Nuclear Reactions

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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

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Notation
Shorthand

• 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

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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
• the fraction of the bombarding particles kinetic
  energy thats retained as kinetic energy of the
  products becomes smaller with increasing mass of
  the target nucleus

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Reaction overview
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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 Vc
• 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)

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Reaction Barrier

• Kinetic energy for reaction in center of mass
• 1/2mv2
• m is mass of compound nucleus
• Projectile and target
• v is from target velocity and mass of compound
  nucleus
• vmpvp/(mpmt)
• Comparison between center of mass and laboratory
  system
• Energy required for conversion to lab system
• Reaction energetic include consideration for
  laboratory system
• Threshold energy (minimum energy for reaction)
• T -Q(mP mT)/mT
• Consider the 14N(a,p)17O reaction
• Find threshold energy
• Q from mass excess
• Q2.425 2.863 7.289 (-0.809) -1.19 MeV
• T -(-1.19)(4 14)/14 1.53 MeV

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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

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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 ??R2, 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

?
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Cross section
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actually,
Cross-sectional area corresponds to collision
with angular momentum lh/(2?)
In the quantum-mechanical treatment, the result
for the total reaction cross section is where Tl
is the transmission coefficient for the reaction
of a neutron with angular momentum l (varies
between 0 and 1) and represents the fraction of
incident particles with angular momentum l that
penetrate within the range of nuclear forces

• Our semiclassical treatment is valid for
  where the only contribution comes from l0
  and the reaction cross section has
  as its upper limit

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Centrifugal Barrier

• Coulomb repulsion will bring the relative kinetic
  energy of the system from ? when the particles
  are very far apart to ?-Vc 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 bm and the relative
  momentum at point of contact is

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Upper limit for the capture of charged particles
can be estimated as the area of the disk of
radius bm

• lm?0 as ??Vc for charged particles lm?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

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Cross section and energy
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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
• exposing 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
• formation 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

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Reactions

• Total Reaction Cross Sections
• summing of all experimentally measured excitation
  functions for individual rxns rarely yields an
  excitation function for ?r
• for 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
• determine (I-Io)/Io
• difficult 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

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• 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
• either 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

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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

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• Represents the nucleus by a square-well potential
  Vo 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
• index of refraction defined as ratio of
  wavelengths
• Modified to allow for absorption of incident
  particle
• change index of refraction to complex number
• damps wave inside potential well, making medium
  somewhat absorbent
• Solve Schrödinger equation

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• Cross sections for elastic scattering and
  reaction for a given l can be expressed in terms
  of amplitude?l, which is a complex number
• Maximum reaction cross section for given l is
  ??/(2?)(2l 1), which occurs for ?l0
• Maximum scattering cross section is 4??2/(4?2)(2l
  1), which occurs for ?l-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

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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

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• 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 dont overlap, behavior of compound
  nucleus essentially governed by properties of
  single quantum state independence hypothesis
• Breit-Weigner one-level formula

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• 1/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
• reinstates symmetrical angular distribution
• assume that overlapping states all have
  essentially the same relative partial widths for
  various possible decay channels of compound
  nucleus
• reinstates independence hypothesis

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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

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• 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 Uc-Sb (if assume b spinless and emitted
  in l0 state maximum kinetic energy becomes
  Uc-Sb-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

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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
• 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
• impulse approximation--collisions with individual
  nucleons in nucleus treated as if they occurred
  with free nucleons--valid

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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
• assumed that for given exciton number, every
  possible particle-hole configuration has equal a
  priori probability
• Gives no information about angular distributions

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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
• 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)
• resonances no longer observable
• with increasing energy, increasing variety of
  reactions possible

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• Deuteron Reactions
• direct reactions in which one of the nucleons is
  stripped off by collision with a nucleus are
  prevalent
• large size and loose binding of deuteron
• neutron comes within range of nuclear forces
  while proton is still outside most of Coulomb
  barrier
• large neutron-proton distance in deuteron
• weakly 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
• determined by energy available, Coulomb barrier,
  density of final states in product nucleus

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High-Energy Reactions

• Mass-Yield Curves
• at low energies, compound-nucleus picture
  dominates, but as energy increases importance of
  direct reactions and preequilibrium 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

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• 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
• originate 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

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• 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
• evident 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
• formation of nucleon isobar (?), which is
  short-lived excited state of nucleon
• short mean-free paths of pions in nuclei

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• 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 at equal
  kinetic energies
• 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

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Heavy Ion Reactions
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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 RR1R2 between
  mass numbers A1 and A2

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• inelastic scattering--scattering in which some of
  projectiles kinetic energy transformed into
  excitation of target nucleus--of greatest
  importance at large impact parameters
• heavy 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
• 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

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• when transfer populates many overlapping states,
  find 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

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• double differential cross sections are
  distinguishing feature
• products 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
• 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 that will
  rotate as result of large angular momentum from
  projectiles
• will dissociate into two fragments
• appreciable fraction of incident kinetic energy
  dissipated and goes into internal excitation

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• 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
• heavy-ion reactions provide only possible means
  for reaching predicted island of stability around
  Z114 to Z184