2007

From Mathsoc wiki

Contents

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Scattering

  • Cross-sections
  • Rutherford scattering
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Nuclear Structure

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Nuclear force

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Nuclear binding and nuclear mass defect

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Semi-Empirical mass formula

According to the semi-empirical mass formula, the mass of a nucleus is given by

m = Z m_p + (A - Z) m_n - \frac{B}{c^2}

where the binding energy B is given by

B = a_v A - a_s A^{2/3} - \frac{a_a}{A} (A - 2Z)^2 - \frac{a_c Z^2}{A^{1/3}} + \delta(A,Z)

where the terms are as follows:

  1. Volume energy term, a_v A due to the fact that B \propto A
  2. Surface energy term, - a_s A^{2/3} due to the less effective binding of surface nucleons. The term is proportional to surface area, which is proportional to R^2 = (A^{1/3})^2.
  3. Asymmetry energy term, - \frac{a_a}{A} (A - 2Z)^2. This term is zero when A = 2Z \Rightarrow Z = N, which is when light nuclei are most tightly bound.
  4. Coulomb energy term, - \frac{a_c Z^2}{A^{1/3}}, due to the Coulomb energy of a charged sphere lowering the binding energy. The Coulomb energy is given by - \frac{3}{5} \frac{(Ze)^2}{4 \pi \epsilon_0 R}.
  5. Pairing energy term \delta(A,Z), which is given by
\delta(A,Z) = \begin{cases}34A^{-3/4} & Z,N \,\, \mbox{even} \\ - 34A^{-3/4} & Z,N \,\, \mbox{odd} \\ 0 & A \,\, \mbox{odd} \end{cases}

which arises as nucleons like to pair with their spins opposed to each other.

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Decay

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Beta Decay

In all forms of beta decay, the parent and daughter nuclides are isobars of each other. On atomic mass parabolae of isobars, nuclides beta decay towards the the nuclide closest to the bottom of the graph, which is stable.

There are three types:

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Electron emission

This involves the emission of a fast electron from the nucleus. The decay is described by:

n \rightarrow p + e^- + \overline{\nu}

where n is a neutron, p is a proton, e^- is an electron and \overline{\nu} is an electron anti-neutrino. Note that this decay can occur for free as well as atomic neutrons. In all cases, the atomic number of the parent nucleus increases from

Z \rightarrow Z + 1.

The condition for the decay to occur is:

m_X > m_{X'} + m_e

or m_X - m_{X'} > m_e so as to make an electron.

In terms of atomic masses, the condition is

M_X > M_{X'}

as here the electron mass is included in the atomic mass of the daughter X'

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Positron emission

This is the emission of a fast positron from the nucleus. Inside the nucleus,

p + \mbox{energy} \rightarrow n + e^+ + \nu

where e^+ is the positron and \nu is an electron neutrino. The energy comes from nuclear rearrangements that occur during the decay, and so this decay cannot happen for a free proton. Here the atomic number decreases

Z \rightarrow Z - 1

The condition for the decay is:

M_X > M_{X'} + 2 m_e

where the factor of two in front of the electron mass is due to the daughter X' having one less proton and so one less electron.

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Electron capture

It is sometimes possible for the nucleus to capture one of the atomic electrons. Inside the nucleus,

p + e^- + \mbox{energy} \rightarrow n + \nu

and the atomic number again decreases

Z \rightarrow Z - 1

The condition for the decay is

M_X > M_{X'}

so electron capture can occur even when positron emission is barred.

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Alpha decay

This is the emission of a fast helium nucleus with kinetic energy of a few MeV from the nucleus.

In general,

X \rightarrow X' + \alpha

and the atomic and mass numbers both decrease

Z \rightarrow Z - 2, A \rightarrow A - 4.

In terms of nuclear masses, the condition for the decay to occur is

m_X > m_{X'} + m_{\alpha}

or in terms of binding energies

B_X < B_{X'} + B_{\alpha}

As the alpha particle is even-even it has a large value, for a light nucleus of \frac{B}{A} \approx 7MeV per nucleon, and so has a large mass defect. It is easier to satisfy the mass and binding energy conditions by ejecting an alpha particle rather than a nucleus with a small value of \frac{B}{A}.

For alpha decay to occur, a plot of \frac{B}{A} vs A shows that the parent nucleus must be a heavy nucleus.

For A > 120, \phi = \frac{B}{A} can be given approximately by

\phi = 9.3 - 0.0075A

which has slope

\frac{d \phi}{d A} = - 0.0075.

The binding energy condition then becomes (in MeV)

A \phi_1 < (A -4) \phi_2 + 28

where \phi_1 and \phi_2 are the values of \phi for X and X'. Now, \phi_1 - \phi_2 = 4 \frac{d \phi}{dA} = -0.03, so we find that

A > \frac{4 \phi_1 - 28}{0.03} - 4

which is satisfied for A > 151. In practice however the decay process is very slow until A > 208 (after which all nuclei are radioactive).

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Heavy element decay chains

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Radioactive decay law

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Analysis of parent-daughter activity relationships

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Nuclear fission

Nuclear fission is the process in which a nucleus splits into 2 fragments of roughly similar mass, together with considerable energy release.

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Spontaneous fission

Like alpha decay, this only occurs for heavy nuclei. The condition for the decay is

m_X > m_1 + m_2

where m_X is the mass of the parent nucleus, and m_1 and m_2 are the masses of the two fragments. In terms of binding energies,

B_X < B_1 + B_2

From this and the semi-empirical mass formula, fission is only energetically possible for A>90. However, it is unmeasurably slow until A \approx 230 (similarly to alpha decay, the timescale is set by process of quantum tunnelling through potential barrier surrounding parent nucleus).

The energy release is approximately 200MeV, or 1 MeV/nucleon. These value increase only slowly with A.

Fission is normally asymmetrical, ie fragment masses unequal. Note that the semi-empirical mass formula fails to predict this, even though energy release is maximised when each fragment has the same mass.

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Induced Fission

Induced fission is the almost instantaneous fissioning of the nucleus when struck by radiation, usually a neutron. It occurs in 2 steps:

U + n \rightarrow U^* \rightarrow 2 fragments + other products, where the U^* denotes an excited state of the nucleus.

The fission products are as follows:

  1. Fission fragments, approximately 160MeV kinetic energy in total.
  2. \gamma-rays emitted immediately during fission process, approximately 6MeV in total.
  3. Prompt neutrons from neutron-rich fragments, emitted approximately 10^{-14}s after fission, approximately 5MeV in total. An average of 2.5 are emitted per fission.
  4. Electrons from 3 or 4 beta decays of each fragment (still neutron-rich), approximately 5MeV in total with the electron anti-neutrinos also emitted by beta decay.
  5. \gamma-rays from fragments emitted after fission and beta decay, approximately 5MeV in total.
  6. Delayed neutrons occasionally emitted from fragments, about a minute after fission during the beta decay process.

Total energy release is approximately 200MeV \pm 10 MeV.

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Nuclear Power

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Fission Reactors

Basic reaction

The basic reaction is: U-235 + neutron \rightarrow 2 fragments + 2.5 neutrons + 200MeV.

There are four possibilities for the emitted neutrons:

  1. Escape from reactor
  2. Capture by non-fissile material
  3. Capture by uranium without fission
  4. Capture by uranium with fission

Natural uranium contains 0.7% U-235 and 99.3% U-238. Only the latter can undergo fission because: neutrons emitted are fast (between 1 and 2 MeV) and the fission cross-sections \sigma_f for fast neutrons are too low. Hence we must enrich to greater than 20% U-235, or else thermalise the neutrons - slow them down to thermal equilibrium with their surroundings, kinetic energy \approx kT = 0.0025 eV at room temperature. For U-235 have \sigma_f \propto \frac{1}{v}.

Critical energy, resonance peaks.

Moderators

A moderator is a bulk material surrounding uranium fuel rods designed to moderate (slow down) fission neutrons to thermal speeds. Commonly used materials are light water, heavy water and graphite.

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