- Beta decay is one of the most easily found kinds of radioactivity. As we have seen, this result of the weak interaction leads to conversion of a neutron into a proton or vice versa, with the necessary charge change being made possible by the emission of a positive or negative beta particle (positron or electron). To ensure conservation of lepton number, each such event is accompanied by emission of an electron neutrino. Alternatively, an orbital electron may be absorbed (electron capture or K-capture), changing a proton into a neutron (this process competes with emission), with emission of an electron neutrino.
- The spectra of electrons or positrons emitted in beta decay is a continuum of energies, up to a maximum value, with most emitted betas having intermediate energies. The emitted, unobserved, neutrinos also have a continuous energy distribution. In K-capture, there is no electron emitted; the neutrino is mono-energetic. (Mono-energetic electrons also emerge from de-excitation of an excited nucleus by internal conversion [see below]; this process leads to electron emission but is quite a different physical mechanism than beta decay.)
- We can predict the general form of the energy spectrum of the (observable) beta particle, but not the absolute decay rate, from a simple theory proposed by Fermi. The form of the spectrum is found simply by considering the density of final states!
- We have seen, using perturbation theory, that the probability
per unit time of a transition into a group of closely spaced states is
given by

where is the density of final states and is the matrix element of the Hamiltonian between states and (Lecture 2). - In beta decay of an even-odd nucleus, the initial state is
simply the wave function of the single odd nucleon (let's say it is a
proton),
. The final
state is the wave function of the new neutron, the emitted
positron and the neutrino:
. The
interaction Hamiltonian is uncertain, but very short range, and Fermi
tried the simplest operator he could think of, namely a constant that
we will call . Thus the matrix element is only non-zero to
the extent that the initial and final wave functions overlap:

- For the lepton wave functions we take plane waves (although
this is not accurate for the beta particle, which is affected by the
Coulomb interaction with the nuclear protons). These must be
normalized, so we again introduce an artificial (large) volume :

and

- Now for lepton energies of a few MeV or less, the wavelengths of
the lepton wave function are of order 1 MeV/
fm. Thus the exponent of each of the exponentials is small over
the extent of the nucleus, and we may replace the two lepton wave
functions by . Then our trial matrix element is just

In general we do not know what the value of the integral is. Let us simply assume that it does not depend on how the released energy is partitioned between the outgoing beta particle and the neutrino, so that for a given decay we may treat it as a constant which we will call . - We have so far ignored the (significant) effect of the Coulomb
interaction between the outgoing beta particle and the nucleus. The
effect of this interaction for non-relativistic particles is to
replace the beta particle plane wave at the origin by the wavefunction
of the beta particle in the electric field of the daughter nucleus of
charge . This is equivalent to multiplying our expression
for by
. It is found that

where and the sign of is opposite to that of the charge on the beta particle. For both electrons and positrons ; for large beta velocity this factor approaches 1 (as it should) but for small the function tends to a large value for electrons (for which the outgoing wavefunction is concentrated around the nucleus) and to a small value for positrons (for which the outgoing wavefunction is made small near the nucleus by the Coulomb repulsion). - So finally the decay rate that we need is

- What density of final states is needed? After the decay, we may
suppose that the final nuclear state is stable, so it has a
well-defined energy, while the initial unstable state has finite width
due to the uncertainty principle
. Suppose that in the decay we can measure precisely the state
of the outgoing electron (its momentum vector, not its position, of
course). Now because of the uncertainty in initial energy, the
outgoing neutrino with have a slightly uncertain energy, and may end
up in any one of a number of closely adjacent states. Thus for the
probability of decay to a
*specific*final electron state, we need the density of final*neutrino*states. - Now recalling that the neutrino is relativistic (and may have
non-zero mass), so that its momentum and energy are related by
, we may use our earlier result
that the density of plane wave states (neglecting spin) in is
, where
and
. Thus the relativistic density of
neutrino states is

- However, because the outgoing neutrino is not observable, we
need to express this in terms of the observable beta particle energy,
, where is the total energy available for
the decay. Thus the probability of decay into a
*single*electron state of energy is

and the total probability of decay into*any*of the final beta particle states within momentum interval - using now the density of*beta*particle states - is

The distribution of electron energies (or momenta) is produced*entirely*by the density of states factors. - This theory of beta decay is usually tested, and the value of
the total decay energy determined accurately, by plotting the value of
against the total or kinetic
energy. Such a plot is called a
*Kurie plot*; it has the virtue that if is small, the measured points are proportional to and the graph crosses 0 where . From such graphs it is found that the rest mass-energy of the neutrino is no more than about 5 eV. - The total transition rate for a decay is found by integrating
the expression above over beta particle energy. The resulting mean
lifetime of a beta-unstable nucleus is

where

neglecting the neutrino mass. The function is dimensionless and has been extensively tabulated. Once the value of the constant has been determined (from an experiment in which the matrix element can be evaluated approximately), other experiments determine experimentally the value of the unknown matrix elements of the decay in terms of the quantity . This*ft-value*is the quantity usually quoted for an experimentally studied beta decay, rather than the value of the matrix element.

- A process that competes with positron beta decay is electron
capture, in which the same conversion of a proton into a neutron is
accomplished by the capture of an
*orbital*electron with emission of a neutrino. We may calculate the rate of this process using very similar reasoning to that discussed above. - Again we use Fermi's golden rule. Now the initial state is a
nuclear proton wave function and an orbital electron wave function,
while the final state is an nuclear neutron and a free neutrino of
(almost) definite energy (the initial state, of finite lifetime, has a
slightly uncertain energy). The initial electron state (assuming the
electron to be a K electron in the innermost shell) is thus

Thus the total rate for this process is

where the initial 2 arises because there are two K-shell electrons, the matrix element is given by

and the density of states is only that of the neutrino,

Because the electron wave function is normalized to its confined volume around the nucleus, only appears to the first power now. Again we simply need to evaluate the electron wave function at the nucleus (i.e. at ), and the only part of the matrix element that we cannot evaluate is the nuclear overlap integral, which is the same integral that appears in the normal beta decay theory. - Putting the pieces together, the transition rate for electron
capture (the inverse of the lifetime) is

where is the energy available in the decay (given to the escaping neutrino). Note that this result is two times larger than in C & G because they have calculated the rate for a single K electron. - The ratio of the decay rate by this process to normal positron
decay is easily calculated because the same nuclear matrix element
appears in both results. We get

where is the (dimensionless) fine structure constant (Lecture 3) and we neglect the mass of the neutrino. We can see that for small available energies this ratio is made large by the smallness of (and is infinite if positron decay is not permitted energetically); it is also relatively large for high because of the factor. - Experiments on electron capture are not as straight-forward as on normal beta decay. The outgoing neutrino is unobservable. Instead, what may be detected is an x-ray from transitions in the electron cloud as the missing K electron is replaced from a higher level, and/or an Auger electron ejected from the excited electron cloud, carrying off the energy released as an L-shell electron drops into the K-shell.

- You have already gone through the (partial) derivation of the
spontaneous emission rate of photons from an excited system (Secs 4.2
and 4.3 in B & J). The final result of this exercise was to derive
the lowest order emission probability per second, or inverse lifetime,
in the dipole approximation,

where is the angular frequency corresponding to the energy difference between states and , and is the matrix element of the position vector between the wavefunctions of states and . - The derivation of this result (repeated in somewhat different
form in C & G) is as valid for nuclear photon emission as for
emission from atoms: B & J Eq. [4.71] is identical to C & G
(12.12). The same approximations are valid, particularly the expansion
of the outgoing wave state
in powers of
. Clearly since nuclei are much smaller than atoms, the
spatial extent of the wavefunction will be much smaller, resulting in
a much smaller value for
, but this is compensated for
by much larger values of than are found for atoms. In
fact the order of magnitude of the emission rate for allowed nuclear
transitions,

is quite a lot*larger*than the characteristic rate for allowed atomic transitions

- Since atoms - in most interesting contexts - collide with one
another and so can de-excite collisionally if radiation is forbidden
(i.e. highly suppressed), they rarely emit much radiation which is not
allowed in the lowest dipole approximation. In contrast, nuclei often
find themselves with no means of de-exciting from an excited state
other than photon radiation. If an excited state differs from all
lower states by several units of angular momentum, radiation will be
possible only by a rather high multipole, and since each higher
multipole of electric radiation is slower at nuclear energies by a
factor of order than the next lower multipole, lifetimes of
radiative decay can sometimes be quite long even by human
standards. In this case we speak not of metastable but of
*isomeric*states; such states may be stable enough to make it into the handbooks.... - Another (electromagnetic) process which competes with radiation,
especially if it is rather forbidden, is
*internal conversion*. In this process, the nucleus transfers its excitation energy directly to an atomic electron (usually a K-shell electron) which is ejected, carrying off the excitation energy. The matrix elements of the two processes are similar, but the increasing concentration of the K-shell electrons near the nucleus as increases means that internal conversion competes increasingly well with photon emission as increases. Note that there is no photon involved here; the energy is transferred*directly*to the atomic electron.

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