When radioactivity was discovered it was discerned that there were three distinct types. They were named alpha, beta and gamma rays. Investigation proved that alpha rays were ionized helium nuclei traveling at high speeds; beta rays were electrons and gamma rays were high powered X-rays.

Measurement of the energy distribution of beta rays showed it to be continuous in contrast to the discreteness of other phenomena at the quantum level. This continuity of the energy spectrum of beta rays presented the additional puzzle that the principle of the conservation of energy was apparently violated. Some physicists, notably Niels Bohr, were ready to give up that principle. Others, notably Wolfgang Pauli, were not willing to do so.

Wolfgang Pauli resovled the problem by postulating that second particle accompanied the ejected electron in beta decay. That particle subsequently became known as a neutrino, the little neutral one, and later as an anti-neutrino to satisfy the rule that when a particle is created it is accompanied by the creation of an anti-particle.

Enrico Fermi

Fermi incorporated Pauli's suggestion in his theory of beta decay, but went beyond the conventional theory to hypothesize a new force that was extremely weak in comparison to electromagnetism. Thus the story of the concept of the nuclear weak force begins with that article written by Enrico Fermi in 1933 to explain beta decay. The available literature tells that Fermi first submitted his article to Nature, the premier science journal of the United Kingdom, and it was rejected with the editor's comment

Your article's speculations are too remote from physical reality to be of interest to the practicing scientists who make up the audience of the journal.

Fermi then submitted his article to the Italian science journal Ricerca Scientifica. It was published in English in that journal in 1934 under the title Tentative Theory of Beta-Radiation. He subsequently wrote an Italian and a German vesion.

It appeared that the rejection by Nature was the editorial blunder of the 20th century. And it was, but the article was hard to follow even though its analysis is masterful. The purpose of this material is to present and explicate Fermi's line of analysis.

Fermi's primary result was a formula for the distribution function of the kinetic energy K of the beta ray particle, which could be an electron or a positron; i.e.,

P(K) = C_ΔL·F(Z, K)·(X−K)²pE

where Z is the number of protons in the final state nucleus and X is the net energy produced by the reaction. The total energy E and momentum p of the beta particle (electron or positron) are given by

E = K+mc²
p² = (E/c)² − (mc)²

where m is the mass of the electron (positron) and c is the speed of light.

C_ΔL is a constant that depends upon the net change in total angular momentum in the reaction (angular momenta of the beta particle plus the (anti)neutrino and their spins). The allowed ΔL for a Fermi transition is 0, whereas for transitions known as Gamow-Teller ΔL can be 0 or ±1.

The function F(Z, K), now known as the Fermi function, in the above formula is dependent upon three quantities

S = (1 − α²Z²)
where α≅1/137 is the
fine structure constant
 
η = ±αZE/(pc)
with + for electrons
and − for positrons
 
g = r_Z/h
where r_Z is the radius
of the final state nucleus

Given those definitions the Fermi function can be defined as

F(Z, K) = 2(1+S)(2pg)^2(S-1)exp(πη)[Γ(S+iη)/Γ(2S+1)]²

where Γ(x) is the gamma function.

When the velocity of the beta particle is small relative to the velocity of light F(Z, K) is approximately

2πη/(1−exp(−2πη))

Fermi developed his theory of beta decay in what is now the standard methodology of particle physics, but was surprisingly sophisticated for the time, 1933.

Fermi's Golden Rule of
Transition Probabilities

The probability per unit time of a system making a transition from an initial state i to a final state f is given by

P(i, f) = (2π/h)|M_if|²ρ

where h is Planck's constant divided by 2π, ρ is the density of energy states. The term M_i,f is defined as

M_i,f = <i|ΔH|f>

where |i> and |f> are the energy eigenfunctions of the initial and final states, respectively, expressed in Dirac's ket notation. ΔH is the perturbation of the Hamiltonian between the initial and final states.

Requirements for theTheory

Fermi states three conditions that an adequate theory of beta decay must satisfy:

• (a) Electrons and (anti)neutrinos can be created or annihilated. He states specifically that this is not the same phenomenon as the creation or annihilation of an electron-positron pair.
• (b) The heavy particles, protons and neutrons, a la Heisenberg, should be considered a two internal quantum of states of a single particle type, now known as a nucleon. He defines an internal coordinate ρ which takes the value of +1 for a neutron and −1 for a proton.
• (c) The Hamiltonian for a system consisting of heavy particles (nucleons) and light particles (leptons) must be such that every transformation of a neutron into a proton must be accompanied by the creation of an electron and an (anti)neutrino. Likewise every transformation of a proton into a neutron must be accompanied by the disappearance of an electron and an (anti)neutrino.

Second Quantization

Fermi makes use of the method of Second Quantization pioneered by Dirac, Jordan and Klein which involves the formulation of creation and annihilation operators which raise and lower the occupation numbers for the energy states of the particle fields. Fermi defines the operators

		| 0	1 |
Q	=	| 0	0 |

and
 

		| 0	0 |
Q*	=	| 1	0 |

The asterisk denotes the transpose of the complex conjugate of a quantity. Thus Q* is the transpose of the complex conjugate of Q. Since Q is real, Q* is just the transpose of Q. The effect of the operator Q is to change a neutron into a proton and that of Q* is the reverse operation.

In Fermi's analysis there are two pairs of creation/annihilation operators; one pair for the beta particle and one pair for the (anti)neutrino. Furthermore for Fermi's analysis the occupation numbers can only be 0 or 1.

Fermi then introduces the functions ψ and φ from Schroedinger's equation as the wave functions for the electron and the (anti)neutrino, respectively. He introduces a complete set of functions {ψ₁, ψ₂, …} for the quantum states of an electron. Then

ψ = Σ ψ_sa_s
ψ* = Σ ψ_s*a_s*

The amplitude coefficients a_s and their complex conjugates a_s* are taken to be operators which act on functions of occupation numbers {N₁, N₂, …} of the single-particle states. Any N_s can take on only the values 0 or 1.

The operator a_s* corresponds to the creation of an electron in quantum state s. Likewise the operator a_s corresponds to the annihilation of an electron in quantum state s.

The actions of the operators a_s and a_s* on a function Ψ of the state occupation numbers {N₁, N₂, …} is depicted as

a_sΨ(N₁, N₂, …) = (−1)^{N₁+N₂+…+N_s-1} N_sΨ(N₁, N₂, …(1−N_s), …)
a_s*Ψ(N₁, N₂, …) = (−1)^{N₁+N₂+…+N_s-1} (1−N_s)Ψ(N₁, N₂, …(1+N_s), …)

At this point I have no explanation for this particular form, but I do note that Fermi's form of the effect of the annihilator operator a_s on the occupancy number of the s-th state as (1−N_s) is correct whereas its usual representation as (N_s−1) is incorrect because if N_s=0 that formula would give the new value of N_s as −1, an unallowed value. On the other hand the effect of the creation operator a_s* of (1+N_s) is equivalent to the usual representation of (N_s+1) which gives the incorrect unallowed value of 2 if N_s=1, but the coefficient of (1−N_s) for Ψ eliminates this case.

Analogous formulas apply for the (anti)neutrinos. First,

φ = Σ φ_sb_s
φ* = Σ φ_s*b_s*

Again the amplitude coefficients b_s and their complex conjugates b_s* are taken to be operators which act on functions of occupation numbers {M₁, M₂, …} of the single-particle states of the (anti)neutrinos. Any M_s can take on only the values 0 or 1.

The operator b_s* corresponds to the creation of an (anti)neutrino in quantum state s. Likewise the operator b_s corresponds to the annihilation of an (anti)neutrino in quantum state s.

The actions of the operators b_s and b_s* on a function Φ of the state occupation numbers {M₁, M₂, …} is

b_sΦ(M₁, M₂, …) = (−1)^{M₁+M₂+…+M_s-1} M_sΦ(M₁, M₂, …(1−M_s), …)
b_s*Φ(M ₁, M₂, …) = (−1)^{M₁+M₂+…+M_s-1} (1−M_s)Φ(M₁, M₂, …(1+M_s), …)

Hamiltonian Functions and
Hamitonian Operators

For the cases being considered the Hamiltonian function of a system is just its total energy expressed in terms of the locations and momenta of the particles of the system. The Hamiltonian operator for a system is formed from the Hamiltonian function by making substitutions for the momenta. In the following H^ will be used to denote the operator corresponding to the Hamiltonian function H.

Generally

H_total = H_heavy + H_light + H_interaction
and, of course
H^_total = H^_heavy + H^_light + H^_interaction

where heavy refers to the neutron and proton and light to a beta particle (electron or positron) and an (anti)neutrino.

Fermi then defines

H^_heavy = ((1+ρ)/2)N^ + ((1−ρ)/2)P^

where ρ equals +1 for a neutron and −1 for proton. N^ and P^ are the energy operators for a neutron and proton, respectively

Let H ₁, H ₂, … and J ₁, J ₂, … be the energies of the stationary energy states of the electron and the (anti)neutrino.

Then

H_light = ΣH_sN_s + ΣJ_σM_σ

To satisfy condition (c) the term

Q*a_s*b_σ*

must be added to the Hamiltonian operator to ensure that when a proton is created from a neutron there are also created an electron in state s and an (anti)neutrino in state σ. Likewise the Hamiltonian operator must contain the term Qa_sb_σ to ensure that when a neutron is created from a proton there are disappearances of an electron and an (anti)neutrino.

The most general interaction Hamiltonian operator satisfying the above requirement is

H_interaction = QΣc_sσa_sb_σ + Q*Σc_sσ*a_s*b_σ*

The simplest such interaction term Fermi represents as

H_interaction = g{Qψ(x)φ(x) + Q*ψ*(x)φ*(x)}

where ψ and φ are quantum states for an electron and an (anti)neurino, respectively, which may be multi-component functions. He later takes ψ and φ to be Dirac four component functions, (ψ₁, ψ₂, ψ₃, ψ₄) and (φ₁, φ₂, φ₃, φ₄).

There are 16 possible products of the two four-component functions. Fermi assserts that because of the requirements of transformations by the Lorentz group the 16 components can be reduced to four components

A₀ = − ψ₁φ₂ + ψ₂φ₁ + ψ₃φ₄ − ψ₄φ₃
A₁ = ψ₁φ₃ − ψ₃φ₄ − ψ₃φ₁ + ψ₄φ₂
A₂ = iψ₁φ₃ + iψ₂φ₄ − iψ₃φ₁ − iψ₄φ₂
A₃ = − ψ₁φ₄ − ψ₂φ₃ + ψ₃φ₂ + ψ₄φ₁

According to Fermi these four quantities constitute the four components of a four-component polar vector and thus an electromagnetic vector potential.

If the velocity of the heavy particle is small compared with the speed of light then the Hamiltonian operator for the interaction of the heavy and light particles can be limited to A₀; i.e.,

H_interaction = g[QA₀ + Q*A₀]

Perturbation Analysis

Fermi's method of anaysis is treat H_heavy+H_light as the unperturbed Hamiltonian and H_interaction as a perturbance. The state of the unperturbed system is specified by

(ρ, n, N₁, …, M₁, …)

where ρ specifies whether the heavy particle is a neutron (ρ=1) or a proton (ρ=−1). Fermi then defines u_n and v_n as the eigenfunctions of the neutron and proton, repectively, of the unperturbed system.

For the application of the Golden Rule Fermi needed matrix elements for the change in the Hamiltonian of the form ΔH^I_F where here I stands for the initial state and F for the final state. As previously stated the Hamiltonian is of torm

H_interaction = QΣc_sσa_sb_σ + Q*Σc_sσ*a_s*b_σ*

Thus when the initial state is a neutron alone and the final state is a proton and an electron ‌ a state s with an (anti)neutrino in state σ the matrix element is given by

±∫v_m*c_sσ*u_ndτ

The sign of this term is given by

(−1)^{N+N₂+…N_s-1+M₁+M₂+…+M_σ-1}

For the Hamiltonian which Fermi adopted for his analysis

c_sσ = gψ_s*Dφ_σ
 
c_sσ* = gψ_sDφ_σ*

where g is a constant and ψ_s and φ_σ are four component column vector eigenfunctions for the electron in state s and the (anti)neutrino in state σ. The notation A denotes the transpose of A. D is the block diagonal matrix

	| 0	−1	0	0 |
	| 1	0	0	0 |
D =	| 0	0	0	−1 |
	| 0	0	1	0 |

With the above expression for c_sσ* the relevant matrix element becomes

±∫v_m*u_nψ_sDφ_σ*dτ

Fermi's Analysis of the
Beta Decay of a Neutron

Fermi introduce the wave function ψ(x) of an electron and its decomposition as

ψ = Σ ψ_Sa_S

where S is the state of the system as given by the state of the heavy particle as a neutron or proton and further by the absence or presence of an electron in energy state s and the (anti)neutrino in in energy state σ. If the system is initially a neutron alone then

a_1,0s0σ = 1
a_−1,1s1σ = 0

For a time interval Δt short enough that a_1,0s0σ remains approximately equal to 1 Schrädinger time dependent equation gives, according to what Fermi calls the usual perturbation theory

d(a_−1,1s1σ)/dt = −(2πi/h)ΔH·exp((−2πi/h)(X − (H_s + J_σ))Δt)

The integration of this relation from 0 to t gives, since a_−1,1s1σ=0 at t=0,

a_−1,1s1σ = ΔH[exp(−(X−(H_s + J_σ))t)−1]/(X − (H_s + J_σ))

Note that

exp(−αt) = cos(αt)−i·sin(αt)
and hence if
cos(αt)≅1
then
−i(exp(−αt)−1) ≅ sin(−αt)

This means that the probability of the beta decay of a neutron is given by

|a_−1,1s1σ|² = |ΔH|²sin²(−(2π/h)(X−(H_s + J_σ))t)/(X−(H_s + J_σ))

Fermi then notes that the de Broglie wavelength for electrons and (anti)neutrinos of energies of a few milllion electron volts (MeV) is larger than the scale of nuclei. He the asserts that as a first approximation the wavefunctions ψ_s and φ_σ can be considered constant within a nucleus. This means the relevant matrix element is given by

ΔH = ±gψ_sDφ_σ*∫v_m*u_ndτ
and hence
|ΔH|² = g²ψ_sDφ_σ*φ_σDψ_s*|∫v_m*u_ndτ|²

Fermi then considered the system quantized over a volume Ω where the normalized (anti)neutrino eigenfunctions are plane Dirac-waves with a density of 1/Ω.

The expected value of the relevant matrix element Fermi gives as

E{ΔH} = (g²/4Ω) |∫v_m*u_n|²[ψ_sψ_s−(μc²/J_σ)ψ_sBψ_s]

where μ is the mass of the (anti)neutrino and B is the Dirac matrix

	| 1	0	0	0 |
	| 0	1	0	0 |
B =	| 0	0	−1	0 |
	| 0	0	0	−1 |

The nmber of (anti)neutrinos with momentum between p_σ and p_σ+dp_σ is given by Fermi as

(8πΩ/h³)p_σ²dp_σ

As was previously noted the probability of beta decay satisfies the equation

|a_−1,1s1σ|² = |ΔH|²sin²(−(2π/h)(X−(H_s + J_σ))t)/(X−(H_s + J_σ))

Note that this calls for a singularity where

X−(H_s + J_σ) = 0

Fermi suggests that this would correspond to a sharp peak in the energy spectrum because all of the equations are approximations. Fermi then define p_σ as the momentum of the (anti)neutrino such that X−(H_s + J_σ) = 0. He then calls for the summation of |a_−1,1s1σ|² over all σ and obtains

t(8π³g²/h⁴)|∫v_m*u_ndτ|²(p_σ²/v_σ) [ψ_sψ_s −(μc²/J_σ)[ψ_sBψ_s ]

where μ is the mass of the (anti)neutrino.

The coefficient of t in the above expression is the probability per unit time of a beta decay.

The condition

H_s ≤ X − μc²

gives the upper limit E₀ on the energy spectrum of the electron in beta decay.

The Rest-Mass of
the (Anti)Neutrino

The rest-mass μ affect the probability of beta decay through the term p_σ²/v_σ. This quantity,according to Fermi, has a dependence on the electron energy E in the vicinity of E₀ of the form

p_σ²/v_σ = (1/c³)(μc² + E₀−E)((E₀−E)² + 2μc²(E₀−E))^½

Fermi then plots the energy distributions for three cases; μ=0, μ small and μ large. The results are shown below.

Fermi then notes

the closest resemblance to the empirical curves is shown by the theoretical curve for μ=0.

Fermi concludes that the mass of the (anti)neutrino is zero or negligibly small. His further analysis procedes on the basis that μ=0.

The Lifetime and Distribution
Curve for Allowed Transitions

Fermi then uses the relativistic eigenfunctions for a hydrogenic atom of atomic number Z to compute the probability distribution of relative momentum η=p/(mc). The result of that computation is

P(η) = g²[256π⁴/(Γ(3+2S))²](m⁵c⁴/h⁷)(4πmcr_Z/h)^2S|∫v_m*u_ndτ|²η^2+2S×
exp(π)γ^α|Γ(1+S+iγα)|²[(1+η₀²)^½ − (1+η²)^½]

where Γ(z) is the Gamma function, γ=Z/137, S=(1−γ²)^½−1, r_Z is the nuclear radius, α=(1+η²)^½/η) and η₀ is the maximum value of η.

(To be continued.)

Experimental Confirmation
of Fermi's Analysis

After the publication of Fermi's article several experimental physicists decided to test his theory by carefully compiling the velocity distribution of beat radiation. They found too many slow electrons for the distributions to fit Fermi's prediction. It was Chien-Shiung Wu (Madame Wu) who, years later, realized what the problem was. The experimentalists, in order to get enough electrons to measure, used big thick pieces of beta emitting material. Wu, working after World War II, used a thin strip of the powerful beta emitting copper isotope Cu-66. The research on nuclear weapons during the war made such materials available. The distributions found by Wu fit the curves predicted by Fermi closely, thus confirming Fermi's analysis and the existence of the weak nuclear force.

References

Robert P.Crease and Charles C. Mann, The Second Creation: Makers of the Revolution in 20th Century Physics, Macmilland and Co,, New York, 1986.

Charles Strachan, The Theory of Beta-Decay, Pergamon Press, Oxford, 1969.

Carsten Jensen, Controversy and Consensus: Nuclear Beta Decay 1911-1934, Birkhäuser Verlag, Berlin, 2000.