11-1 Nuclear Reactions Notation Energetics of Nuclear Reactions Reaction Types and Mechanisms §Barriers §Scattering Nuclear Reaction Cross Sections Reaction Observables Rutherford Scattering Elastic Scattering Direct Reactions Compound Nuclear Reactions Photonuclear Reactions Heavy Ion Reactions High Energy Reactions

11-2 Notation Number of nucleons (except in reactions involving creation or annihilation of antinucleons), charge, energy, momentum, angular momentum, statistics, and parity conserved Q is the energy of the reaction §positive Q corresponds to energy release, negative Q to energy absorption §Q terms given per nucleus transformed Shorthand:

11-3 Energetics Q may even be calculated if the masses of involved nuclei are not known §if the product nucleus is radioactive and decays back to the initial nucleus with known decay energy Q of a reaction is not necessarily equal to the needed kinetic energy of the bombarding particles for the reaction to occur §nucleus conservation of momentum requires that some of the particles’ kinetic energy be retained by the products as kinetic energy àthe fraction of the bombarding particle’s kinetic energy that’s retained as kinetic energy of the products becomes smaller with increasing mass of the target nucleus

11-4 Reaction overview

11-5 Barriers for Charged Particles Coulomb repulsion between charged bombarding particles and the nucleus §repulsion increases with decreasing distance of separation until charged particle comes within range of nuclear forces of the nucleus §gives rise to the previously discussed potential barrier of height V c §probability of tunneling through barrier drops rapidly as energy of particle decreases §Coulomb barriers affect charged particles both entering and leaving the nucleus àcharged particles emitted from nuclei have considerable kinetic energies (greater than 1 MeV)

11-6 Reaction Barrier Kinetic energy for reaction in center of mass §1/2mv 2 àm is mass of compound nucleus *Projectile and target *v is from target velocity and mass of compound nucleus Øv=m p v p /(m p +m t ) §Comparison between center of mass and laboratory system àEnergy required for conversion to lab system Reaction energetic include consideration for laboratory system Threshold energy (minimum energy for reaction) §T ≥ -Q(m P + m T )/m T Consider the 14 N( ,p) 17 O reaction §Find threshold energy àQ from mass excess *Q= – – (-0.809) = MeV §T= -(-1.19)(4 + 14)/14 = 1.53 MeV

11-7 Elastic Scattering Simplest consequence of a nuclear collision §not a “reaction” Particles do not change their identity during the process and the sum of their kinetic energies remains constant As energy of bombarding particle is increased, the particle may penetrate the Coulomb barrier to the surface of the target nucleus §elastic scattering will also have a contribution from nuclear forces May be considered to arise from optical-model potential Reaction cross section is the cross section for all events other than (potential) elastic scattering

11-8 Cross Section Limits Although it might be expected that a nucleus that interacts with everything that hits it would have a reaction cross section of  R 2, this is only correct at high energies §wave nature of incident particle causes upper limit of reaction cross section to be: Collision between neutron and target nucleus characterized by distance of closest approach of two particles if there were no interaction between them §this distance, b, is called the impact parameter 

11-9 Cross section

11-10 Our semiclassical treatment is valid for where the only contribution comes from =0 and the reaction cross section has as its upper limit Cross-sectional area corresponds to collision with angular momentum h/(2  ): In the quantum-mechanical treatment, the result for the total reaction cross section is where T is the transmission coefficient for the reaction of a neutron with angular momentum (varies between 0 and 1) and represents the fraction of incident particles with angular momentum that penetrate within the range of nuclear forces actually,

11-11 Centrifugal Barrier Coulomb repulsion will bring the relative kinetic energy of the system from  when the particles are very far apart to  -V c when the two particles are just touching The trajectory of the particle is tangential to the nuclear surface when it approaches with the maximum impact parameter b m and the relative momentum at point of contact is:

11-12 m  0 as  V c for charged particles; m  0 as  0 for neutrons §Coulomb barrier causes the transmission coefficient for charged particles to approach zero under these circumstances, whereas that for the neutron remains finite §vanishing cross section for charged particles of energies approaching that of Coulomb barrier to be compared with upper limit of  (R+ /(2  )) 2 Upper limit for the capture of charged particles can be estimated as the area of the disk of radius b m

11-13 Cross section and energy

11-14 Types of Experiments Excitation Functions §relation between variation of a particular reaction cross section with incident energy §shape can be determined by stacked-foil method àexposing several target foils in same beam with appropriate energy- degrading foils interposed §provide information about probabilities for emission of various kinds of particles and combinations of particles in nuclear reactions àformation of a given product implies what particles were ejected from the target nuclide §possible to get crude estimates of kinetic energies §will not yield any information about angular distribution of emitted particles

11-15 Reactions Total Reaction Cross Sections §summing of all experimentally measured excitation functions for individual rxns rarely yields an excitation function for  r àfor most target-projectile combinations, some rxns lead to stable products and thus cannot be measured by activation technique §measure attenuation of beam to determine  r àdetermine (I-I o )/I o àdifficult to apply because target must be kept thin enough to minimize energy degradation of beam, but thin target produces a small attenuation in intensity, which is hard to measure with accuracy Partial Spectra §focuses attention on energy and angular distributions of the emitted particles

11-16 §information collected experimentally by detection of emitted particles in energy-sensitive detector placed at various angles  with respect to incident beam §limitation lies in lack of knowledge about other particles that may be emitted in same event àeither use energy so low that probability for emission of more than one particle is negligible or by having several detectors and demanding coincidences among them before an event is recorded Radiochemical Recoil Measurements §to obtain angular distributions and kinetic-energy spectra for heavier fragments and product nuclei §combines the activation technique with angular and energy measurements provided the product of interest is radioactive

11-17 Optical Model Attempt to understand cross sections for nuclear reactions in which the interactions of the incident particle with the nucleons of the nucleus are replaced by its interaction with a potential-energy well used fruitfully in interpretation of elastic-scattering and total- reaction cross sections at energies down to a few million electron volts

11-18 Represents the nucleus by a square-well potential V o MeV deep and R fm wide Kinetic energy of neutron entering nucleus will be higher inside the well than outside §refraction at the nuclear surface àindex of refraction defined as ratio of wavelengths Modified to allow for absorption of incident particle §change index of refraction to complex number àdamps wave inside potential well, making medium somewhat absorbent Solve Schrödinger equation

11-19 Cross sections for elastic scattering and reaction for a given  can be expressed in terms of amplitude , which is a complex number Maximum reaction cross section for given  is  /(2  )(2 +1), which occurs for  =0 Maximum scattering cross section is 4  2 /(4  2 )(2 +1), which occurs for  =-1 Appearance of resonances in elastic-scattering cross sections at energies corresponding to single-particle states of the incident particle in effective potential Mean free path:

11-20 Compound-Nucleus Model Assumes incident particle, upon entering the target nucleus, amalgamates with it in a way that it kinetic energy is distributed randomly among all nucleons §resulting nucleus in excited quasi-stationary state and called compound nucleus Nuclear reaction divided into 2 distinct independent steps: §capture of incident particle with random sharing of energy among nucleons in compound nucleus §“evaporation” particles from excited compound nucleus Excitation energy U:

11-21 Probability for de-excitation by  emission much higher than neutron escape in slow-neutron reaction §excitation energy of compound nucleus from by capture of slow neutron only slightly higher than binding energy of neutron in compound nucleus With low-energy neutrons, relative probabilities of various possible events should be completely determined by quantum state of compound nucleus §if resonances don’t overlap, behavior of compound nucleus essentially governed by properties of single quantum state; “independence hypothesis” Breit-Weigner one-level formula:

/v Law: in the region where    o  §if no close resonances, capture cross section may be quite small and follow this law Statistical Assumption §assume interference terms have random signs and thus cancel out àreinstates symmetrical angular distribution §assume that overlapping states all have essentially the same relative partial widths for various possible decay channels of compound nucleus àreinstates independence hypothesis

11-23 Statistical Model--Evaporation Theory Particles emitted with considerably less than maximum energy available Statistical assumption implies that statistical equilibrium exists during compound-nucleus reaction §relative numbers of compound nuclei and sets of particles that correspond to various decay channels determined by their relative state densities Statistical model able to predict energy spectrum of evaporated particles and excitation functions for various products in terms of certain average nuclear properties §principle of detailed balance

11-24 Important consequences of including angular-momentum effects §compound nuclei can be formed in states of rather high angular momentum §yrast levels: lowest-energy states of given spin  existence of yrast level at excitation U(I) means maximum kinetic energy of emitted particle is less than U c -S b (if assume b spinless and emitted in =0 state maximum kinetic energy becomes U c -S b -U(I)) Odd-Even Effects on Level Densities §pairing effects on level density becomes less marked with increasing excitation §level density of odd-odd nucleus at given excitation energy is greater than that of adjacent even-odd or odd-even nucleus, which is greater than adjacent even-even nucleus

11-25 Direct Interaction Assumes incident particle collides with only a few nucleons in target nucleus, thereby ejecting some of them Include event in which only part of incident complex particle interacts with target nucleus §transfer reactions àstripping reaction and pickup process àuseful for determination of energies, spins, and parities of excited states of nuclei Knock-On reactions §mean free path  for incident particle large compared to average spacing between nucleons àimpulse approximation--collisions with individual nucleons in nucleus treated as if they occurred with free nucleons--valid

11-26 Preequilibrium Decay Intermediate model between compound-nucleus and direct- interaction model Particles can be emitted prior to attainment of statistical equilibrium Successive two-body interactions invoked, but without spatial considerations §focuses on total number of excitons in each step àassumed that for given exciton number, every possible particle-hole configuration has equal a priori probability Gives no information about angular distributions

11-27

11-28 Low-Energy Reactions with Light Projectiles Slow-Neutron Reactions §purest example of compound-nucleus behavior à1/v law governs most neutron cross sections in region of thermal energies §neutrons available from nuclear reactions only and produced with appreciable kinetic energies Reaction Cross Sections §Coulomb barrier makes it impossible to study nuclear reactions with charged particles of kinetic energies below the million eV region (except with lightest nuclei) àresonances no longer observable àwith increasing energy, increasing variety of reactions possible

11-29 Deuteron Reactions §direct reactions in which one of the nucleons is stripped off by collision with a nucleus are prevalent àlarge size and loose binding of deuteron §neutron comes within range of nuclear forces while proton is still outside most of Coulomb barrier àlarge neutron-proton distance in deuteron àweakly bound deuteron can be broken up, leaving proton outside barrier Competition among Reactions §depends on relative probabilities for emission of various particles from compound nucleus àdetermined by energy available, Coulomb barrier, density of final states in product nucleus

11-30 Mass-Yield Curves §at low energies, compound-nucleus picture dominates, but as energy increases importance of direct reactions and preequilibrium emission increase àabove 100 MeV, nuclear reactions proceed nearly completely by direct interactions §products down to mass number 150 are spallation products §those between mass numbers 60 and 140 are fission products Cascade-Evaporation Model High-Energy Reactions

11-31 Spallation Products §products in immediate neighborhood (within 10 to 20 mass numbers) of target element found in highest yields §yields tend to cluster in region of  stability in case of medium-weight products and increasingly more to the neutron-deficient side of stability with increasing Z of products High-Energy Fission §single broad peak in mass-yield curve instead of double hump seen in thermal-neutron fission §many neutron-deficient nuclides, especially among heavy products àoriginate from processes involving higher deposition energies, have lower kinetic energies, do not appear to have partners of comparable mass, arise from spallation-like or fragmentation reactions

11-32 Fragmentation §involved in phenomena observed with projectiles in GeV region that cannot be explained with two-step model of intranuclear cascades followed by evaporation and fission àevident from recoil properties--ranges and angular distributions differ from those of fission products and cannot be accounted for by cascade-evaporation calculations Reactions with Pions §scattering of pions by nucleons exhibits pronounced, broad resonance centered around 180 MeV àformation of nucleon isobar (  ), which is short- lived excited state of nucleon §short mean-free paths of pions in nuclei

11-33 §possibility of pion absorption by pair of nucleons, resulting in total energy of pion to be shared by two nucleons (pion capture) àtwo-step reaction: formation of , followed by  -nucleon scattering leading to two ground-state nucleons §reaction patterns of pions of given kinetic energy resemble those induced by protons with kinetic energy higher by 140 MeV (the pion rest energy) àat higher energies, proton- and pion-induced spallation patterns become similar at equal kinetic energies §neutron-rich products become more prominent in  - -induced, proton-rich products in  + -induced reactions àexpected from change in N/Z ratio of target-projectile combination if pion is absorbed

11-34 Heavy Ion Reactions

11-35 Heavy-Ion Reactions In addition to mechanisms invoked for light-ion reactions (elastic and inelastic scattering, compound-nucleus formation, direct ineractions), the deeply inelastic reaction is important §impact parameter of collision, kinetic energy of projectile, and masses of target and projectile nuclei determine which mechanisms predominate Elastic and Inelastic Scattering, Coulomb Excitation §elastic-scattering measurements used to obtain information on interaction radii R=R 1 +R 2 between mass numbers A 1 and A 2

11-36 §inelastic scattering--scattering in which some of projectile’s kinetic energy transformed into excitation of target nucleus--of greatest importance at large impact parameters àheavy ions valuable because can excite high-spin states in target nuclei because of large angular momenta §high charges, so they can be at high energies and still be below Coulomb barrier heigths and excite nuclei by purely electromagnetic interactions (Coulomb excitation) Transfer Reactions §stripping and pickup reactions prevalent with heavy ions àtake place at impact parameters just below those at which interactions are purely Coulombic §angular distributions show oscillatory, diffraction-like pattern when transfer reaction to single, well-defined state observed

11-37 §when transfer populates many overlapping states, find single peak at characteristic angle (grazing angle) àprojectile trajectory essentially controlled by Coulomb forces §one-nucleon transfer reactions have thresholds below Coulomb-barrier energies and cross sections rise as energy increased §excitation functions of multinucleon transfer reactions rise with increasing energy Deeply Inelastic Reactions §processes in which relatively large amounts of nuclear matter transferred between target and projectile and which show strongly forward-peaked angular distributions à“grazing contact mechanism”

11-38 §double differential cross sections are distinguishing feature àproducts with masses in vicinity of projectile mass appear at angles other than classical grazing angle, with relatively small kinetic energies §total kinetic energies of products strongly correlated with amount of mass transfer àthe more the A of product and projectile differ in either direction, the lower the kinetic energy §at impact parameters intermediate between those for purely Coulombic interactions and those leading to compound-nucleus formation, short-lived intermediate complex formed that will rotate as result of large angular momentum from projectiles àwill dissociate into two fragments àappreciable fraction of incident kinetic energy dissipated and goes into internal excitation

11-39

11-40 Compound-Nucleus Reactions §compound-nucleus formation can only take place over a restricted range of small impact parameters àcan define critical angular momentum above which complete fusion cannot occur à  cf /  R decreases with increasing bombarding energy §light heavy ions produce compound nuclei on neutron-deficient side of  stability belt §heavy ion of energy above Coulomb barrier brings enough excitation energy to evaporate several nucleons §heavy-ion reactions provide only possible means for reaching predicted island of stability around Z=114 to Z=184