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Chapter 15
β Decay
Note to students and other readers: This Chapter is intended to supplement Chapter 9 of Krane’s excellent book, ”Introductory Nuclear Physics”. Kindly read the relevant sections in Krane’s book first. This reading is supplementary to that, and the subsection ordering will mirror that of Krane’s, at least until further notice.
β-particle’s are either electrons^1 or positrons that are emitted through a certain class of nuclear decay associated with the “weak interaction”. The discoverer of electrons was Henri Becquerel, who noticed that photographic plates, covered in black paper, stored near ra- dioactive sources, became fogged. The black paper (meant to keep the plates unexposed) was thick enough to stop α-particles, and Becquerel concluded that fogging was caused by a new form of radiation, one more penetrating than α-particles. The name “β”, followed naturally as the next letter in the Greek alphabet after α, α-particles having already been discovered and named by Rutherford.
Since that discovery, we have learned that β-particles are about 100 times more penetrating than α-particles, and are spin- 12 fermions. Associated with the electrons is a conserved quantity, expressed as the quantum number known as the lepton number. The lepton number of the negatron is, by convention +1. The lepton number of the positron, also the anti- particle of the negatron, is -1. Thus, in a negatron-positron annihilation event, the next lepton number is zero. Only leptons can carry lepton number. (More on this soon.) Recall, from Chapter 13 (Chapter 6 in Krane), our discussion of the various decay modes that are associated with β decay:
(^1) Technically, the word “electron” can represent either a negatron (a fancy word for e−) or a positron
(e+). I’ll use “electron” interchangeably with this meaning, and also e−. Usually the context determines the meaning.
2 CHAPTER 15. β DECAY
A ZXN^ −→^
A Z+1X
′ N − 1 +^ e
− (^) + ν e β − (^) decay A ZXN^ −→^
A Z− 1 X
′ N +1 +^ e
- (^) + ν e β
- (^) decay A ZXN^ −→^
A Z− 1 X
′ N +1 +^ νe^ electron capture (ε)^ (15.1)
We see from the above processes that there are other particles called neutrinos. Neutrinos are also spin- 12 leptons (part of the larger fermion family). They are very nearly massless (but proven to have mass^2 ). The electron neutrino is given the symbol νe, and has lepton number +1. The antineutrino, the νe, has lepton number -1. A sketch of the organization of fundamental particles is given in Figure 15.1.
Figure 15.1: The particle physics classification of bosons and fermions, with the sub- classifications of baryons and fermions shown.
Three views of β decay
There are three ways of viewing β decay. The first is the “radiological physics view” expressed by (15.1). The next is the “nuclear physics view”, where we recognize that the decays of the nuclei are actually caused by transformations of the nucleon constituents, as expressed in (15.2).
n −→ p + e−^ + νe β−^ decay p −→ n + e+^ + νe β+^ decay p + e−^ −→ n + νe electron capture (ε) (15.2)
A free neutron will decay with a meanlife, τ = 885.7(8)s, about 11 minutes. A free proton is basically stable. Once these nucleons are bound in a nucleus, however, conservation of energy, with the availability of lower energy states, dictates whether or not these processes are free to proceed.
Then, there is the more microscopic view, the “particle physics view” expressed in (15.3),
d −→ u + e−^ + νe β−^ decay u −→ d + e+^ + νe β+^ decay u + e−^ −→ d + νe electron capture (ε) (15.3) (^2) A direct measurement of neutrino mass suggests that its upper limit is mνe < 2 .2eV. Indirect measure-
ment of the neutrino mass suggest that 0.04eV < mνe < 0 .3eV. For the more massive lepton family groups, mνμ < 180keV, and mντ < 15 .5MeV.
4 CHAPTER 15. β DECAY
Qn = 0.782568(84)[MeV] (15.5)
Accounting for proton recoil, the exact relationship between the electron endpoint energy and Q, is given by:
T (^) emax = (mp + me)c^2
[
2 Qnmpc^2 [(mp + me)c^2 ]^2
]
T (^) emax ≈
Qn 1 + me/mp
Putting in numerical values, was calculate T (^) emax = 0.782142(84)[MeV], which agrees with the direct measurement of T (^) emax = 0.782(13)[MeV].
We can calculate the proton’s recoil energy by using Conservation of Energy:
T (^) pmax = Qn − T (^) emax
T (^) emax ≈ Qn
1 + me/mp
T (^) emax ≈ Qn(me/mp). (15.7)
This evaluates numerically to T (^) pmax ≈ 0 .426(84)[keV].
Q for β−-decay
For β−-decay
A ZXN^ −→^
A Z+1 X
′ N − 1 +^ e
− (^) + ν e (15.8)
Going back to the definition of Q in terms of nuclear masses, and ignoring, henceforth, the mass of the neutrino:
Qβ− =
[
mN (AZXN ) − mN ( (^) Z+1AX N′ − 1 ) − me
]
c^2 , (15.9)
where the subscript “N” connotes nuclear (not atomic) masses.
The relationship between the nuclear (no subscript “N”) and atomic mass is:
15.1. ENERGY RELEASE IN β DECAY 5
m(AZXN )c^2 = mN (AZXN )c^2 + Zmec^2 −
∑^ Z
i=
Bi , (15.10)
where Bi is the binding energy of the i’th atomic electron.
Substituting (15.10) in (15.9), to eliminate the (less well known) nuclear masses results in:
Qβ− =
[
m(AZXN ) − Zme
]
c^2 −
[
m( (^) Z+1AX N′ − 1 ) − (Z + 1)me
]
c^2 − mec^2 +
[ Z
i=
Bi −
Z∑+
i=
B′ i
]
[
m(AZXN ) − m( (^) Z+1AX N′ − 1 )
]
c^2 +
[ Z
i=
Bi −
Z∑+
i=
B i′
]
[
m(AZXN ) − m( (^) Z+1AX N′ − 1 )
]
c^2 +
[ Z
i=
(Bi − B i′) − B′ Z+
]
noting that the electron masses have canceled in this case. The factor
∑^ Z
i=
Bi −
Z∑+
i=
B i′ =
∑^ Z
i=
(Bi − B i′) − B Z′+
is the difference in the energy of the electronic orbital configuration of the parent and daugh- ter nuclei. Generally, this difference can be ignored. However, in the case of large Z nuclei, it can amount to about 10 keV. For accurate determinations of Q, the difference in atomic electron binding energy must be accounted for.
Q for β+-decay
Similar considerations for β+^ decay lead to:
Qβ+ =
[
m(AZXN ) − m( (^) Z−A 1 X N′ +1) − 2 me
]
c^2 +
[ Z
i=
Bi −
Z∑− 1
i=
B′ i
]
Here we note that the electron rest-mass energies do not completely cancel. However, the discussion regarding the electron binding energy remains the same.
Q for electron capture
For electron capture:
15.2. FERMI’S THEORY OF β DECAY 7
15.2 Fermi’s theory of β decay
Fermi’s theory of β decay starts with a statement of Fermi’s Golden Rule #2 for transition rate, λ:
λ =
2 π ℏ
|Vif |^2 ρ(Eif ) , (15.14)
where V is a potential that causes the transition from an initial quantum state Ψi (the parent nucleus in the this case) to a final one, Ψf , that includes wavefunctions of the daughter nucleus, the electron and its neutrino. Vif ≡ 〈Ψf |V |Ψi〉 is the transition amplitude.
The derivation of Fermi’s Golden Rule #2 is generally reserved for graduate courses in Quantum Mechanics, but a version of the derivation is available in Chapter 13, for your interest.
What concerns us now, is to calculate the density of final states, ρ(Eif ), for the β-transition. This derivation figures so prominently in the β-spectrum, and the endpoint energy.
Starting in Chapter 13, the density of states is derived for non-relativistic particles with mass, relativistic particles with mass (the electron in this case), and massless particles (the neutrino in this case).
We start with (13.21). The number of states, N, of a particle in the final state with energy E is given by:
dN =
π 2
n^2 dn. (15.15)
where n =
n^2 x + n^2 y + n^2 z , and (nx, ny, nz ) are the quantum numbers of a free particle in n infinite box potential, with side L. the momentum and the n’s are related by:
pi = niπℏ/L. (15.16)
Putting (15.16) into (15.15) gives:
dN =
2 π^2
L^3
ℏ^3
p^2 dp. (15.17)
Or, dividing by dE,
dN dE
2 π^2
L^3
ℏ^3
p^2
dp dE
8 CHAPTER 15. β DECAY
We should point out that (15.18) is valid for all particles, massless, relativistic and non- relativistic, since (15.16) is universal.
All we need do now is relate momentum to energy to compute the density factors. For the neutrino, which we are now treating as massless,
pν = Eν /c dpν = dEν /c dNν dEν
2 π^2
L^3
ℏ^3 c^3
E^2 ν (15.19)
For the electron, that must be treated relativistically,
pe =
E e^2 − (mec^2 )^2 /c dpe = [Ee/(c
E^2 e − (mec^2 )^2 )]dEe dNe dEe
2 π^2
L^3
ℏ^3 c^3
E e^2 − (mec^2 )^2 Ee
dNe dTe
2 π^2
L^3
ℏ^3 c^3
Te(Te + 2mec^2 )(Te + mec^2 ) (15.20)
For β decay we have two particles in the final state, so we can express the rate of decay to produce an electron with momentum p as:
dλβ dp
2 π ℏ
|Vif |^2
dNe dp
dNν dEif
If q is the momentum of the neutrino,
Eif = Te + cq dEif = c(dq) (Te fixed). (15.22)
Thus,
dλβ dp
2 π ℏc
|Vif |^2
2 π^2
L^3
ℏ^3
p^2
2 π^2
L^3
ℏ^3
q^2 δ(Eif − [Te + Tν ]). (15.23)
Where the δ-function accounts specifically for the conservation of energy.
10 CHAPTER 15. β DECAY
exp(i~q · x/ℏ) = 1 +
i~q · x ℏ
~q · x ℏ
Thus the leading-order term of (15.25) is:
M if^0 = 〈ψX′ |Oβ |ψX 〉. (15.27)
If M if^0 6 = 0, the β decay is called an “allowed” transition, and the rate is relatively prompt. If M if^0 = 0, then we must go to higher order terms in (15.26). These are called “forbidden” transitions, and occur, but at much slower rates. (More on this topic later.)
Krane likes to adopt the following shorthand. For allowed transitions, we see that:
dλ^0 β dp
= g^2
|M if^0 |^2 2 π^3 ℏ^7 c
p^2 q^2. (15.28)
If we have N(t) β-emitters in a sample, the momentum spectrum of electrons that may be measured is:
N^0 (p)dp = N(t)dλ^0 β =
g^2 N(t)
|M if^0 |^2 2 π^3 ℏ^7 c^5
p^2 q^2 dp. (15.29)
If N(t) changes little over the course of the measurement of the spectrum (the usual case):
N^0 (p)dp = C(0)p^2 q^2 dp , (15.30)
where we have gathered all constants with inside the large parentheses in (15.29) into a global constant C(0), that is determined experimentally. It can be determined through the a normalization condition,
∫ dp N^0 (p) ≡ 1.
Conventional forms: N^0 (p), N^0 (Te)
N^0 (p) expressed in (15.30) contains p and q, that are related by conservation of energy. In terms of single momentum variable,
N^0 (p)dp =
C(0)
c^2
p^2
[
Q −
(cp)^2 + (mec^2 )^2 + mec^2
] 2
dp , (15.31)
15.2. FERMI’S THEORY OF β DECAY 11
using relativistic kinematic relationships. The maximum possible p occurs when the neutrino component drops to zero. This is easily found to be:
pmax =
c
Q^2 + 2Qmcc^2. (15.32)
An even more common expression is to show N^0 in terms of Te.
We find this by saying:
N^0 (Te)dTe = N^0 (p)dp = N^0 (p)
dp dTe
dTe , (15.33)
Applying relativistic kinematic relationships, we find:
N^0 (Te)dTe =
C(0)
c^5
T (^) e^2 + 2Temec^2 (Te + mec^2 )(Q − Te)^2 dTe. (15.34)
Here the β-endpoint at Q = Te is evident.
Accounting of ”forbiddeness” and nuclear Coulomb effect.
There are two other attributes of β-spectra we must take account of, before we start using the theoretical spectral shape to assist in analyzing data.
The first of these has to do with the interaction of the daughter’s Coulomb charge with the resultant electron or positron in the final state. This nuclear charge has no effect, of course, on the neutral neutrino. Going back to (15.25), we wrote the electron wavefunction as a free plane wave. In actual fact, that was a fairly crude approximation. These plane waves are distorted significantly by the attraction the β−^ would feel, and the repulsion that the positron would feel. Incidentally, there is no effect on our conclusions regarding “allowed” or “forbidden”.
Accounting for this is quite involved, but not beyond our capabilities. We would have to go back to (15.25) and write the electron wave functions in terms of free particle solutions to the Coulomb potential. (In NERS 311 we learn a lot about bound states of the Coulomb potential.) I have never seen detailed discussion of this in even graduate-level texts, and interested students are usually told to seek out the papers in the literature. The result is, however, that the β-spectra are multiplied by a correction factor, the Fermi function, that depends on the charge of the daughter nucleus, Z′, and the electron momentum and sign, F ±(Z′, p). The effect it has could have been anticipated from classical considerations. The electron spectra is dragged back toward lesser values, while the positron spectra are pushed toward higher values. See Figure (9.3) in Krane.
15.3. EXPERIMENTAL TESTS OF FERMI’S THEORY 13
15.3 Experimental tests of Fermi’s theory
Kurie plots: Shape of the β spectrum
To employ (15.35) to analyze β spectra, one plots:
√ N(p) SL(p, q)F ±(Z′, p)
vs. Te, (15.36)
using the initial assumption that L = 0, so that SL(p, q) = 1. If the data points fall an a straight line (statistical tests may be necessary), once can easily obtain the Q-value from the x-intercept. This type of plot is called a Kurie plot (named after Franz Kurie.) one has also identified, from the shape, that this is an allowed transition.
If the Kurie plot is not straight, one must successively test shape factors until a straight line match is obtained. Once the shape factor is determined, the level of forbiddeness is determined, and the Q-value may be extrapolated from the data unambiguously.
Total decay rate: The f t 1 / 2 , log 10 f t values
Putting in the Coulomb and shape factors into (15.28) allows us to determine the total decay rate for a β-decay process,
λβ = g^2
|MifL |^2 2 π^3 ℏ^7 c
∫ (^) pmax
0
dp SL(p, q)F ±(Z′, p)p^2 q^2
= g^2
m^5 e c^4 |MifL |^2 2 π^3 ℏ^7
[
(mec)^5
∫ (^) pmax
0
dp SL(p, q)F ±(Z′, p)p^2 q^2
]
≡ g^2
m^5 e c^4 |MifL |^2 2 π^3 ℏ^7
fL(Z′, Q) , (15.37)
where the dimensionless integral in large square brackets, is a theoretical factor that may be pre-computed and employed in the data analysis. This is conventionally written in terms of halflife, t 1 / 2 = log(2)/λβ.
Thus,
fL(Z′, Q)t 1 / 2 ≡ f t 1 / 2 =
loge(2)2π^3 ℏ^7 g^2 m^5 e c^4 |MifL |^2
14 CHAPTER 15. β DECAY
This is known colloquially as the f t value. (Pronounced eff tee.) The f t’s can be quite large, and sometimes the “log f t” value is quoted. (Pronounced log eff tee.) The precise definition is log 10 (f t 1 / 2 ).
Mass of the neutrino
Our applications of β-decay ignore the neutrino mass, but they turn out to be critically important for cosmology.
There is one important fact: they do have mass, but it is very small.
The table below shows the current state of the mass determinations of the three generations of leptons, e, μ, and τ.
lepton flavor neutrino symbol mass (eV) e νe 0. 04 −→ 2. 2 μ νμ < 1. 70 × 105 τ ντ < 1. 55 × 107
15.4 Angular momentum and parity selection rules
Classification of transitions in β decay
The e and the ν in the final states of a β decay each have intrinsic spin- 12. Conservation of total angular momentum requires that:
~IX = ~IX′ + ~L + S ,~ (15.39)
where I~X , ~IX′ are the total angular momenta of the parent and daughter, respectively, and ~L, S~ are the total orbital and total spin angular momentum, respectively, of the eν pair.
Therefore, the ∆I can be ±L. or ±|L± 1 |. If L = 0, then ∆I = ±1. There are only two cases for lepton spin alignment. S = 0, when the eν intrinsic spins anti-align, is called a Fermi transition. S = 1, when the eν intrinsic spins align, is called a Gamow-Teller transition. Generally, as L ↑, λ ↓, t 1 / 2 ↑ , because there is much less overlap of the eν wavefunctions with the nucleus.
The entire characterization scheme is given in Table 15..
Nomenclature alert!
16 CHAPTER 15. β DECAY
Examples of allowed β decays
This is straight out of Krane.
(^14) O(0+) → (^14) N∗(0+) must be a pure Fermi decay since it is 0+ (^) → 0 +. Other examples are (^34) Cl→ (^34) S, and 10 C→ (^10) B∗.
(^6) He(0+) → (^6) Li(1+), a 0+ (^) → 1 + (^) transition. This must be a pure Gamow-Teller decay. Other
similar examples are 13 B(^32 −)→^13 C(^12 −), and 230 Pa(2−)→^230 Th∗(3−).
n(^12
) → p(^12
) This is a mixed transition. The F transition preserves the nucleon spin direction, the GT transition flips the nucleon spin. (Show drawing.)
β decay can either be of the F type, the GT type or a mixture of both. We may generalize the matrix element and coupling constant as follows, for allowed decays:
gM^0 = gF MF^0 + gGT MGT^0 = gF 〈ψX′ | 1 |ψX 〉 + gGT 〈ψX′ |O↑↓|ψX 〉 , (15.40)
where O↑↓ symbolizes an operator that flips the nucleon spin for the GT transition. The operator for the F transition is simply 1 , (i.e. unity), and just measures the overlap between the initial and final nuclear states.
The fraction of F transitions is:
fF =
g^2 F |MF^0 |^2 g F^2 |MF^0 |^2 + g^2 GT |MGT^0 |^2
y^2 1 + y^2
where,
y ≡
gF MF^0 gGT MGT^0
Tables of y values are given in Krane on page 290.
15.4.1 Matrix elements for certain special cases
This section is meant to explain several things given without explanation in Krane’s Chap- ter 9.
15.5. COMPARATIVE HALF-LIVES AND FORBIDDEN DECAYS 17
Mif =
2 , for superallowed 0 +^ → 0 +^ transitions
This was stated near the top of the text on Krane’s p. 284.
We know that a 0+^ → 0 +^ allowed transition (super or regular), must be an F transition. In the case that it is also a superallowed transition, we can write explicitly:
Mif =
ψX′ (0+)
[e(↑)ν(↓) + e(↓)ν(↑)]
∣ψX (0+)
where the intrinsic spins of the eν pair are shown explicitly. This spin wavefunction is properly normalized with the
2 as shown.
Separating the spins part, and the space part,
Mif =
〈ψX′ |ψX 〉〈(e(↑)ν(↓) + e(↓)ν(↑))|~ 0 〉 =
since 〈ψX′ |ψX 〉 = 1 for superallowed transitions, and 〈e(↑)ν(↓)|~ 0 〉 = 〈e(↓)ν(↑))|~ 0 〉 = 1.
Using this knowledge, one can measure directly, gF from 0+^ → 0 +^ superallowed transitions. Adapting (15.38) for superallowed transitions,
g^2 F =
loge(2)π^3 ℏ^7 m^5 e c^4
f t 1 / 2
meas
giving a direct measurement of gF via measuring f t. Table 9.2 in Krane (page 285) shows how remarkable constant f t is for 0+^ → 0 +^ superallowed transitions. This permits us to establish the value for gF to be:
gF = 0. 88 × 10 −^4 MeV · fm^3. (15.46)
15.5 Comparative half-lives and forbidden decays
Not covered in NERS312.
15.6 Neutrino physics
Not covered in NERS312.
