2

1/2

−

5/2

1/2

3/2

7/2

1/2

−

Find the conﬁgurations of the protons and neutrons in the incomplete shells

of the one-particle shell model for these nuclei and predict the quantum

numbers of their ground states and ﬁrst excited levels. Compare your results

with the table.

b) The spins of odd-odd nuclei are generally given by a vector addition of the

total angular momenta of the two unpaired nucleons. Which possible nuclear

spins and parities should

Li and

K have? Experimentally these nuclei have

the quantum numbers 1

and 4

−

3. Shell model

a) Find the gap between the 1p

1/2

und 1d

5/2

neutron shells for nuclei with mass

number A ≈ 16 from the total binding energy of the

O (111.9556 MeV),

O (127.6193 MeV) and

O (131.7627 MeV) atoms [AM93].

b) How does this agree with the energy of the ﬁrst excited level of

O(cf.

Fig. 17.7)?

c) What information does one obtain from the energy of the corresponding state

d) How do you interpret the diﬀerence in the total binding energies of

O und

F? Estimate the radius of these nuclei.

e) The ﬁrst excited state of

F is below the equivalent state of

O. A possible

explanation of this is that the unpaired nucleon has a diﬀerent spatial ex-

tension (smaller?, larger?) in the ﬁrst excited state than in the ground state.

What do you expect from considering the quantum numbers?

284 17 The Structure of Nuclei

4. Shell model

It is conspicuous that many of the nuclei which possess long lived isomer states

have N or Z in the ranges 39 ···49 and 69 ···81 [Fe49, No49, Go52]. Why is

this?

5. Magnetic moment

The

Sc nucleus has a low lying level with J

(I)=7

(0) and an excitation

energy of 618 keV.

a) Which shell model conﬁguration would you assign this state to?

b) What magnetic moment would you expect?

6. The Goldhaber experiment

152

Sm possesses a state with excitation energy 0.963 MeV and quantum numbers

−

which decays via an E1 transition into the ground state.

a) How large is the recoil energy of the nucleus?

b) Compare this energy with the width of the state which is equivalent to an E1

one particle transition probability. Can a so-emitted photon be absorbed by

another nucleus? What happens is we take the inﬂuence of thermal motion

into account?

c) Show that this energy loss is compensated if the excited

152

Sm nucleus was

produced in an electron capture decay of

152

Eu and the photon was emitted

in the recoil direction of the

152

Sm nucleus.

The energy of the emitted neutrino is 0.950 MeV.

7. Coupling strength of β-decay

A maximal energy of E

max

kin

= 1810.6 ± 1.5 keV is measured in the β-decay

O →

N+e

+ ν

(Fig. 2.6) [EL92, Wi78]. A phase space function f(Z



)

of 43.398 is calculated from this [Wi74]. What half-life should

O have? The

experimental value is t

1/2

= 70 606 ± 18 ms [Wi78].

18 Collective Nuclear Excitations

We showed in Sect. 17.3 that the nuclear ground states may be well described

if we assume that the nucleons are in the lowest shell model orbits. The single

particle picture, we further showed for the case of a single valence nucleon

or nucleon hole, works very well if shells are nearly full or empty. Excited

states are then understood as being created by a valence nucleon jumping

into a higher shell model state; a direct analogy to our picture of the atom.

As well as such straightforward single particle excitations, more complicated

phenomena can take place in the nucleus. Collective excitations provide some

of the most beautiful aspects of nuclear dynamics.

Collective excitations of many body systems can be phenomenologically

understood as ﬂuctuations around a state of equilibrium. These may be ﬂuc-

tuations in the density or shape. The type of collective excitation strongly

depends upon the composition of the system and the manner in which its

components interact with each other. We want now to show the connection

between nuclear collective excitations and the forces inside and the structure

of the nucleus.

Electromagnetic transitions provide us with the most elegant way to in-

vestigate collective excitations in nuclei. We will therefore ﬁrst consider how

electromagnetic transitions in nuclei may be determined, so that we can then

say to what extent collective eﬀects are responsible for these transitions.

The ﬁrst measurements of photon absorption in nuclei led to the discov-

ery that the lion’s share of the the absorption is by a single state. The ﬁrst

description of this giant dipole resonance state was of an oscillation of the pro-

tons and neutrons with respect to each other. Later on it was discovered that

the transition probability for electric quadrupole transitions of lower energy

states was much higher than a single particle picture of the nucleus predicts.

The transition probability for octupole transitions also predominantly comes

from single states which we call octupole vibrations.

The single particle and collective properties of nuclei were regarded for

a long time as distinct phenomena. A uniﬁed picture ﬁrst appeared in the

1970’s. We want to illustrate this modern framework through the example

of giant dipole resonances. What we will discover can be easily extended to

quadrupole and octupole oscillations.

Another important collective eﬀect is the rotation of deformed nuclei.

Such rotations form a most pleasing chapter, both didactically and aestheti-

cally, in the story of γ spectroscopy.

286 18 Collective Nuclear Excitations

18.1 Electromagnetic Transitions

Electric dipole transitions. The probability of an electric dipole transition

can be somewhat simplistically derived by considering a classical Hertz dipole.

The power output emitted by the dipole is proportional to ω

. The rate of

photon emission, i.e., the transition probability, may be obtained by dividing

the power output by the photon energy ω. One so ﬁnds

3πε







xψ

∗

xψ



, (18.1)

where we have replaced the classical dipole ex by the matrix element. This

result may also be obtained directly from quantum mechanics.

In the following derivation we want to treat the electromagnetic transitions

semiclassically, i.e., we will not concern ourselves with quantising the radiation

ﬁeld or spin.

Consider ﬁrst an excited nuclear state ψ

which through γ emission enters a

lower lying state, ψ

. The golden rule says that the transition probability is

dW =

2π



|ψ

int

|ψ

|

d(E) . (18.2)

int

describes the interaction of the moving charge with the electromagnetic ﬁeld

and (E) is a phase space factor that describes the ﬁnal state density at total

energy E. For photon emission we have E = E

.Sinceγ radiation is generally not

spherically symmetric, we consider the phase space in a solid angle element dΩ

around the momentum vector. As in (4.16) we set

d(E)=

V |p|

d|p| dΩ

(2π)

. (18.3)

For the photon we have E = c ·|p| and dE = c · d|p|, which implies

d(E)=

V dΩ

(2πc)

. (18.4)

The H

int

operator can be obtained by considering the classical Hamiltonian for

the interaction between a charge e, which emits the photon, and the electromagnetic

ﬁeld A =(φ/c, A)[Sc95]:

H =

(p − eA)

+ eφ. (18.5)

Note that we have here assumed a point-like charge. The term quadratic in A is

negligible and we may write

H =

−

pA + eφ . (18.6)

18.1 Electromagnetic Transitions 287

The ﬁrst term corresponds to free movement of the charged particle and the last

two describe the interaction

int

= −

pA + eφ , (18.7)

which, for a point-like particle, is just given by the scalar product of the electric

four-current

j =(e · c, ev) (18.8)

and the electromagnetic ﬁeld

A =(φ/c, A) . (18.9)

In an electromagnetic decay eφ does not contribute to the transition probability,

since real photons are transversely polarised and monopole transitions are hence

forbidden.

If one replaces the momentum p by the operator p = − i∇ and interprets the

vector A as the wave function of the photon, one obtains the matrix element

ψ

int

|ψ

 = −

ie

xψ

∗

∇ψ

A . (18.10)

The gradient ∇ may be replaced by the commutator of the coordinate x with the

Hamilton operator, since for stationary states

+ V (x) (18.11)

we have the following relation:

x H

−H

x =

i

p =



∇. (18.12)

In this way we have

−



xψ

∗

(xH

−H

x) ψ

A =



− E

)

xψ

∗

x ψ

A , (18.13)

and the matrix element has the standard form for multipole radiation.

In the semiclassical derivation of γ emission, one writes the photon wave func-

tion as

A =



2ε

ωV

ε cos(kx − ωt) , (18.14)

where ε is the polarisation vector of the photon, E

= ω is its energy and k the

wave vector. That this is indeed correct may be easily checked by calculating the

electromagnetic radiation energy in a volume V using A from (18.14):

ω = V ·

„

= Vε

with E = −

∂A

∂t

, (18.15)

where the bar represents time averaging. With this result we now may write the

transition probability as:

2π



2ε

ωV



xψ

∗

xψ

ikx

V dΩ

(2πc)

8π



xψ

∗

x e

ikx

dΩ. (18.16)

288 18 Collective Nuclear Excitations

The wavelengths of the gamma rays are large compared to a nuclear radius.

The multipole expansion

ikx

=1+ikx + ··· (18.17)

is very useful, since, generally speaking, only the lowest transition that the quantum

numbers allow needs to be taken into account. Only very occasionally are two

multipoles of equal strength in a transition. If one now sets e

ikx

≈ 1, integrates

(18.16) over the solid angle dΩ and the polarisation one obtains (18.1).

Electric dipole (E1) transitions always connect states with diﬀerent parities.

The photon carries away angular momentum || =1 and so the angular

momenta of the initial and ﬁnal states may at most diﬀer by one unit.

Since transitions from one shell into the one immediately above play the

most important role in collective excitations, we now introduce the standard

notation for the wave function. A closed shell shall be denoted by the symbol

|0 (“vacuum wave function”). If a particle in the state φ

of the closed shell

jumps into the state φ

of the next shell a particle-hole state is created,

which we symbolise by |φ

−1

. The dipole matrix element

φ

−1

|ex|0 = e



xφ

∗

xφ

(18.18)

describes the transition of a nucleon from the state φ

to the state φ

.Since

|0 is a full shell state it must have spin and parity J

, hence the

excited particle-hole state after the electric dipole transition must have the

quantum numbers J

−

Magnetic dipole transitions. The transition probability of a magnetic

dipole (M1) transition is obtained by replacing the electric dipole in (18.1)

by a magnetic one:

3π





xψ

∗

µψ



where µ =

(L + gs) .

(18.19)

Here L is the orbital angular momentum operator and s is the spin operator.

Higher multipoles. If the electric dipole transition is forbidden, in other

words if both states have the same parity or the vectorial addition of the

angular momenta is inconsistent, then only higher multipole radiation can be

emitted. The next highest multipoles in the transition probability hierarchy

are the above magnetic dipole (M1) transition and the electric quadrupole

(E2) transition [Fe53]. Both are second order in the expansion (18.17). The

parity of the initial and ﬁnal states must be identical in electric quadrupole

transitions and the triangle inequality |j

−j

|≤2 ≤ j

+ j

must be fulﬁlled

by the angular momenta. While the transition probability for dipole radiation

is, from (18.1), proportional to E

, for electric quadrupole radiation it goes

as E

. This is because there is a new factor of ikx in the matrix element and

18.2 Dipole Oscillations 289

|k| is proportional to E

. The energy-independent part of the matrix element

has the form r

(θ, ϕ).

18.2 Dipole Oscillations

Photon absorption in nuclei. A broad resonance, which was already

known in the 1950’s, dominates the absorption of gamma rays by nuclei. The

experimental techniques for investigating this resonance were rather awkward

since no variable energy gamma sources existed.

The method of in ﬂight positron annihilation, which was developed in the

1960’s, ﬁrst permitted detailed measurements of the gamma cross-sections.

Positrons, which have been produced through pair creation from a strong

bremsstrahlung source, are selected according to their energy and focused

upon a target. They then partially annihilate with the target electrons and

produce bremsstrahlung as an unwanted by-product (Fig. 18.1).

Such a gamma spectrum is shown in Fig. 18.2. A peak can be clearly

distinguished from the bremsstrahlung at the maximal possible energy and

this is presumed to come from the e

−

annihilation. The energy dependence

of γ-induced cross-sections can be thoroughly investigated by varying the

energy of the positrons. As well as the total cross-section, the cross-section

for the photoproduction of neutrons (nuclear photoeﬀect)

X(γ,n)

A−1

X (18.20)

Electron beam

Photons

Positrons

Fig. 18.1. Experimental set up for in ﬂight positron annihilation (from [Be75]). An

electron beam hits a target (T

). The bremsstrahlung that is produced converts into

electron-positron pairs. The positrons are then selected according to their energy

by three dipole magnets (M

) before hitting a second target (T

). Some

of them annihilate in ﬂight with target electrons. A further magnet (M

) deﬂects

all charged particles and only photons arrive at the experimenter’s real target (S).

290 18 Collective Nuclear Excitations

100

200

300

400

100

200

300

0.6 0.8 1.0 1.2

e–

e+ –e–

Counts

100

200

300

400

[MeV]

Fig. 18.2. The photon spectrum

from in-ﬂight electron positron anni-

hilation [Be75]. This is later used for

(γ,n) reactions. The background of

bremsstrahlung from positrons hitting

the target is determined by aiming a

mono-energetic beam of electrons at

the target. The cross-section for ﬁxed

photon energies is found by performing

experiments with the two diﬀerent pho-

ton beams and subtracting the count-

ing rates.

is of especial importance. This is in fact the major part of the total cross-

section. The photoproduction of protons is, by contrast, suppressed by the

Coulomb barrier. In what follows we will limit ourselves to the (γ, n) reaction.

We have chosen σ(γ, n) for neodymium isotopes as an example (Fig. 18.3).

Various observations may be made.

– The absorption probability is centred in a resonance which we call a giant

resonance.

– The excitation energy of the giant resonance is roughly twice the separation

between neighbouring shells. This is astounding since, for reasons of parity

and angular momentum conservation, many more single particle transitions

are possible between one shell and the next than between a shell and the

next but one.

– While a narrow resonance is observed in absorption by

142

Nd, this splits

into two resonances as the mass number increases.

– The integrated cross-section is about as big as the sum over all expected

cross-sections for the transition of a single nucleon from the last closed

shell. This means that all the protons and neutrons of the outermost shell

contribute coherently to this resonance.

18.2 Dipole Oscillations 291

300

250

200

150

100

[mb]

150

148

146

145

144

143

142

6 8 10 12 14 16 18 20 22

[MeV]

Fig. 18.3. Cross-section for γ-induced emission of neutrons in neodymium isotopes

[Be75]. The curves have been shifted vertically for the sake of clarity. Neodymium

isotopes progress from being spherically symmetric to being deformed nuclei. The

giant resonance of the spherically symmetric

142

Nd nucleus is narrow, while that of

the deformed

150

Nd nucleus shows a double peak.

A qualitative explanation of the giant resonances comes from the oscil-

lation of protons and neutrons with respect to each other (Fig. 18.4). The

150

Nd is deformed and has a cigar-like shape. The two maxima for this nu-

cleus correspond to oscillations along the symmetry axis (lower peak) and

orthogonal to it (higher peak).

We will attempt to justify this intuitive picture of giant resonances and

their excitation energies in the framework of the shell model.

The giant dipole resonance. Consider once again the example of the

doubly magic

O nucleus. B Let us assume that photon absorption leads to

a nucleon in the 1p

3/2

or 1p

1/2

shell being promoted into the 1d

5/2

,1d

3/2

or 2s

1/2

shell. If this nucleon drops back into the 1p shell, it can pass on

its excitation energy through recoils to other nucleons, which may then, for

example, be themselves excited out of the 1p shell into the 1d or 2s shell.

If the nuclear states that are produced by the excitation of a nucleon into a