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Nuclear Reactions . Readings: Modern Nuclear Chemistry, Chapter 10; Nuclear and Radiochemistry, Chapter 4 Notation Energetics of Nuclear Reactions Reaction Types and Mechanisms Barriers Scattering Nuclear Reaction Cross Sections Reaction Observables Scattering Direct Reactions

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nuclear reactions
Nuclear Reactions

Readings: Modern Nuclear Chemistry, Chapter 10; Nuclear and Radiochemistry, Chapter 4


Energetics of Nuclear Reactions

Reaction Types and Mechanisms



Nuclear Reaction Cross Sections

Reaction Observables


Direct Reactions

Compound Nuclear Reactions

Photonuclear Reactions

Heavy Ion Reactions

High Energy Reactions

nuclear reactions1
Nuclear Reactions
  • Nucleus reactions with a range of particles
    • nucleus, subatomic particle, or photon to produce other nuclei
    • Short time frame (picosecond)
  • First nuclear reaction from Rutherford
    • What reaction was this?
  • Number of terms conserved
    • Number of nucleons
      • except in reactions involving creation or annihilation of antinucleons
    • charge,
    • Energy
    • momentum
    • angular momentum
    • 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


  • Energetically many orders of magnitude greater than chemical reactions
    • Know this from radioactive decay
  • 14N(a,p)17 O Q=-1.193 MeV
    • Convert energy to per molar basis
      • 1 eV = 1.60E-19 J
      • =7.18E23 MeV/mole= 7.18E29 eV/mole
      • =115E3 J/mole
  • Reactions so large that mass change is observable
  • Q value can be experimentally measured to provide a route to determine particle mass of reactants
  • Q can 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
    • 106Pd(n,p)106Rh
    • 106Rh beta decay to 106 Pd, Q=3.54 MeV
    • 106Pd +np+106Rh + Q
    • 106Rh 106Pd + b- + nanti + 3.54 MeV
    • n=1H+Q+3.54 MeV, solve for Q
    • Q=n-(1H+3.54 MeV)
    • Q=(8.0713-(7.2890+ 3.54)) MeV
    • Q=-2.76 MeV
  • Q of a reaction is not necessarily equal to kinetic energy of the bombarding particles for the reaction to occur
    • conservation of momentum
      • Some of the particles’ kinetic energy must be retained by products as kinetic energy
  • Amount retained as kinetic energy of products
    • Based on projectile mass
    • Becomes smaller with increasing target mass
energetics reaction barrier
Energetics: Reaction Barrier
  • Need to consider comparison of laboratory and center of mass frame
    • Laboratory frame conservation of momentum considers angle of particles
    • Center of mass
      • Total particle angular momentum is zero
  • Kinetic energy carried by projectile (Tlab) is not fully available for reaction
  • Tcm must be carried away by center of mass
    • Available energy is Tlab - Tcm = T0
  • For reaction to occur Q + T0 must be achieved
    • Basis for threshold reaction
reaction barrier
Reaction Barrier
  • Threshold energy (minimum energy for reaction)
    • Fraction of the bombarding particle’s kinetic energy retained as kinetic energy of the products becomes smaller with increasing mass of target
  • Consider the 14N(a,p)17O reaction
    • Find threshold energy
      • Q from mass excess
        • Q=2.425 + 2.863 – 7.289 – (-0.809) = -1.19 MeV
    • T= -(-1.19)(4 + 14)/14 = 1.53 MeV
  • Reaction barrier also induced by Coulomb interaction
    • Need to have enough energy to react
    • Can be above threshold energy
    • Utilize equations for Coulomb barrier (3.36 MeV)
    • 3.36x (18/14) = 4.32 MeV alpha needed for reaction (CM and lab frame)
barriers for charged particles
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
    • Probability of tunneling through barrier drops rapidly as energy of particle decreases
    • Coulomb barriers affect charged particles both entering and leaving the nucleus
      • Charged particles emitted from nuclei have considerable kinetic energies (greater than 1 MeV)
      • Also seen with position emission
elastic scattering
Elastic Scattering
  • Simplest consequence of a nuclear collision
    • Not a “reaction”
      • no exchange of nucleons or creation of particles
  • 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 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
    • cross section for all events other than (potential) elastic scattering
cross section limits
Cross Section Limits
  • Reaction cross section of R2 is approximated at high energies
    • Wave nature of incident particle causes upper limit of reaction cross section include de Broglie wavelength
  • Collision between neutron and target nucleus characterized by distance of closest approach
    • B is impact parameter
cross sections
Cross sections
  • Angular momentum of system is normal to the relative momentum p
  • b any value between 0 and R
  • Quantum mechanics requires angular momentum component in a particular direction be an integer in units ħ, l =0,1,2,…
  • b corresponds angular momentum
cross section
Cross section
  • Sum all l from 0 to lmax
  • R is interaction radius
  • Nuclear reaction cross sections can be several orders of magnitude larger than the nuclear geometrical cross section of
    • Manifest by slow-neutron reactions

Quantum-mechanical treatment Tl is the transmission coefficient for the reaction of a neutron with angular momentum l (varies between 0 and 1) represents the fraction of incident particles with angular momentum l that penetrate within the range of nuclear forces

cross section and energy
Cross section and energy
  • General trends for neutron and charged particles
    • Charged particle cross section minimal at low energy
    • Neutron capture cross section maximum at low energy
types of experiments excitation functions
Types of Experiments: Excitation Functions
  • relation between variation of particular reaction cross section with incident energy
  • shape can be determined by stacked-foil method
    • exposing several target foils in same beam with appropriate energy-degrading foils interposed
  • possible to get crude estimates of kinetic energies
  • will not yield any information about angular distribution of emitted particles
types of experiments excitation functions1
Types of Experiments: Excitation Functions
  • provide information about probabilities for emission of various kinds of particles and combinations of particles in nuclear reactions
    • formation of given product implies what particles were ejected from the target nuclide
  • Range of cross sections can be evaluated
Total Reaction Cross Sections

summing of all experimentally measured excitation functions for individual rxns rarely yields an excitation function for r

some rxns lead to stable products

cannot be measured by activation technique

measure attenuation of beam to determine r

determine (I-Io)/Io

difficult to apply

target must be kept thin enough to minimize energy degradation of beam

thin target produces a small attenuation in intensity, hard to measure with accuracy

Partial Spectra

focuses attention on energy and angular distributions of emitted particles

  • Information collected experimentally
    • detection of emitted particles at with respect to incident beam
  • Limitation lies in lack of knowledge about other particles that may be emitted in same event
    • Unable to detect all particles
      • Low energy for only one particle emission
      • several detectors with coincidences
  • Radiochemical Recoil Measurements
    • 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
direct interaction
Direct Interaction
  • Assumes incident particle collides with only a few nucleons in target nucleus
    • Some of the nucleons ejected
  • Include event in which only part of incident complex particle interacts with target nucleus
    • transfer reactions
      • stripping reaction and pickup process
      • useful 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 in nucleus
      • High energies, low wavelength
    • impulse approximation
      • collisions with individual nucleons in nucleus treated as if occurred with free nucleons valid
low energy reactions with light projectiles
Low-Energy Reactions with Light Projectiles
  • Slow-Neutron Reactions
    • purest example of compound-nucleus behavior
      • 1/v law governs most neutron cross sections in region of thermal energies
    • neutrons available from nuclear reactions only and produced with appreciable kinetic energies
      • Range of energies can be obtained
  • Reaction Cross Sections
    • Coulomb barrier prevents study of nuclear reactions below 1 MeV
      • resonances no longer observable
      • with increasing energy, increasing variety of reactions possible
low energy reactions
Low-Energy Reactions
  • Deuteron Reactions
    • Prevalence of one nucleon stripping
      • large size and loose binding of deuteron
    • neutron comes within range of nuclear forces while proton is still outside most of Coulomb barrier
      • Inherent in large neutron-proton distance in deuteron
      • weakly bound deuteron can be broken up
        • proton outside barrier
  • Competition among Reactions
    • depends on relative probabilities for emission of various particles from compound nucleus
      • determined by number of factors
        • energy available
        • Coulomb barrier
        • density of final states in product nucleus
high energy reactions
High-Energy Reactions
  • Mass-Yield Curves
    • at low energies, compound-nucleus picture dominates
      • as energy increases importance of direct reactions and preequilibrium(pre-compound nucleus) emission increase
      • above 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
    • Above 100 MeV reactions
    • energy of the incident proton larger than interaction energy between the nucleons in the nucleus
    • Wavelength less than average distance between nucleons
      • proton will collide with one nucleon at a time within the nucleus
        • high-energy proton makes only a few collisions in nucleus
        • Produces nucleons with high energy
high energy reactions1
High Energy Reactions
  • Spallation Products
    • products in immediate neighborhood of target element found in highest yields
      • within 10 to 20 mass numbers
    • yields tend to form in two regions
    •  stability for medium-weight products
    • neutron-deficient side of stability with increasing Z of products
    • Used to produce beam of neutrons at spallation neutron source
      • Heavy Z will produce 20-30 neutrons
  • 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
      • originate from processes involving higher deposition energies
      • lower kinetic energies
      • do not appear to have partners of comparable mass
      • arise from spallation-like or fragmentation reactions
high energy reactions2
High Energy Reactions
  • Fragmentation
    • observed with projectiles in GeV region
    • cannot be explained with two-step model of intranuclear cascades followed by evaporation and fission
      • evident from recoil properties
      • ranges and angular distributions differ from those of fission products
      • cannot be accounted for by cascade-evaporation calculations
  • Reactions with Pions
    • Strong interaction particles between nucleons
    • scattering of pions by nucleons exhibits pronounced, broad resonance centered around 180 MeV
      • formation of nucleon isobar (), which is short-lived excited state of nucleon
    • short mean-free paths of pions in nuclei
high energy reactions3
High Energy Reactions
  • possibility of pion absorption by pair of nucleons, resulting in total energy of pion to be shared by two nucleons (pion capture)
    • two-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)
    • at higher energies, proton- and pion-induced spallation patterns become similar
  • neutron-rich products become more prominent in --induced, proton-rich products in +-induced reactions
    • expected from change in N/Z ratio of target-projectile combination if pion is absorbed
heavy ion reactions
Heavy-Ion Reactions
  • Range of heavy ion reactions
    • elastic and inelastic scattering
    • compound-nucleus formation,
    • direct interactions
    • deeply inelastic reaction
  • Reactions influence by parameter
    • impact parameter of collision
    • kinetic energy of projectile
    • masses of target
    • projectile nuclei
  • Elastic and Inelastic Scattering, Coulomb Excitation
    • elastic-scattering measurements used to obtain information on interaction radii
    • R=R1+R2between mass numbers A1 and A2
heavy ion reactions1
Heavy Ion Reactions
  • inelastic scattering
    • scattering in which some of projectile’s kinetic energy transformed into excitation of target nucleus
      • greatest importance at large impact parameters
    • heavy ions valuable
      • can excite high-spin states in target nuclei because of large angular momenta
  • Can experience Coulomb excitation
    • high charges
    • below Coulomb barrier heights and excite nuclei by purely electromagnetic interactions
  • Transfer Reactions
    • stripping and pickup reactions prevalent with heavy ions
      • take 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
heavy ion reactions2
Heavy Ion Reactions
  • when transfer populates many overlapping states
    • single peak at characteristic angle (grazing angle)
    • projectile 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
      • “grazing contact mechanism”
deeply inelastic reactions
Deeply Inelastic Reactions
  • double differential cross sections are distinguishing feature
    • products with masses in vicinity of projectile mass appear at angles other than classical grazing angle
      • relatively small kinetic energies
  • total kinetic energies of products strongly correlated with amount of mass transfer
    • the 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
    • will rotate as result of large angular momentum from projectiles
  • Product will dissociate into two fragments
    • appreciable fraction of incident kinetic energy dissipated and goes into internal excitation
compound nucleus reactions
Compound-Nucleus Reactions
  • compound-nucleus formation can only take place over a restricted range of small impact parameters
    • can define critical angular momentum above which complete fusion cannot occur
    • 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
    • 5-10 MeVdeexcitation for neutron evaporation
  • heavy-ion reactions provide only possible means for reaching predicted island of stability around Z=114 to Z=184
  • U is excitation energy, MAand Ma masses of target and projectile, Ta is projectile kinetic energy, Sa is projectile binding energy in compound nucleus
photonuclear reactions
Photonuclear reactions
  • Reactions between nuclei and low- and medium-energy photons dominated by giant resonance
    • Excitation function for photon absorption goes through a broad maximum a few MeV wide
      • Due to excitation of dipole vibrations of protons against neutrons in the nucleus
  • Resonance peak varies smoothly with A
    • 24 MeV at 16O
    • 13 MeV at 209Bi
  • Peak cross sections are 100-300 mb
  • (, p), (, n), (,a) reactions
review notes
Review Notes
  • Understand Reaction Notation
  • Understand Energetics of Nuclear Reactions
    • Q values and barriers
  • Understand the Different Reaction Types and Mechanisms
    • Particles
    • Energy
  • Relate cross sections to energy
  • Describe Photonuclear Reactions
  • Describe the different types of nuclear reactions shown on 9-29.
  • Provide notations for the following
    • Reaction of 16O with 208Pb to make stable Au
    • Formation of Pu from Th and a projectile
  • Find the threshold energy for the reaction of 59Co and an alpha that produces a neutron and a product nuclei
  • What are the differences between low and high energy reactions?
  • How does a charged particle reaction change with energy? A neutron reaction?
pop quiz
Pop Quiz
  • Provide the Q value, threshold energy, and Coulomb barrier for the compound nucleus reaction of 18O with 244Cm