Like the alchemist’s dream of chemical transformation Nuclear physics has found the means of transmutation

Nuclear Reactions or, if you prefer Besides his famous scattering of  particles off gold and lead foil, Rutherford observed the transmutation: or, if you prefer Though a more compact form is often used: Target (Projectile, Detected Particle) Residual Nucleus Whenever energetic particles (from a nuclear reactor or an accelerator) irradiate matter there is the possibility of a nuclear reaction

enclosed in brackets between the target and daughter nuclei the projectile and emitted particle are enclosed in brackets between the target and daughter nuclei or, if you prefer Though a more compact form is often used: Target (Projectile, Detected Particle) Residual Nucleus Convenient as the bracketed part can be used by itself to refer to a particular class of reactions like (a , p) or (n, g).

Classification of Nuclear Reactions divided roughly into the two groups compound nucleus reactions and direct reactions.

Classification of Nuclear Reactions direct reactions are of the types scattering reaction projectile and scattered (detected) particle are the same elastic scattering residual nucleus left in ground state inelastic scattering residual nucleus left in excited state Even inelastic proton scattering is strongly peaked in the forward direction with the cross-section only gradually changing with higher energies.

 Ef , pf Ei , pi EN , pN The simple 2-body kinematics of scattering fixes the energy of particles scattered through .

Measuring Ef at a fixed angle  The predicted elastic scattering calculated from the kinematics of a 2-body collision. Only Ef should be observed at . Number of particles with Ef 8.0 8.5 9.0 9.5 10.0 Energy Ef of scattered particle (MeV)

Classification of Nuclear Reactions Addition details on direct reactions inelastic scattering individual collisions between the incoming projectile and a single target nucleon; the incident particle emerges with reduced energy pickup reactions incident projectile collects additional nucleons from the target O + d  O + H (d, 3H) Ca + He  Ca +  (3He,) 16 8 15 8 3 1 41 20 3 2 40 20 stripping reactions incident projectile leaves one or more nucleons behind in the target 90 40 Zr + d  Zr + p 91 40 (d,p) (3He,d) 23 11 Na + He  Mg + d 3 2 24 12

Restricted to highest energy projectiles (>20-MeV) charge exchange reactions A proton (neutron) enters the nucleus, but emerges as a neutron (proton) “exchanging charge” with one of the nucleons Direct reactions are often described as “surface” reactions involving interactions with individual nucleons within the target occurring very rapidly (on the order of 10-22 seconds) Restricted to highest energy projectiles (>20-MeV) (from accelerators)

There is evidence for an altogether different mechanism taking place at more moderate energies slower process – taking 10-16 to 10-18 seconds involving an intermediate, unstable, short-lived state outliving direct contact of projectile & target final products ejected isotropically compared to the “forward-peaked” cross-section we described for the direct processes for incident energies <20-MeV governs all “natural” processes where the projectiles are natural decay products

The most obvious evidence for long lived intermediate states in nuclear reactions is the strongly resonant nature of nuclear cross-sections. This is illustrated in the figure at right which shows the indium total cross section for neutrons.

The energy of these long lived states is defined to a few eV and if we apply the uncertainty relation dEdt ~ h we see this implies a lifetime of ~ 10-16 s which is very long compared to the time it takes a nucleon to traverse a nucleus ~ 10-22 s. Consider the time for a relativistic a to cross/pass through a medium mass (A=125u) nucleus Recall: Direct reactions do not involve the formation of some intermediate state; their characteristic time of interaction is more like 10-22 s. Variations in their cross-sections, as a function of energy, are spread over a few MeV.

What could possibly distinguish low from high energy collisions? Let’s compare de Broglie wavelength for 1 MeV (“low” energy) proton 5-10 MeV ’s (typical for nuclear reaction studies) 100-MeV 

High energy projectiles are better focused on individual target nucleons Their collisions are billiard ball-like and result in Direct Reactions. Low energy projectiles cannot see anything finer than the whole target nucleus The resultant collisions are with the entire nucleus, and the energy is shared across it. The intermediate state (itself never directly observed) is referred to as the Compound Nucleus.

Total ,4n Cross Section ,3n ,2n ,n Alpha particle energy The above figure shows the contributing cross-sections for the reactions and so on…

With increasing energy Total ,4n Cross Section ,3n ,2n ,n Alpha particle energy With increasing energy more and more neutrons are likely to be knocked free it becomes less and less likely for an individual neutron to be freed This is NOT explicable as an -nucleon collision, the  shares its energy across the nucleus, elevating it to some excited state.

Consider a neutron (or -particle) with energy E V = 0 Consider a neutron (or -particle) with energy E within a square well potential. R V = V0 At r = R there is a large confining force which as we saw in Problem Set #1 can produce reflections. Let’s consider the simple ℓ=0 case, where we can substitute: with u satisfying: for inside the well.

Since uinside must vanish at the origin: inside the well: Since uinside must vanish at the origin: outside the well: For scattering off this potential, we want to consider final states that are traveling outwards outside the well: for r > R

The energy of the inside states must be expressed as a complex number! Continuity requires at r = R or The energy of the inside states must be expressed as a complex number! In scattering experiments we are not dealing with simple “stationary states” with real-valued energy eigenvalues. but with E = E0 + i/2

A stationary state of energy E0 that decays! because the probability of finding the particle in that state: These are “virtually bound” states or resonances.

For a short range or abrupt-sided potential there exist quasi-bound or virtual single particle states of positive energy. Long range potentials (like the coulomb potential) have no such states.

The projectile can be momentarily trapped in one of these excited states, sharing its energy through interactions with the nucleons inside the nucleus: raising some of them into excited states, itself dropping into lower states. This is the formation of the many particle excited state which is the compound nucleus. At this stage all memory of the original mode of excitation is lost. At a later time when a decay occurs, the energy is once more concentrated in a single particle.

The Optical Model where and now here To quantum mechanically describe a particle being absorbed, we resort to the use of a complex potential in what is called the optical model. Consider a traveling wave moving in a potential V then this plane wavefunction is written where If the potential V is replaced by V + iW then k also becomes complex and the wavefunction can be written and now here

now describes a traveling wave of decreasing amplitude: A decreasing amplitude means that the transmitted particle is being absorbed.

In most cases V >> W. To make an estimate of the mean free path (the “attenuation length”) we will assume that condition:

replacing the kinetic energy term (E - V) by the expression mv2/2 gives 2p/h = 2/ where  is the de Broglie wavelength of incident projectile

or the mean free path - the distance over which the intensity ( * ) is attenuated to 1/eth its initial value – is given by or since this gives a (mean) distance where the mean free time - the time by which the intensity ( * ) is attenuated to 1/eth its initial value

The SHELL MODEL we relied on earlier has a potential depth ~40 MeV. If the attenuation distance is of the same order as the nuclear radius (3-6 fm) then W  6-8 MeV. To describe both diffraction and scattering phenomena with the optical model requires an imaginary potential of a few MeV. NOTE: this is entirely consistent with the lifetime of the virtual single particle state before absorption of about 10-22 sec. ~ 10-22 

Writing the compound nucleus reaction as the cross-section (which remember expresses a probability) can be expressed as the width for decay to b the total decay width the cross-section for absorbing a and forming the compound nucleus b is the “partial width”    is the total width (ħ times the total decay probability).

The behavior of the cross-section with energy depends on the relative sizes of  & the spacing between energy levels. For low excitation of a nucleus the energy levels are relatively well spaced and the cross-section exhibits resonance while at higher energies of excitation the width  will overlap several energy levels and the cross-section varies much more slowly with energy. This is the continuum. The energy at which the resonance  continuum transition occurs depends upon A. For A ~ 20 ~10 MeV while for A ~ 200 ~100 keV.

the 1st exponential gives the normal oscillatory time dependence For well separated levels, an individual state will decay as exp(- t/ ħ) RECALL: we write such a wave function as the 1st exponential gives the normal oscillatory time dependence of a wavefunction with E0 (here the energy above ground state). The second exponential gives the decay of the state. Since it is decaying, such a state is not a solution of Schrödinger’s equation with a static potential (not a stationary state). It can,however,be considered a superposition of such states This convolution you’ll recognize as the Fourier transform!

The function A(E) can be obtained by Fourier transform which yields Giving the cross-section the form The constant C depends upon the phase space available to the incident particle where p is the momentum of the incident particle, ℓ its orbital angular momentum, and a the partial width for the decay back to the initial state.

All together where This is the Breit-Wigner formula for a single level reaction cross-section. For example - in the case of elastic scattering (a = b) at the maximum of the resonance (E = E0) the cross-section is And if no other processes to compete with (a=) the maximum possible elastic cross-section.

For the inelastic cross-section (b = -a) at the maximum of its resonance (E = E0) with a maximum value of when a = /2. When the separation of energy levels is much smaller than the total width there are many levels contributing to a given process. This mixture of levels described as a compound nucleus.

Furthermore the compound nucleus The projectile is captured by the target forming an intermediate state the compound nucleus In a compound nucleus all sense of direction of the incident particle is lost the produced particles are ejected in an essentially isotropic distribution in the center of mass frame. Furthermore The subsequent decay of this intermediate state is largely independent of its mode of formation. A given compound nucleus may be formed by any of several reactions but the probability of a certain type of final state is only dependent upon the amount of excitation energy.

[ Ne]* 20 10

the relative cross-sections for the different processes 63Cu(p,n)63Zn Here are plotted the yields of decay products from the compound nucleus 64Zn formed by 2 different routes Note: the relative cross-sections for the different processes 63Cu(p,n)63Zn and 60Ni(,n)63Zn remain ~constant across the plotted  energies.