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 reactionpositive Q corresponds to energy release, negative Q to

energy absorptionQ terms given per nucleus transformed

QHOHeN 11

178

42

147

Shorthand: OpN 1714 ),(

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 energythe 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

2McE

11-4

Reaction overview

11-5

Barriers for Charged Particles

• Coulomb repulsion between charged bombarding particles and the nucleusrepulsion 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 nucleuscharged particles emitted from nuclei have

considerable kinetic energies (greater than 1 MeV)

11-6

Reaction Barrier• Kinetic energy for reaction in center of mass

1/2mv2

m is mass of compound nucleus * Projectile and target* v is from target velocity and mass of compound nucleus

v=mpvp/(mp+mt) 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(,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

11-7

Elastic Scattering

• Simplest consequence of a nuclear collisionnot 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 nucleuselastic 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 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

2)( Rr

lb

pbL lb,...2,1,0l

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): 121 2222 llll

ml

mr Rll0

2222 112

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,

0

2 12l

lr Tl

11-11

Centrifugal Barrier

2

22

2 R

l

2

2

1 2

)1(

R

llV

R

ZZV Aac

2

•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:

2/1

2/12/12/1 122

c

c

VVp

11-12

• m0 as Vc for charged particles; m0 as 0 for

neutronsCoulomb 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

2/1

1

c

m

VRb

c

r

VR 12

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

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 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

11-15

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

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 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 Measurementsto obtain angular distributions and kinetic-energy spectra

for heavier fragments and product nucleicombines 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 Vo MeV deep and R fm wide

• Kinetic energy of neutron entering nucleus will be higher inside the well than outsiderefraction at the nuclear surface

index of refraction defined as ratio of wavelengths

• Modified to allow for absorption of incident particlechange index of refraction to complex number

damps wave inside potential well, making medium somewhat absorbent

• Solve Schrödinger equation

02

2

EV

m

p

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 42/(42)(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:

1

22

o

oo

V

WW

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 nucleonsresulting 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:

aaaA

A STMM

MU

11-21

• Probability for de-excitation by emission much higher than neutron escape in slow-neutron reactionexcitation 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 nucleusif resonances don’t overlap, behavior of compound

nucleus essentially governed by properties of single quantum state; “independence hypothesis”

• Breit-Weigner one-level formula:

222

2/1212

12

o

BbAa

aA

CBCA II

I

11-22

• 1/v Law: in the region where oif 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 outreinstates symmetrical angular distribution

assume that overlapping states all have essentially the same relative partial widths for various possible decay channels of compound nucleusreinstates independence hypothesis

vn /1,

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 reactionrelative 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 propertiesprinciple of detailed balance

11-24

• Important consequences of including angular-momentum effectscompound nuclei can be formed in states of rather high

angular momentumyrast 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 =0 state maximum kinetic energy becomes Uc-Sb-U(I))

• Odd-Even Effects on Level Densitiespairing effects on level density becomes less marked with

increasing excitationlevel 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 nucleustransfer reactions

stripping reaction and pickup processuseful 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 nucleonsimpulse 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 considerationsfocuses 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

11-27

11-28

Low-Energy Reactions with Light Projectiles

• Slow-Neutron Reactionspurest 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 SectionsCoulomb 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 observablewith increasing energy, increasing variety of

reactions possible

11-29

• Deuteron Reactionsdirect reactions in which one of the nucleons is stripped off by

collision with a nucleus are prevalentlarge size and loose binding of deuteron

neutron comes within range of nuclear forces while proton is still outside most of Coulomb barrierlarge neutron-proton distance in deuteronweakly bound deuteron can be broken up, leaving

proton outside barrier• Competition among Reactions

depends on relative probabilities for emission of various particles from compound nucleusdetermined by energy available, Coulomb barrier,

density of final states in product nucleus

11-30

• Mass-Yield Curvesat low energies, compound-nucleus picture dominates, but

as energy increases importance of direct reactions and preequilibrium emission increaseabove 100 MeV, nuclear reactions proceed nearly

completely by direct interactionsproducts down to mass number 150 are spallation

productsthose between mass numbers 60 and 140 are fission

products• Cascade-Evaporation Model

High-Energy Reactions

11-31

• Spallation Productsproducts in immediate neighborhood (within 10 to 20

mass numbers) of target element found in highest yieldsyields 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 Fissionsingle broad peak in mass-yield curve instead of double

hump seen in thermal-neutron fissionmany neutron-deficient nuclides, especially among heavy

productsoriginate 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

• Fragmentationinvolved in phenomena observed with projectiles in

GeV region that cannot be explained with two-step model of intranuclear cascades followed by evaporation and fissionevident from recoil properties--ranges and

angular distributions differ from those of fission products and cannot be accounted for by cascade-evaporation calculations

• Reactions with Pionsscattering of pions by nucleons exhibits pronounced,

broad resonance centered around 180 MeVformation of nucleon isobar (), which is

short-lived excited state of nucleonshort 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)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 reactionsexpected 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 importantimpact parameter of collision, kinetic energy of

projectile, and masses of target and projectile nuclei determine which mechanisms predominate

• Elastic and Inelastic Scattering, Coulomb Excitationelastic-scattering measurements used to obtain

information on interaction radii R=R1+R2 between mass numbers A1 and A2

3/12

3/11 AArR o

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 parametersheavy 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

11-37

when transfer populates many overlapping states, find single peak at characteristic angle (grazing angle)projectile trajectory essentially controlled by

Coulomb forcesone-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 Reactionsprocesses in which relatively large amounts of

nuclear matter transferred between target and projectile and which show strongly forward-peaked angular distributions“grazing contact mechanism”

11-38

double differential cross sections are distinguishing featureproducts 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 transferthe more the A of product and projectile differ in

either direction, the lower the kinetic energyat 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 projectileswill dissociate into two fragments appreciable fraction of incident kinetic energy

dissipated and goes into internal excitation

11-39

11-40

• Compound-Nucleus Reactionscompound-nucleus formation can only take place

over a restricted range of small impact parameterscan define critical angular momentum above

which complete fusion cannot occurcf/R decreases with increasing bombarding

energylight heavy ions produce compound nuclei on

neutron-deficient side of stability beltheavy 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