The first points to note are that free neutrons are unstable particles with a half life of about 10 minutes and that nuclei formed by neutron capture are often unstable with respect to beta-decay
For example ^{58}Fe + n ^{59}Fe ^{59}Co + e^{-} +
Neutron capture processes naturally divide into two classes
^{16}O + ^{16}O | ^{31}S + n | |
---|---|---|
or | ||
^{14}N + ^{4}He | ^{18}F | |
^{18}F | ^{18}O + e^{+} + | |
^{18}O + ^{4}He | ^{22}Ne | |
^{22}Ne + ^{4}He | ^{25}Mg + n |
In stellar explosions, supernovas, the time scale is of the order of 10^{4} seconds and the explosively produced neutron flux is such that the r-process occurs.
There is also a distinction between the isotopes that can be
produced in the two processes. While some can be
obtained by either, there are some which can only be produced by the s-process,
with relatively few neutrons, some that can only be produced by
the r-process, all those heavier than lead or bismuth and
generally those with the largest neutron excess, and a few that can
be produced by neither, those with the largest number of
protons.
These latter are known as bypassed nuclei and may have been
produced by proton capture. Altogether there are about 30
nuclei of this type which are produced by secondary processes
acting on s- and r- process material - they are much less abundant.
Suggested processes are
(Z, A) + n (Z, A+1) + |
---|
(Z, A+1) (Z+1, A+1) + ^{-} + |
---|
t = 1/(n) = 3 x 10^{22}(1/n) s = 10^{15}(1/n) years |
---|
Suppose that an s-process is occurring with a neutron density n. The rate of production of nucleus A is given by :
dN_{A}/dt = n{(_{A-1})N_{A-1} - (_{A})N_{A}} |
---|
dN_{A}/dt = _{th}n{(_{A-1})_{th}N_{A-1} - (_{A})_{th}N_{A}} |
---|
d = _{th} n dt |
---|
dN_{A}/d = _{A-1}N_{A-1} - _{A}N_{A} |
---|
There is also a natural termination to the s-process sequence since ^{209}Bi is the most massive stable nucleus. Neutron capture by ^{209}Bi leads to a nucleus which decays by alpha particle emission to ^{206}Pb. Thus the sequence of equations would be
dN_{56}/d | = | - _{56}N_{56} |
---|---|---|
dN_{A}/d | = | _{A-1}N_{A-1} - _{A}N_{A} |
dN_{206}/d | = | _{205}N_{205} - _{206}N_{206} + _{209}N_{209} |
dN_{A}/d < 0 | if | _{A}N_{A} > _{A-1}N_{A-1} |
---|---|---|
dN_{A}/d > 0 | if | _{A}N_{A} < _{A-1}N_{A-1} |
_{A}N_{A} ~ _{A-1}N_{A-1} | ie | dN_{A}/d = 0 |
---|
A | N_{A} % | class | (10^{-31} m^{2}) | N |
---|---|---|---|---|
144 | 2.8 | p | 119±55 | 342±158 |
147 | 14.94 | r,s | 1173±192 | 17600±2900 |
148 | 11.24 | s | 258±48 | 2930±540 |
149 | 13.85 | r,s | 1622±279 | 22500±3900 |
150 | 7.36 | s | 370±72 | 2770±535 |
152 | 26.90 | r,s | 411±71 | 11100±1900 |
154 | 22.84 | r | 325±61 | 7430±1400 |
Here
(N)_{148}/(N)_{150} = 1.06 ± 0.27 |
---|
(N)_{148}/(N)_{150} = 1.02 ± 0.06 |
---|
Overall the evidence for the s-process production of heavy elements is good and we can add that observation of the spectra of technetium (Z = 43 ) in the light from stars is further evidence. This element has no stable isotopes - the longest lived has a half-life of 26 x 10^{6} years which is much shorter than the lifetime of stars - thus the element must have been produced in the star in which it is observed.
6.2 The r-process - The r-process is much less straightforward than the s-process both from the point of view of calculating expected abundances and the supporting evidence from observed abundances. The major factor pointing to its probability as a process is the existence of elements heavier than ^{209}Bi for the production of which no other explanation is known.
To obtain the expected abundances due to the r-process the contribution due to the s-process, calculated in the way described above, is first subtracted from the observed values. A sketch of the residues from such a proceedure is shown below.
The shift of the peaks to lower A value can be understood when it is remembered that stable nuclei are not produced directly by the r-process, only indirectly by beta-decay from the neutron rich nuclei. In beta-decay neutrons are transformed into protons and so A is conserved. The value of A equivalent to a neutron number of 126 say is lower in the neutron rich nuclei than it is in the stable nuclei. This point and the differing routes of the s and r-processes are illustrated below.
To help understand the r-process we should look once again at the nuclear binding energies. We write the nuclear mass as
M(A, Z) c^{2} = Z m_{p} c^{2} + (A - Z) m_{n}c^{2} - B(A, Z) |
---|
The rapid neutron absorption process
M(A, Z) + m_{n} M(A+1, Z) |
---|
B(A, Z) = B(A+1, Z) |
---|
Q(A+1, Z) = B(A+1, Z) - B(A, Z) ~ 2 MeV |
---|
A problem in calculating these halt-points in the r-process is that the form of the binding energy as a function of A and Z is not known for nuclei far from the region of stability (ie for extremely neutron rich nuclei). There is a fairly simple model for the stable nuclei and it takes the form
B = | a_{v}A | - a_{s}A^{2/3} | - a_{c}Z(Z - 1)/A^{1/3} | - a_{a}(N - Z)^{2}/A | - a_{p}A^{-3/4} |
---|---|---|---|---|---|
latent heat of vapourisation |
surface energy |
coulomb repulsion |
asymmetry | pairing |
Clearly it is a considerable extrapolation to use such a model with these values in the region of neutron rich nuclei. There is a further effect and that is the abrupt change in binding energy which occurs at neutron shell closures (magic numbers) but the expression above takes no account of this. Thus further terms have to be added and also others which simulate the changes to stability which are associated with the deformation of the nucleus. These considerations render the whole process rather uncertain - although model calculations have been made in particular to calculate the time spent at the main halt-points. Some estimated values are
N = 50 | t_{1/2}~ 0.8 s |
---|---|
N = 82 | t_{1/2}~ 0.4 s |
N = 126 | t_{1/2}~ 0.4 s |
When discussing binding energy earlier we made the point that heavier nuclei energetically favoured fission. This becomes more and more likely in the mass region 230 < A < 270 where spontaneous fission finally terminates the r-process and also yields further seed nuclei to be built up by neutron absorption. Thus the r-process can be thought of as cyclic and for neutron densities n ~ 10^{30} m^{-3} and temperatures T ~ 10^{9} K the cycle time is estimated to be about 5 s.
We have seen that for each value of Z there will be a waiting point beyond which the process of neutron absorption cannot continue until a beta-decay occurs. Thus for the abundance at each Z value we can write
dn_{Z}/dt = _{Z-1}n_{Z-1} - _{Z}n_{Z} |
---|
At the neutron magic numbers (N = 50, 82, 126) there will be a sequence of waiting points because the binding energy of the next neutron is so low. There will be a succession of beta-decays until a region in which the neutron binding energy is significant is reached. Then the absorption of neutrons will continue. From the time estimates given above it can be seen that these waiting steps are considerable and the resulting pile up of nuclei leads to the peaks in the abundance curve. The lines of constant A from these progenitors indicate that beta-decay will finally produce peaks in stable nuclei abundances at A ~ 130 and 195 in good agreement with observations. These peaks remain visible in the overall abundance curve which was shown in the introduction to the course. That illustration is repeated here with the peaks marked out. The s-process produces somewhat sharper peaks at the magic neutron number nuclei themselves - these are also indicated and serve to emphasise how much the r-process peaks are shifted.