Nuclear applications - physics.wisc.edu induced fission of 235 92U produces a distribution of fission products. A slow neutron is captured producing an excited vibrating state 235

December 09 Modern Physics

Nuclear applications


Outline

Nuclear Reactions Cross Sections Interactions of neutrons Nuclear Fission Nuclear Reactors Nuclear Fusion Interaction of particles with matter Radiation Damage in Matter Radiation Detectors Uses of Radiation


Nuclear reactions

Nuclear reactions conserve: -the number of nucleons, -the total electric charge -energy -linear momentum -angular momentum

Notation:


Energy considerations

In this fusion reaction, a neutron strikes a deuterium (D) nucleus, fuses with it to form a tritium (T) nucleus, and the binding energy is released as a high energy photon (gamma ray). Relativistic energy conservation reads for D at rest


Kinematics A reaction with Q>0 is exothermic - energy is released - and can proceed with minimal initial kinetic energy although Coulomb repulsion can be another barrier.

A reaction with Q<0 is endothermic and requires the input of kinetic energy in the initial state particles.

The threshold kinetic energy of ‘a’ for Q<0 in a frame with X at rest is


Derivation of threshold energy Consider a+X=>b+Y where X is a target at rest.

Energy conservation:

In the center of momentum frame

So we write (non-relativistic) energy conservation as


Transform to the cm frame The velocity of the cm in the lab frame is given by

In the cm, X has the negative of this velocity so

Given p from this equality (knowing masses and velocities of a and X), we can find the momentum of the final particles in the cm:

(see next slide)


Energy conservation

The left hand side must be >0. If Q>0 (exothermic), since p2>0, energy is released and appears as KE of b and Y. If Q<0, kinetic energy must be injected to achieve |p’|>0.

The initial kinetic energy to just permit production of the final state is that for which the final particles are produced at rest, ie p’=0.


Threshold condition

If ‘a’ is a single nucleon and X a heavy nucleus, the threshold energy is essentially -Q.


Reaction cross section Consider particles incident upon a thin foil. The rate R at which reactions occur is proportional to the density of targets n and to the thickness x:


Attenuation length If the reaction eliminates the incident particle from the beam via absorption or scattering, the change in the number of beam particles in a distance dx is given by:


Classical cross sections

The classical cross section for a spherical object of radius A to interact via a contact force with a spherical object or radius B is the area

A+B


Order of magnitude The cross section for a high energy nucleon to interact with a large nucleus is approximately the classical projected area of the nucleus.

The density of nuclei in normal stuff is 1/ angstrom3

so the nuclear attenuation length is of order


Non-classical cross sections

If the interaction is not a contact interaction, the geometrical interpretation is fuzzy. A nucleus is semi-transparent to electrons and neutrinos and the cross sections for their interactions on nuclei very small compared to the size of the nuclei.

Particles are governed by wave mechanics and the scattering cross section depends on the ratio of the incident particle wavelength to the nuclear radius and hence depends on energy significantly.


Example

A fast neutron beam passes through a piece of aluminum foil of thickness 0.1 mm. If the cross section for scattering is geometrical, estimate the fraction scattered out of the beam.


Example A fast neutron beam passes through a piece of aluminum foil of thickness 0.1 mm. If the cross section for scattering is geometrical, estimate the fraction scattered out of the beam.

Aluminum has a mass number of 27-ish so radius

The density of nuclei is

The fraction scattered is


Fission

Fission was discovered in 1938. Bombardment of 235 92U resulted in 56 Ba and also 57 La.


Fragment distribution Neutron induced fission of 235 92U produces a distribution of fission products.

A slow neutron is captured producing an excited vibrating state 235 92U* which breakups into typically two large fragments with several neutrons emitted.


Fission yield

The difference in binding energy per nucleon for A=235 and A =100 is about 1 MeV. Several hundred MeV is released per fission as kinetic energy of the products.


Chain reaction The neutrons released in neutron induced fission may induce further fissions and a chain reaction!

Key is the proviso that the neutrons are not lost to absorption by lighter elements or escape. The net amplification K of neutrons must exceed 1.


Chicago pile

The first successful test of a self sustaining nuclear reaction took place in 1942 under the stadium at the University of Chicago in an effort led by Enrico Fermi.


Control of fission Natural uranium contains 7% of fissile 235 92U and over 90% 238 92U which absorbs neutrons without fission. A critical mass is required. Other absorbers inserted mechanically control absorption.

For 235 92U, the fission cross section is large only for thermal neutrons so a moderator must be used to slow the neutrons down.


Nuclear power station

Controlled fission is the basis for nuclear power generators.


Nuclear fusion Fusion of light elements to form more stable nuclei also releases energy.

Fusion requires overcoming the electric repulsive barrier between nuclei to reach contact between nuclei, to be within range of the nuclear force.


Thermal induced fusion We can estimate the temperature such that light nuclei have sufficient kinetic energy to achieve contact and fusion from the electrostatic potential energy at a separation R approximately equal to the sum of the radii:

Room temperature is 1/40 eV so T~40 million times hotter! Fusion is achieved in the center of the Sun.


Fusion reactions Solar processes

Power generation


Fusion power Practical power generation in a hot D-D and D-T plasma requires that the energy produced exceed that lost (to gamma rays and neutrons). Generated power is a strong function of temperature and the critical ignition temperature may be calculated to be equivalent to only 10’s of keV.


Lawson number The energy required to heat the plasma is proportional to ion density n. The energy released is proportional to collision rate and hence to n2 and to the confinement time. Equating these leads a critical density for a given confinement time:


Toroidal plasma design

In a tokamak, a toroidal magnetic field going around the doughnut combines with a poloidal field due to circulating plasma current to confine a hot plasma in “vacuum.”


International Thermonuclear Experimental Reactor

http://www.iter.org/

Goal is to demonstrate self-sustaining power generation. See web site for status.


Madison Symmetric Torus

The Plasma Physics group at UW-Madison is a national leader in plasma confinement research.

http://plasma.physics.wisc.edu/mst/html/mst.htm


Inertial confinement

An alternate “brute force” approach to achieve fusion is inertial confinement - implosion of a ~mm fuel pellet by focused beams of laser light or accelerated particles.


Schematic inertial confinement fusion power station


Interactions of charged particles with matter

A heavy/ fast charged particle passing an atom repells or attracts the electrons and imparts a momentum impulse sideways to its path.

q

dp = F dt

Impulses sufficiently fast and large produce ionization.The energy loss to electrons slows the particle down and a trail of ions and electrons remains.

(small r less probable)

(large r more probable)


dE/dx The rate of energy loss per meter of material traversed (dE/dx) depends on electron density and on energy. At low velocity, the force acts too gradually to result in ionization; at high energy the speed is c and the interaction time constant so dE/dx approximately energy independent.


Range Protons and alpha particles slow down through dE/dx energy loss and come to rest.

The range or distance before stopping in air is 10 cm for an alpha of 10 MeV.

Condensed matter is 1000 times more dense and the range correspondingly smaller.


Interaction of photons with matter

Optical and X-ray photons interact with atomic electrons via absorption (PE effect) and Compton (elastic) scattering. The cross section for photon interactions decreases dramatically with energy. Gamma rays with energy >1 MeV can produce an electron-positron pair e+e-.


X-ray absorption For picometer wavelength X-rays, the attenuation length in water is a few cm.

The differential absorption length is the basis for X-ray contrast images in medicine.


Gamma ray absorption For gamma rays, the attenuation length in for example lead is a few mm.

Generally, absorption by pair production varies with Z2.


Radiation dose Radiation exposure damages materials - ionization is created, bonds disrupted, nucleui destroyed. The damage is determined to an extent by the total energy deposited and this is measured in units of 1 rad = dose to deposit 10-2 J/kg.


Biological radiation damage As a measure of the relative importance for biological tissues, RBE factors are assigned to different particles and 1 rem = dose(rad)*RBE.


Ionization detection The ion chamber uses an electric field to collect the ionization produced in a gas.


Geiger counter In the Gieger counter, the high electric field near the wire produces an avalanche (plasma/spark) discharge so one (constant size) pulse per ionization trail event.


Photomultiplier tube Optical photons may be detected one by one with a photomultiplier tube. A photemissive surface sources one electron per absorbed photon. Multiplication of a single electron is achieved via a cascade of collisions with metal plates.


Bubble chamber In a bubble chamber, ions nucleate the growth of bubbles in a near boiling tank of liquid hydrogen or other liquid. The tank is depressurized during particle passage. Photographs of the bubble trails provide striking images of elementary particle interactions.


Radiation in medicine Radiation in various forms is used in medicine.

Radioactive Na and I are used as tracers of blood flow and biological processing.

Positron Emission Tomography (PET) uses an injected radioactive glucose. Positrons annihilate to back to back gammas, detected and reconstructed.


Radiation therapy Radiation beams are used to intentionally damage and eliminate cancerous tissues .


Summary

Nuclear reactions naturally tend to produce stable intermediate mass nuclei like Fe.

Fission and (we hope) fusion are clean energy sources.

Radiation from nuclear sources, creation of particle beams, and radiation detection are applied in imaging and as tracers, in medicine and many other arenas.

Documents

Nuclear applications - physics.wisc.edu induced fission of 235 92U produces a distribution of fission products. A slow neutron is captured producing an excited vibrating state 235