r/β+andβ−spectra of^64 Cu.β−in these decays.^7 There is a relatively high probability for the
beta and neutrino each to carry off roughly half the kinetic energy,
the reason being identical to the kind of phase-space argument
discussed in sec. 5.4.2, p. 328. Therefore in each case we get a
bell-shaped curve stretching from 0 up to the energyQ, withQ
being slightly different in the two cases.
So we expect the two bell curves to look almost the same except
for a slight rescaling of the horizontal axis. Yes — but we also see
markedly different behavior at low energies. At very low energies,
there is almost no chance to see aβ+with very low energy, but
quite a high probability for aβ−.
We could try to explain this difference in terms of the release of
electrical energy. Theβ+is repelled by the nucleus, so it gets
an extra push on the way out the door. Aβ−should be held
back as it exits, and so should lose some energy. The bell curves
should be shifted up and down in energy relative to one another,
as observed.
But if we try to estimate this energy shift using classical physics,
we come out with a wildly incorrect answer. This would be a
process in which the beta and neutrino are released in a point-
like event inside the nucleus. The radiusrof the^64 Cu nucleus
is on the order of 4 fm (1 fm = 10−^15 m). Therefore the en-
ergy lost or gained by theβ+orβ−on the way out would be
U∼k Z e^2 /r∼10 MeV. The actual shift is much smaller.
To understand what’s really going on, we need quantum mechan-
ics. A beta in the observed energy range has a wavelength of
about 2000 fm, which is hundreds of times greater than the size
of the nucleus. Therefore the beta cannot be much better local-
ized than that when it is emitted. This means that we should really
use something more liker∼500 fm (a quarter of a wavelength) in
our calculation of the electrical energy. This givesU∼0.08 MeV,
which is about the right order of magnitude compared to obser-
vation.
A byproduct of this analysis is that aβ+is always emitted within
the classically forbidden region, and then has to tunnel out through
the barrier. As in example 17, we have the counterintuitive fact
about quantum mechanics that a repulsive force canhinderthe
escape of a particle.
(^7) Redrawn from Cook and Langer, 1948.
Section 13.3 Matter as a wave 909