Topic 6 - Neutron Capture and the Production of Heavy Elements

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 58Fe + n 59Fe 59Co + e- +

Neutron capture processes naturally divide into two classes

  1. the s-process - slow neutron capture, where the produced nucleus decays to a stable nuclide before further neutron capture occurs.
  2. the r-process - rapid neutron capture, where the neutron flux is so high that the nucleus captures many neutrons before it can decay.
There is a clear distinction between the two processes. In stars the neutron flux is produced and maintained by nuclear interactions like the above cited

16O + 16O 31S + n
or

14N + 4He 18F
18F 18O + e+ +
18O + 4He 22Ne
22Ne + 4He 25Mg + n

These can occur during helium burning - remember that 14N is a common end product of the CNO cycle. The flux is relatively low and so there is ample time for beta-decay to occur between successive neutron capture reactions. These are the conditions in which the s-process provides nucleosynthesis. The typical time scale is 104 years or so.

In stellar explosions, supernovas, the time scale is of the order of 104 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

  1. (p,) and (p,n) reactions which could occur during explosive burning.
  2. (,n) reactions - at temperatures of about 109 K thermal radiation can have sufficient energy to eject neutrons from nuclei.
  3. Inverse beta-decay - at high temperatures there will be a certain number of e+e- pairs in thermal equilibrium and e+ capture by a nucleus can produce a more proton rich species.

Section of the chart of nuclides

Summary
  1. The path of s-process production follows through the beta stable nuclides - coloured squares - stepping up in Z through beta-decay.
  2. The r-process nuclides are obtained via successive beta-decays from nuclei with a large excess of neutrons.
  3. Some nuclides are shielded from the r-process and so can only be produced by the s-process.
  4. Two bypassed nuclei are shown.
  5. An effect of nuclear structure is seen in that there are fewer stable nuclides with odd N or Z.
6.1 The s-process - If nuclei are placed in a neutron flux then the capture process (n,) produces isotopes of progressively larger A

(Z, A) + n (Z, A+1) +

When this process creates an unstable nucleus then the path switches to the next Z value via beta-decay

(Z, A+1) (Z+1, A+1) + - +

then neutron capture continues as before. Generally the present day abundance of a particular nuclide will have been the result of both s and r- processes but for those which can be produced only by the s-process we would expect some relationship with neutron capture cross-sections. Where the cross-section is small we expect a build-up leading to larger abundance.
To a high degree of accuracy, neutron cross-sections are inversely proportional to neutron velocity - this is true throughout the range of velocities relevant to the s and r processes. Thus the product can be considered as a constant and for a heavy nucleus it is about 3 x 10-23 m3 s-1. Then for a neutron density n m-3 such a nucleus will capture a neutron in a time t given by

t = 1/(n) = 3 x 1022(1/n) s = 1015(1/n) years

which implies a neutron density of 1011 m-3 for a characteristic time of 104 years. On the other hand the r-process requires a density of 1022 to 1030 m-3 which can only happen if a significant fraction of matter in the locality is converted into neutrons.

Suppose that an s-process is occurring with a neutron density n. The rate of production of nucleus A is given by :

dNA/dt = n{(A-1)NA-1 - (A)NA}

where the two terms on the right represent production and destruction respectively. Since is constant we replace with the thermally averaged cross-section th and by th (~ 2200 ms-1).

dNA/dt = thn{(A-1)thNA-1 - (A)thNA}

and defining a neutron exposure as.

d = th n dt

we obtain

dNA/d = A-1NA-1 - ANA

Thus the process is represented by a large number of differential equations linked together in the way indicated. The value of NA at any time influences the rate of production of the neighbouring elements. In order to use these equations to calculate abundances some boundary conditions are required. The starting conditions are usually taken to be the iron group nuclei and since in solar abundances we find N(53Fe ) ~ N(57Fe) ~ 0.1N(56Fe) the starting material is usually taken to be 56Fe.

There is also a natural termination to the s-process sequence since 209Bi is the most massive stable nucleus. Neutron capture by 209Bi leads to a nucleus which decays by alpha particle emission to 206Pb. Thus the sequence of equations would be

dN56/d = - 56N56
dNA/d = A-1NA-1 - ANA
dN206/d = 205N205 - 206N206 + 209N209

with initial conditions NA(0) equals N(0) at A = 56 and is zero for all other values of A.
From the main body of these equations it is clear that

dNA/d < 0 if ANA > A-1NA-1
dNA/d > 0 if ANA < A-1NA-1

Thus the system is self regulating - decreasing NA if it is high and increasing NA if it is relatively low - tending towards the balance

ANA ~ A-1NA-1 ie dNA/d = 0

This equality is called the local approximation because it is not expected to work where the cross-section is particularly low - for example in the region of the magic numbers of neutrons, N = 50, 82, 126. In the plot of the neutron capture cross-sections below the low values at the magic numbers can be clearly seen.


The extent to which the local approximation is valid between the magic numbers can be inferred from the next figure which shows the product of the solar abundances times the cross-sections N. This figure is only indicative as many of the points have sizeable errors but there is some evidence of flat regions between magic numbers.


Somewhat more convincing evidence can be obtained from looking at the isotopes of samarium in particular

A NA % class (10-31 m2) 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

while more accurate direct measurements of the ratio of the cross-sections yield

(N)148/(N)150 = 1.02 ± 0.06

which is in excellent agreement with the local approximation.

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 106 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 209Bi 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.


There are strong indications of peaks at A = 80, 130 and 195 which are about ten units below the atomic numbers equivalent to magic numbers of neutrons A = 90 (N = 50), 140 (N = 82), 208 (N = 126). As we have indicated above the neutron capture cross-section is exceptionally small at these values of N.

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.


The magic numbers N = 82, 126 are indicated and the two lines for A = 130, 195 give some idea of just how far from the region of beta stability the r-process nuclei are formed. On the other hand the s-process follows this latter very closely.

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) c2 = Z mp c2 + (A - Z) mnc2 - B(A, Z)

in terms of energy with B the binding energy.

The rapid neutron absorption process

M(A, Z) + mn M(A+1, Z)

is repeated over and over until the next neutron is not bound - that is to say it does not contribute to the binding energy or

B(A, Z) = B(A+1, Z)

In fact the process stops somewhat before this point because of photonuclear emission due to the thermal photons. For example under the conditions of neutron density 1030 m-3 and temperature 109 K the process stops at

Q(A+1, Z) = B(A+1, Z) - B(A, Z) ~ 2 MeV

under which circumstances the probability of photoneutron emission is just equal to the probability of neutron capture.

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 = avA - asA2/3 - acZ(Z - 1)/A1/3 - aa(N - Z)2/A - apA-3/4

latent heat of
vapourisation
surface
energy
coulomb
repulsion
asymmetry pairing

Typical values in MeV are:

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 t1/2~ 0.8 s
N = 82 t1/2~ 0.4 s
N = 126 t1/2~ 0.4 s

There are also some other longer than average half life values which total to about 0.4 s while the remaining 50 or so beta-decays which must occur contribute a further 0.5 s bringing the total time to ~ 2.5 s. The actual process would be longer because of the neutron capture rate and fission cycling if it occurs.

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 ~ 1030 m-3 and temperatures T ~ 109 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

dnZ/dt = Z-1nZ-1 - ZnZ

where Z is the decay constant at the waiting point for charge Z [ recall that radioactive decay is exponential, governed by N = No exp(-t) ]. This set of coupled differential equations is very similar to those for the s-process except that now we see that nZ will be inversely proportional to the beta-decay constants rather than the neutron absorption cross-sections. We have seen from the expression for the binding energy that a nucleus with odd N is less tightly bound than a neighbouring isotope (ie same Z) with even N thus the waiting points tend to be at even N nuclei where neutron absorption would produce a weakly bound nucleus which would rapidly lose the additional neutron.

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.


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