Nuclear Reactions A Z. Radioactivity, Spontaneous Decay: Nuclear Reaction, Induced Process: x + X Y + y + Q Q > 0. Exothermic Endothermic

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1 Radioactivity, Spontaneous Decay: Nuclear Reactions A Z 4 P D+ He + Q A 4 Z 2 Q > 0 Nuclear Reaction, Induced Process: x + X Y + y + Q Q = ( m + m m m ) c 2 x X Y y Q > 0 Q < 0 Exothermic Endothermic 2

2 13 13 x + X Y + y + Q p + C N + n + Q The energy to drive an endothermic reaction is usually supplied by x. Since momentum must be conserved in the nuclear reaction, x must be given a kinetic energy larger than Q. m M E = + threshold M Q

3 Cross Section p + C N + n + Q If probability of this reaction is a function of the kinetic energy of the proton. One way to express this probability is as a cross section, σ = R where R is the number of reactions that occur per unit time per nucleus and I is the number of particles incident per unit time per unit area. When 13 C is bombarded with energetic protons, there are several possible results and each process can be described by a partial cross section scattering C( p, p) C σ pp, ( pn, ) capture ( p, α) / I C( p, n) C( p, γ ) C( p, α) p, α The probability of some event can be expressed as the total cross section: σ = σ pp, + σ pn, + σ p, γ + σ p, α +... N N B σ σ σ pn, p, γ

4 Compound Nucleus Bombardment by an energetic particle may produce a compound nucleus. It exists for a short but non-zero amount of time. B + α N* C + d N + n N* C + p N + γ N* N + γ C + d N* B + α C + p N* N + n The left reaction is called the entrance channel and the right reaction is called the exit channel. The relative probabilities (or branching ratios) of producing various products will not depend on how the compound nucleus was produced The cross section for C( p, n) N will be σ pn, = σ cpn, the cross section for creating the compound nucleus times the probability that it will decay to 13 N.

5 B + α N * The energy dependence of the cross section displays peaks whose positions tell us the nuclear energy levels in 14 N and whose width tell us the lifetime of those excited states, E t = Γτ.

6 Energy Loss Spectroscopy Inelastic scattering events leave the nucleus in an excited state.

7 Neutron Capture Fast moving neutrons will collide many times and lose energy in these collisions. When their energy is of the order of kt, they have been thermalized and are called thermal neutrons. Thermal neutrons will eventually be captured by a nucleus. n+ M M + 1 A A+ 1 γ 0 Z Z Neutron capture usually produces an unstable nucleus, which may decay by alpha decay, beta decay, or fission. Nuclear reaction cross sections are most conveniently measured in barns where 1 barn = ( 10fm) = 10 m n+ Ag Ag + γ

8 A heavy nucleus spontaneously splits into two smaller nuclei. The combined mass of the daughter nuclei is less than the mass of the parent nucleus. Bombarding uranium with neutrons can cause it to undergo fission, splitting into two nearly equal parts and releasing about 200 MeV of energy! * 0 n+ 92U 92 U ( 1 ) Ba+ Kr + 3 n 236 U is fissile but 239 U is not. Since 99.3% of natural uranium is 238 U and only 0.7% is 235 U, uranium must be enriched to be used for fission. 0 Nuclear Fission

10 When Uranium undergoes fission, it also emits neutrons which may cause another Uranium nucleus to undergo fission. Put enough uranium of the right isotope together (called critical mass) and you can have a self-sustained chain reaction.

11 A self-sustained reaction requires the reproduction factor,. k 1 A moderator is used to slow down the emitted neutrons to increase the rate of neutron capture. Heavy water is a common moderator because 2H is light enough to be a good moderator, but has a small cross-section for neutron capture.

14 Super-heated moderator water separate from turbine water. Cadmium control rods raised and lowered to keep reaction critical. Possible because some neutrons are not prompt but delayed. Containment structure to prevent accidental releases.

15 Breeder Reactors * Produce fissile plutonium-239 from non-fissile uranium-238. * Fuel doubles in 7 to 10 years. U + n U* U + γ U Np + β + ν Np Pu + β + ν Plutonium-239 Breeder Reactors * more neutrons per fission * captures fast neutrons so no moderator needed * fewer delayed neutrons make control more difficult * higher operating temperature * liquid sodium used for heat transfer * loss of coolant much more serious problem

2 3 4 Fusion H+ H He + n + 17. 6MeV Fusion of two light nuclei into a heavier nucleus. Nuclear reactions which power stars. Less difficult to obtain fuel, possibly. Less problem with nuclear waste.

16 2 3 4 Fusion H+ H He + n MeV Fusion of two light nuclei into a heavier nucleus. Nuclear reactions which power stars. Less difficult to obtain fuel, possibly. Less problem with nuclear waste. Higher efficiency as measured by energy released per nucleon. Requires large energies to get two nuclei close enough for fusion to occur. Too much energy if we bombard one nucleus with the other. Need to get thermal energies high enough to overcome repulsion. 8 T ~ 10 K kt ~ 10 kev At these temperatures, the reactants exist as a plasma of ions and electrons. In the sun, the gravitational force provides confinement. In man-made fusion reactors, we use magnetic confinement.

17 Inertial Confinement - ion or electron beams bombard a pellet of frozen deuterium or tritium

18 Proton Cycle Proposed by Hans Bethe, the chain reaction that powers the sun. + H+ H H + e + ν MeV H+ H He + γ MeV He+ He He + 2 H + γ MeV e

19 Neutron Activation Analysis A Z M( n, γ ) A+1 Z During neutron irradiation, the production rate of A+1 M is R0 = N0σ I where N is the number of A 0 Z M in the sample, σ is the neutron capture cross section and I is the neutron flux. The decay rate (also the rate of gamma production) will be M t t Rt ( ) = Nt ( ) = R( e λ λ λ 1 ) = Nσ I( 1 e ) 0 0 If the irradiation proceeds for more than several half-lives, saturation will occur: R( ) = λn( ) = R = N σ I Z 0 0 The number of A Z M in the sample can then be determined by measuring the gamma production rate: R N = ( ) 0 σ I

20 U Nuclear Magnetic Resonance = µ B For hydrogen atoms, the energy difference between the two spin states of the nucleus will be ( ) E = 2 µ B= hf z p where f is the frequency of EM waves needed to resonantly excite the transition from the lower to the upper state. At a magnetic field of ~ 1 T, this frequency is around 42.5 MHz. Since the magnetic field splitting the states is the sum of the external and internal magnetic fields, this frequency is extremely sensitive to the hydrogen atom s environment. Magnetic Resonance Imaging (MRI) maps the resonant frequency as a function of position. Using a spatially varying external field (whose direction can be changed) allows one to distinguish signal from various locations in the field.

Computer Assisted Tomography (CT or CAT scan) Short-lived gamma emitters that bind to particular organs can be used to image specific areas of the body using a gamma scan.

21 Computer Assisted Tomography (CT or CAT scan) Short-lived gamma emitters that bind to particular organs can be used to image specific areas of the body using a gamma scan. Like traditional x-ray images, gamma scans give a projection. Three dimensional images of a transverse slice, a tomograph, can be obtained by rotating the source and detector around the person and using a computer to calculate the image. The combination of a gamma scan and the rotating detectors of the tomograph is called SPECT, single photon emission computer tomography.

The annihilation usually produces two 0.511 MeV photons which travel in opposite directions.

22 Positron Emission Tomography (PET scan) Use of a positron emitter instead of a gamma emitter can enhance resolution. The emitted positron has a short mean free path before undergoing annihilation with an electron. The annihilation usually produces two MeV photons which travel in opposite directions. By detecting both photons in a coincidence counting apparatus, more accurate information about the location of the source is obtained.

23 Carbon Dating N( n, p) Carbon-14 is produced in the atmosphere by the reaction. It is incorporated in carbon dioxide molecules and then into all living tissue. It decays back to nitrogen-14 by the reaction T12 / = 5730 y with a half-life of. C N + β + νe All living things have a ratio of carbon-14 to carbon-12 of [ 14 C] [ 12 ] C =. Once they die, the ratio decreases as time passes. The ratio can be used to date the time of death. It s only useful for times in the 100 to 100,000 year time scale. Other ratios can be used for other time scales [ 238 ] [ 235 ] [ 87 U U Rb ] [ 206 ] [ 238 ] [ 87 Pb U Sr] 12,,, [ 40 K] [ 40 Ar] C

24 Accelerator Mass Spectrometry

25 Particle Induced X-ray Emission (PIXE)

Chapter 12: Nuclear Reaction

Chapter 12: Nuclear Reaction A nuclear reaction occurs when a nucleus is unstable or is being bombarded by a nuclear particle. The product of a nuclear reaction is a new nuclide with an emission of a nuclear