Some nuclei are stable, some are not. A stable nucleus is one that will remain in its current state indefinitely unless an outside agent interacts with it; an unstable nucleus is one that will spontaneously change to another. Various modes of decay will be covered below. In general, stable nuclei have approximately equal number of neutrons as protons, and a strong excess of one or the other will result in an unstable nucleus. The ratio of neutrons to protons in a stable nucleus is thus around 1:1 for small nuclei (Z < 20). The ratio increases slowly with atomic number up to about 1.58 at high Z. There are only two stable nuclei with Z > N (more protons than neutrons): 1H and 3He, each of which has one more proton than it has neutrons.
There are some tendencies, or rules, that stable nuclei observe. By definition A = N + Z, i.e., it is the total number of nucleons in a nucleus.
| 236Ra | -> | 222Rn | + α + γ + Q |
| Z = 88 | 86 | (number of protons) | |
| N = 138 | 136 | (number of neutrons) | |
| N / Z = 1.5682 | 1.5814 |
Nuclei hold together because the mass energy associated with the atom
is smaller than the sum of the mass energies of the individual nucleons
and electrons that constitute that atom. Thus for a helium atom with mass
4.00260 amu (atomic mass units, a unit equal one-sixteenth of the mass
of the oxygen-16 atom) the mass decrement is the difference between the
masses of the individual nucleons that make it up and that value of 4.00260
amu. This difference is
2 * 1.007276 + 2 * 1.008650 + 2 * 0.0005486 - 4.00260 = 0.00260 amu.
This is a substantial difference relative to the small mass of the helium
and accounts for the stability of the helium nucleus. In fact, it is so
stable that other comparatively unstable nuclei often decay by emitting
helium nuclei, also known as alpha (α) particles.
The first type of decay we will consider is alpha decay. The general
class of this reaction at the nuclear level is
AX -> (A-4)Y + 4He
+ γ + Q
where Y is the nucleus with two fewer protons than X, and the
photon (gamma ray) may or may not be produced. Q is intended to keep track
of kinetic energy released in the reaction.
Alpha decays occur only with heavy (Z>83) nuclei except for two nuclei
that look almost like double-alpha particles themselves: 8Be
and 8B. The 8Be nucleus is literally a pair of
alpha particles mashed together, and 8B isn't much different,
so it is unsurprising that they can adopt this mode of decay that otherwise
is found only in heavy nuclei.
Alpha particles are emitted with some kinetic energy Q. This energy
is higher for starting nuclei with short half-lives than for those with
long half-lives. The kinetic energy is characteristic of the alpha decay
in question, so for any given decay path there is only one energy that
will be found.
Alpha particles are charged and heavy and consequently deposit large amounts
of energy in the medium through which they travel. So they are high-linear
energy transfer particles. They are rarely radiologically useful, because
they cannot penetrate far enough to be helpful either as therapeutic agents
or as diagnostics.
This occurs in the opposite case--where N/Z is too low for the range
of Z in which the nucleus begins. Here a proton is converted to a neutron,
a positive electron or positron:
proton -> neutron + e+ + neutrino
with class reaction
AX -> AY +
β+ +
ν + γ + Q
This time the nucleus Y has one less proton than the starting
nucleus X.
Thus this type of decay is similar to negative decay. But it has some
special properties for two reasons. First, the resulting nucleus has one
less proton in it, so the atom will need to yield up an electron to remain
neutral. This becomes part of the overall mass-balance equation of the
system. The other sense in which positron decay is different is that the
emitted charged particle--the positron--is a stranger in conventional matter.
In most cases it will quickly undergo annihilation by interacting with
an ordinary, negatively charged, electron, to produce a pair of photons:
β+ + e- ->
2γ
In practice the pair of photons will be produced with equal energies
and will travel in almost exactly opposite directions. This is because
the energies input into this reaction are almost entirely the mass energies
of the positron and the electron--0.511 MeV each, resulting in photon energies
of about 0.511 MeV each. This corresponds to a photon wavelength of 2.46
pm. These photons will play a role in the radiation biophysics of elements
that decay by positron emission.
This is a different mechanism by which neutron-poor species can become
more stable. In this case the nucleus recruits an ordinary orbital electron
from one of the inner shells and in doing so converts a proton to a neutron.
Thus the class reaction is
AX + e- -> AY
+ ν + γ + Q
where, as in positron decay, the nucleus Y has one less proton than
the starting nucleus X.
The resulting atom will be neutral "automatically" if the original
atom was, since the above equation already results in charge conservation.
The only problem in the resulting atom is that there's a hole in an inner
shell. Atoms with vacancies in inner shells are unstable, and restabilize
by means of transitions in which free electrons or outer-shell electrons
move "down" to fill the vacancy. Typically these conversions result in
emissions of one or more photons. These photons are known as K-capture
X-rays, assuming that a K-shell orbital is the one that had been vacated
and re-filled.
This isn't quite the whole story in radioactive events. The output nucleus resulting from several of the above mechanisms is often left in an excited state, and the nucleus can decay by gamma (photon) emission. But it can also decay by a transfer of the nucleus's excitation energy to an orbital electron, which is then ejected. As in an EC event, this requires grabbing a free electron to fill the inner shell, with resulting emission of a gamma ray. The text provides further details.
This type of electron emission constitutes a minor variation of the "internal conversion" theme in which an L- or M-shell electron is recruited instead of a free electron. The name is pronounced "Oh-Jay".
Spontaneous fission is not really covered in the text, but it is in principle a form of radioactive decay. The energy emitted is large compared to some of the other mechanisms described here. Spontaneous fission occurs only when the starting nucleus can by a straightforward mechanism split into two roughly equal fragments with more favorable binding-energy properties than the starting nucleus. Relatively few nuclei do this, and none of them is therapeutically or diagnostically useful.
The product nucleus (typically written as Y in the above reaction schemes) is not necessarily stable. If the product is itself unstable, it will decay further, so that the a multi-step pathway of decay occurs. Thus calculations of activity for a starting nucleus need to take into consideration all of the components of that pathway, each of which will have a characteristic half-life (see below).
Many nuclei can decay by more than one mechanism. An unstable nucleus might decay in some cases by positron emission, while in others by electron capture; another might decay by alpha emission in some cases and by a series of beta (positive and negative) emissions in other cases. The probabilities of the various transitions that a specific nucleus can undergo are properties of that nucleus, and can be calculated with some precision; as a result, the percentage of decay by a particular approach is a statistically observable value. As a result, the multi-step decay schemes just mentioned can involve branches, and decay calculations can get complicated. The text has details.
Activity, as we'll discuss in a moment, is determined by the rate of disintegrations of a nuclide. The old unit for activity was the Curie (Ci). It was originally defined in terms of the activity associated with 1 g of radium, but that number actually went up with time, as the ability of chemists to purify radium improved. The fixed definition was set at 3.7*1010 disintegrations per second. The modern unit for activity is the becquerel, which is simply one disintegration per second. Thus 1 Ci = 3.7*1010 becquerel, or 1 becquerel = 2.703*10-11 Ci.
It's useful to recognize that disintegrations don't really have units at all; they're simply counted. Therefore a becquerel is really a unit of inverse time, like the Hertz. When we specify activity, we're not saying anything about how a nuclide decays. As Alpen says, "Whether one emission acoompanies a disintegration or whether four emissions result from a single disintegration, it is counted as a single event for the determination of decay."
The underlying reality of radioactivity is that it is a random process
in which the probability that any given atom will decay is fixed.
Therefore the number of atoms ΔN that will decay
in a particular time interval Δt will depend only on the
nature of the nucleus and the number N(t) of nuclei present at
that time. If we double the number of atoms in the sample, we will double
the number that will disintegrate. In the limit of small
Δt we can express this as
-dN/dt = λN
where λ is a constant characteristic of the nuclide.
This is a terribly simple differential equation that may be solved
by restating it as
dN/N = -λdt
We can integrate both sides of this equation to get
ln|N| = -λt + C
We don't actually need the absolute value signs here, because the number
of atoms is always non-negative, so
lnN = -λt + C
To compute the value of the constant C,
we set the value of N to N0 at time t = 0.
Thus lnN0 = -λ(0) + C,
so C = lnN0 and the activity equation is
lnN = -λt + lnN0, or
lnN - lnN0 = -λt
A basic identity of logarithms says that lnA - lnB = ln(A/B)
for B ≠ 0, so
lnN/N0 = -λt
We can raise e to the power of both sides of this equation:
exp(lnN/N0) = exp(-λt)
But exp(ln(A)) = A for any A, so
N/N0 = exp(-λt), or
N = N0exp(-λt)
We define activity as the rate at which a nuclide decays. So if N(t)
is the number atoms of a nuclide present at time t,
then the activity A can be expressed as
A = -dN/dt
where the negative sign reminds us that we are losing atoms of our starting
type.
We define τ = mean life as the average lifetime for all the atoms
in a starting population of emitters. By definition of an average,
τ = (1/(0 - N0))
∫N00 t dN
but since dN = -λNdt and
N = N0exp(-λt),
τ = (-1/N0)
∫0&infin
t(-λN0)exp(-λt)dt
where the limits of integration are zero and infinity because the number
of atoms is N0 at time t = 0 and it is zero
after infinite time has elapsed.
The two minus signs cancel and the N0 in the numerator
and the denominator cancel, so
τ =
λ∫0∞
texp(-λt)dt
This integral can be determined fairly easily.
Let u(λ) =
∫0∞
exp(-λt)dt
We know what this integral equals: we substitue v = λt
so du = dv / λ
Thus u(λ) =
(1/λ)∫0∞
exp(-v)dv = 1/λ
Therefore du/dλ = -&lambda-2.
But we can also take our original definition for u(λ)
and differentiate with respect to λ
inside the integral to get
du/dλ =
∫0∞
(-t)exp(-λt)dt =
-∫0∞
texp(-λt)dt =
-λ-2; thus
τ =
&lambda*(-du/d&lambda) = &lambda * &lambda-2
= 1/λ
Therefore the mean life is the reciprocal of the activity constant.
Hence τ =
t1/2 / 0.693147 =
1.4427t1/2
Specifics regarding radioactivity and interaction with matter