Basdevant J.-L., Rich J., Spiro M. Fundamentals in Nuclear Physics: From Nuclear Structure to Cosmology

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4.3 Weak interactions 201

−

neutron

200 400 600

800

electron kinetic energy (keV)

−N

) / (N +N

)

0.06

0.04

0.02

0.00

Fig. 4.10. The neutron β decay asymmetry for polarized neutrons. About 5%

more neutrons are emitted in the direction of the neutron spin than opposite the

direction of the neutron spin. This indicates that parity is violated in β-decay. This

is demonstrated in the bottom ﬁgure where a spinning neutron decays with the

electron emitted in the direction of the neutron (spin) angular momentum. Viewed

in the mirror, the spin is reversed but the direction of the electron is not. The

excess of electrons emitted in the direction of the neutron spin becomes, viewed

in the mirror, an excess of electrons emitted opposite the direction of the neutron

spin. What is viewed in the mirror does not correspond to the real world, indicating

that physics in the real world does not respect parity symmetry.

202 4. Nuclear decays and fundamental interactions

This is, again, an indication of parity violation since helicity is reversed if

viewed in a mirror. It turns out that ν

emitted in β

−decay or electron-

capture have negative helicity. The experimental demonstration of this will

be discussed in Sect. 4.3.4.

Finally, we mention the origin of the factor 2.4:

2.4G

∼ cos θ

(1 + 3g

)/2 . (4.81)

This formula can only be understood within the framework of the complete

relativistic theory but we see that the constant g

∼ 1.25 comes from the

spin-dependent couplings. Its non-integer value can be understood from the

underlying theory of quark decay. The Cabibbo angle,cosθ

=0.975 ±0.001,

comes from the mixing of quarks as discussed in Sect. 4.4.3.

4.3.2 β-decay of nuclei

As we already emphasized, β-radioactivity of nuclei stems from the funda-

mental processes

n → pe

−

, p → ne

. (4.82)

As illustrated in Fig. 4.11, it is useful to think of nuclear β-decay as neutron

or proton β-decay to empty orbitals. The decaying nucleons are considered

as moving in a ﬁxed nuclear potential due to the other nucleons. If there

is only one Pauli-unblocked nucleon that is in a position to decay (e.g.

in Fig. 4.11), the matrix element is the same as (4.66) except that the nu-

cleon wavefunctions are not plane waves but rather normalized bound state

wavefunctions:

(A, Z + 1)e

−

|(A, Z) =

2.4G



rψ

∗

(r)ψ

(r) exp[−i(p

+ p

) · r/¯h] . (4.83)

As in neutron decay, we write 2.4G

to simulate a more complicated spin-

dependent matrix element that can be calculated from the underlying rela-

tivistic theory.

The matrix element (4.83) involves the Fourier transform of the product

of the initial and ﬁnal nucleon wavefunctions. A useful approximation comes

about by noting that the typical lepton wavelengths, 2π¯hc/1MeV ∼ 10

are much greater than nuclear sizes, A

1/3

fm, i.e. much greater than the extent

of the nucleon wavefunctions in (4.83). We can therefore replace the lepton

wavefunctions by their value at r = 0 and we have

(A, Z + 1)e

−

|(A, Z) =

2.4G



r ψ

∗

(r)ψ

(r) . (4.84)

The matrix element is proportional to the overlap of the initial and ﬁnal state

nucleons.

4.3 Weak interactions 203

+14

O 14

N 1/2

C 1/2

Fig. 4.11. Independent particle picture of nuclear β-decay. The top panel shows

the decay

N →

. The decay can be considered to be the β-decay of the

valence proton that moves in the ﬁeld of an inert

C core. The decay is shown

with no spin-ﬂip but both ﬂip and non-ﬂip processes are possible. The bottom

panel shows the decay

O →

. This decay is predominantly to the ﬁrst

excited state (0

)of

N followed by the radiative decay to the ground state (1

)

(see Fig. 4.12) . For the decay of

O, there are two valence protons, either of which

can decay to a neutron.

The calculation of the decay rate then proceeds in the same manner as for

free neutron decay except that momentum conservation no longer holds. This

is because the initial and ﬁnal nuclei are represented by a ﬁxed potential in

which the decaying nucleon is conﬁned. The ﬁxed nuclear potential therefore

has eﬀectively an inﬁnite mass. This is just as in single-particle scattering

in a ﬁxed potential where there was also no momentum conservation. The

diﬀerentialdecayrateis

dλ =

8π

¯h



2.4G

(¯hc)



|M|

δ(∆mc

− E

, (4.85)

where ∆m is the diﬀerence in nuclear masses and the matrix element M is

M =



∗

(r)ψ

(r)d

r . (4.86)

The diﬀerential rate (4.85) is the same as the neutron rate (4.68) except the

−m

is replaced with m

A,Z

−m

A,Z±1

= ∆m and a factor |M|

is added.

204 4. Nuclear decays and fundamental interactions

N Mg S

100%

24%

28%

47%

100%

97%

0.06%

99.3%

0.6%

70.6 s

0.72 My

6.36 s

32.0 m

1.5 s

E (MeV)

Fig. 4.12. The β-decays

O →

Al →

Mge

, and

Cl →

O has three allowed decays, to the ground state (Q

=5.14 MeV) and to the

2.31 MeV and 3.95 MeV excited states. The highest branching fraction is for the

super-allowed decay to the 0

excited state.

Al has two β-decaying states, the

ground state and the 0+ (E =0.228 MeV) isomeric state. The later decays

via a super-allowed decay to the ground state of

Mg (Q

=4.233 MeV). The

ground state has no allowed decays, explaining its long lifetime, t

1/2

=7.2 ×10

yr.

For

Cl, the roles of the isomer (E =0.146 MeV) and ground states are reversed

with the ground state decaying through the super-allowed mode (Q

=5.49 MeV).

The isomer has two allowed β-modes to the 2

excited states of

S. The radiative

transition to the ground state of

Cl has a 47% branching fraction making this a

mixed β-radiative-decay.

Integrating over the neutrino momenta, we ﬁnd the electron energy spec-

trum

dλ

¯h



2.4G

(¯hc)



|M|

(∆mc

− E)



− m

E. (4.87)

The spectrum is the same as that in neutron decay with the replacement

− m

→ ∆m.

By integrating over the electron energy we get the total decay rate.

λ =

¯h



2.4G

(¯hc)



|M|



∆mc

(∆mc

− E)



− m

EdE. (4.88)

4.3 Weak interactions 205

Σ n

−4

np 14

26m

AlAl

ΣΛ

100

Q(MeV)

λ (sec

−

)

Fig. 4.13. β transition rates for super-allowed β-decays as a function of the max-

imum kinetic energy of the ﬁnal electron. The decays include elementary particles

as well has nuclei where symmetry requires the integral (4.86) to be unity. The

solid line shows the phase-space integral with the dashed line corresponding to the

high-energy value Q

. The super-allowed β

emitters,

26m

Al,

Cl,

38m

Sc,

Mn, and

Fe fall slightly below the line because Coulomb corrections re-

duce the rate. The strangeness-changing decays like Λ → p fall rather far below the

extrapolation because these decays are suppressed by the Cabibbo factor sin

as discussed in Sect. 4.4.3.

TherateasafunctionofQ

=(∆m − m

is shown in is shown in Fig.

4.13 in the case |M|

∼ 1. For Q

 m

, the rate goes like the ﬁfth power

of the decay energy:

λ ∼|M |

. (4.89)

Generally, one has |M|

 1, reducing considerably the rate below the

rate calculated by extrapolating from free-neutron decay. However, in certain

circumstances, isospin symmetry requires that the initial and ﬁnal wavefunc-

tion overlap nearly perfectly so that |M|

∼ 1. Such decays are called super-

allowed decays. Three examples are shown in Fig. 4.12 and their rates are

shown in Fig. 4.13. Note that for β

(β

−

) decays, the Coulomb modiﬁcations

discussed below make the |M|

= 1 rates a bit below (above) that expected

by extrapolating from neutron decay.

In order to calculate more reliable decay rates, it is necessary to modify

(4.84) to take into account the spin of the nucleons and leptons. As in the case

of neutron decay, there are two important terms, the Fermi term proportional

206 4. Nuclear decays and fundamental interactions

to the overlap integral (4.84), and the Gamow–Teller term proportional to

the matrix element of σ between initial and ﬁnal state nuclei.

Just as in radiative decays, there are selection rules governing which com-

binations of initial J

and ﬁnal J

spins are possible. The Fermi term will

vanishes if the angular dependences of the initial and ﬁnal wavefunctions are

orthogonal so we require

Fermi : J

= J

. (4.90)

The Gamow–Teller term can change the spin but vanishes if the initial and

ﬁnal angular momenta are zero:

GT : J

= J

± 1 J

= J

= 0 forbidden . (4.91)

Additionally, in both cases, the parity of the initial and ﬁnal nuclei must be

the same. Transitions that respect the selection rules are called “Allowed”

decays. “Forbidden” decays are possible only if one takes into account the

spatial dependence of the lepton wavefunctions, i.e. using (4.83) instead of

(4.84) The examples of forbidden decays in Fig. 4.12 illustrate the much

longer lifetimes for such transitions.

p (MeV/c)

0.2

0.2 0.6 1.0

1.4 1.8

1.8

1.4

1.00.6

Fig. 4.14. The β

−

and β

spectra of

Cu [44]. The suppression the of the β

spectrum and enhancement of the β

−

at low energy due to the Coulomb eﬀect is

seen.

4.3 Weak interactions 207

Finally, it is necessary to take into account the fact that in the presence

of a charged nucleus, the energy eigenfunctions of the ﬁnal state e

are not

plane waves but are suppressed (enhanced) near the nucleus for e

−

This has the eﬀect of suppressing β

decays at low positron energy and of

enhancing β

−

decays at low electron energy. This eﬀect is clearly seen in Figs.

4.13 and 4.14. The spectrum (4.87) is modiﬁed as

dλ

∝ F (Z



) E



− m

(∆mc

− E

)

, (4.92)

where F (Z



) the “Coulomb correction.” Tabulated values can be found

in [43].

4.3.3 Electron-capture

All atomic nuclei that can decay by β

emission can also capture an atomic

electron thereby decaying via

(A, Z)e

−

→ (A, Z − 1) ν

. (4.93)

This process is illustrated in Fig. 4.15. The neutrino energy is

=(m(A, Z) − m(A, Z − 1)]c

+2m

= Q

β+

+2m

. (4.94)

Electron capture is the only decay mode possible if Q

β+

< 0andQ

β−

< 0,

i.e. if the nuclear masses of neighboring isobars diﬀer by less the the electron

mass.

The eﬀective Hamiltonian is

(A, Z − 1)ν

|(A, Z)e

−

 =

2.4G

3/2



rψ

∗

(r)ψ

(r) exp i[p

· r/¯h] . (4.95)

As in the case of nuclear β-decay, we can usually make the approximation

that the neutrino and electron wavefunctions are constant over the nucleus

so that

(A, Z +1)ν

|(A, Z)e

−

 =

2.4G

3/2

(r =0)



r ψ

∗

(r)ψ

(r) .

This gives a decay rate

λ =

π(¯hc)

(2.4G

)

|ψ

(0)|

|M|

. (4.96)

The inner-most atomic electrons experience only the nuclear electric ﬁeld so

their wavefunctions are hydrogen-like except that the potential is ∼ Ze

/r in-

stead of ∼ e

/r. The eﬀective Bohr radius is then a factor Z smaller implying

that the wavefunction at the origin is a factor Z

larger:

|ψ

(0)|

∼

. (4.97)

208 4. Nuclear decays and fundamental interactions

mlk

(A,Z)

(A,Z−1)

a) b)

Fig. 4.15. Electron capture. After the nuclear transformation, the atom is left

with an unﬁlled orbital, which is subsequently ﬁlled by another electron with the

emission of photons (X-rays). As in the case of nuclear radiative decay, the X-ray

can transfer its energy to another atomic electron which is then ejected from the

atom. Such an electron is called an Auger electron.

The decay rate is then

λ =

π(¯hc)

(2.4G

)

|M|

. (4.98)

Compared with nuclear β-decay, the Q dependence is weak, Q

rather than

. This means that for small Q

, electron-capture dominates over β

decay,

as can be seen in Fig. 2.13. The strong Z dependence coming from the de-

creasing electron orbital radius with increasing Z means that electron-capture

becomes more and more important with increasing Z.

Finally, we note that nuclear decay by electron capture leaves the atom

with an unﬁlled atomic orbital. This orbital is ﬁlled by other atomic electrons

falling into it and radiating photons. The photons are in the keV (X-ray)

range since the binding energy of the inner most electron of an atom of

atomic number Z is

E ∼ 0.5Z

=0.01 Z

keV . (4.99)

4.3 Weak interactions 209

810

12 14

30−

counts

E (keV)

t > 50 days

t < 50 days

l−capture

k−capture

Fig. 4.16. The spectrum of Auger electrons from the electron-capture decay

Ge as measured by the Gallium Neutrino Observatory (continuation of the

GALLEX project). The spectrum is measured with a small gas-proportional counter

ﬁlled with a Xe − GeH

mixture. A small number of the germanium nuclei are ra-

dioactive

Ge nuclei (t

1/2

=11.4day) produced by solar neutrinos as described in

Sect. 8.4.1. Two peaks, corresponding to K- and L-capture, are observed in the ﬁrst

50 days of counting. After 50 days, most of the

Ge has decayed and the counts

are due to ambient radioactivity due to impurities in the counter.

As in the case of nuclear radiative decays, these photons can transfer their

energy to another atomic electron that is then ejected from the atom. Such

an electron is called an “Auger electron.” Since the only high-energy particle

emitted in electron capture is a neutrino, X-rays and Auger electrons are

generally the only sure way to signal a decay be electron capture. A typical

spectrum is shown in Fig. 4.16

4.3.4 Neutrino mass and helicity

Studies of nuclear β-decay have revealed two interesting properties of neu-

trinos. The most obvious is its small mass. This is seen in the fact the the

energy spectrum of electrons is consistent with (4.87) which was calculated

assuming m

= 0. A non-zero neutrino mass modiﬁes this distribution in two

ways. First, the maximum electron energy is lowered

(max) = (m

A,Z

− m

A,Z+1

− m

. (4.100)

210 4. Nuclear decays and fundamental interactions

Second, the shape of the electron energy spectrum (4.92) is modiﬁed, espe-

cially near the “end point,” i.e. the maximum electron energy:



− m

dλ

∝ (E

− E

)



− E

)

− m

, (4.101)

where E

=(m(A, Z) − m(A, Z



))c

is the maximum electron energy in the

case m

= 0. A plot of the square root of the left-hand side vs. E

is, for

= 0, a straight line intersecting zero at E

= E

. Such a plot is called a

Kurie plot. As shown in Fig. 4.17, this causes the Kurie plot to curve down

near the endpoint.

d λ / dE

1/2

Fig. 4.17. The Kurie plot near the electron maximum energy in the case of m

(dashed line) and m

= 0 (solid line).

Very precise measurements of the electron energy spectrum have been

made for tritium β-decay

H →

He e

−

=18.54 keV . (4.102)

Tritium is chosen since the low value of Q

facilitates accurate measurements

of E

near the end point.

The most precise experiment to date [45] is shown in Fig. 4.18 which mea-

sures the electron spectrum with an electrostatic spectrometer. The agree-

ment of the observed spectrum with that expected for massless neutrinos,