phys490: nuclear physicsns.ph.liv.ac.uk/phys490/chapter13.pdf · 13. nuclear reactions. a variety...

3/5/2020 PHYS490 : Advanced Nuclear Physics : E.S. Paul 1

PHYS490: Nuclear Physics

Advanced Nuclear Physics

Chapter 13



13. Nuclear Reactions

A Variety Of Reactions



Introduction

In a typical nuclear reaction a (light) projectile a “hits” a (heavy) target A producing fragments b (light) and B (heavy)

Schematically this can be written

a + A b + B

In this nuclear “transmutation” we need to consider both kinetic energy and binding energy (E = mc2)


The Impact Parameter…

A measure of how “close” the reactants come together


…Impact Parameter

Central collisions occur for small b, e.g. fusion

Peripheral collisions occur at large b, e.g. elastic and inelastic scattering, transfer reactions

Deep inelastic or massive transfer reactions occur at intermediate values of b

Reactions can be classified

by the impact parameter b


Collision Kinematics

The Q value is:

[ (MA + Ma) - (MB + Mb) ] c2

Exothermic (Q > 0) reactions give off energy – kinetic energy of reaction products

Endothermic (Q < 0) reactions require an input of energy to occur. By considering the kinetic energy available in the centre-of-mass frame, the threshold energy is:

Ta > |Q| [ (Ma + MA) / MA ]


The Compound Nucleus Consider the reactions:

a + A C* a + A*

b + B*

γ + C*

The incident particle a enters the nucleus A and suffers collisions with the constituent nucleons, until it has lost its incident energy, and becomes an indistinguishablepart of the excited compound nucleus C*

The compound nucleus ‘forgets’ how it was formed and its subsequent decay is independent of its formation: “Bohr’s Hypothesis of Independence”


Compound Nucleus Example Consider a beam of alpha particles of energy 5 MeV/A

(or MeV per nucleon) impinging on 60Ni:

+ 60Ni 64Zn*

At this (kinetic) energy, the incident particle is non-relativistic, β = v/c = 0.1, and it will take the alpha particle ~10-22 s to travel across the target nucleus

In a compound nucleus, the first emission of a nucleon or gamma ray takes > 10-20 s

Hence the alpha particle traverses the compound nucleus hundreds of times and loses its identity !


Geometric Cross-Section In the classical picture, the projectile and target nuclei

will fuse if the impact parameter b is less than the sum of their radii

A disk of area π(R1 + R2)2 is swept out

This area defines the geometric cross-section

Remember: units of cross-section are area (1 barn = 100 fm2;

1 fm = 10-15 m)


Coulomb Excitation (Coulex)


Coulomb Excitation

Coulomb Excitation (Coulex) is the excitation of a target nucleus by the long-range electromagnetic (EM) field of the projectile nucleus, or vice versa

The biggest effect occurs for deformed nuclei with high Z: In these nuclei, rotational bands can be excited to more than 20 ħ

In pure Coulex, the charge distributions of the two nuclei do not overlap at any time during the collision.


Coulex Example 248Cf bombarded by 5.3 MeV/A 208Pb

The beam energy is kept low – below the Coulomb Barrier– so that other reactions, e.g. fusion, do not compete

In this example:

Beam energy = 5.3 x 208 MeV = 1.1 GeV

The Coulomb Barrier (in the lab frame) is:

{ Z1Z2e2 / [4πε0(R1 + R2)] } x {(A1 + A2) / A2}

C-o-M barrier = 1.3 GeV

Coulex Gamma–ray Spectrum

Rotational states in 248Cf were excited up to 30 ħ



Intermediate Energy Coulex

At higher beam energies (> 30 MeV/A), well above the Coulomb Barrier, Coulex can still take place but in competition with other violent reactions

The process is now so fast that only the first excited states (2+ for even-even nuclei) are populated

Intermediate energy Coulex is characterised by straight line trajectories with impact parameters larger than the sum of the radii of the two colliding nuclei


IE Coulex Example (GSI RISING Experiment)

A gold target (179Au) bombarded by a 140 MeV/Aradioactive 108Sn beam

The beam energy is:

140 x 108 MeV = 15.1 GeV

At this energy, β = v/c =0.48 – the projectile is travelling at half the speed of light !

Note: 108Sn is not stable – cannot make a target, but can generate a short-lived Radioactive Ion Beam (RIB)


Intermediate Energy Coulex

Ideally suited for use with fragmentation beams (Ebeam > 30 MeV/u)

Large cross sections (~100 mb)

Can use thick targets (~100 mg/cm2)


Neutron-Rich Sulphur Isotopes

Energy spectra in target (top) and projectile (bottom) frames of reference for 40S + 197Au at MSU

β = v/c = 27%

H. Scheit et al. Phys. Rev. Lett. 77, 3967 (1996)


Coulex Cross Sections

For Intermediate Energy Heavy Ions, the Coulomb excitation cross section can be approximated as:

σπλ = [Z1e2/ħc] B(πλ;0λ) [πR2/e2R2λ(λ-1)]

for λ ≥ 2

Here Z1 is the charge of the projectile and R is the sumof the radii of target and projectile

The cross section is peaked at forward angles within the angular range

Δθ ≈ 2Z1Z2e2/RE

where E is the beam energy

Octupole Nuclei

B(E3) values have been deduced in radium isotopes (RIBs) implying octupole (pear-shape) collectivity

2020 Phys. Rev. Lett.


Measuring the relative gamma-ray intensities in heavy nuclei (i.e. Coulex cross-sections) gives an insight into electromagnetic matrix elements


Neutron Capture


Neutron Capture Low-energy neutron-capture

cross-sections exhibit peaks or resonances corresponding to a compound system

An example is the capture of a 1.46 eV neutron by 115In to form a highly excited state (6.8 MeV !) in 116In

The high excitation energy in 116In arises due to the binding energy of the neutron


Neutron Capture Cross-Sections At 1.46 eV, the measured total cross-section for neutron

capture by 115In is σ ≈ 2.8 x 104 barn

However the geometrical cross-section (πR2) is only ≈ 1.1 barn

Quantum effect: we need to consider the de Broglie wavelength (λ/2π) instead of the nuclear radius – slow neutrons have a large wavelength and hence a long-rangeinfluence

The cross-section becomes:σ = πR2 σ = π(λ/2π)2


De Broglie Wavelength The momentum of the neutron is:

pn = √{2mnE} = √{2 x 939 x 1.46 x 10-6}= 0.052 MeV/c

The de Broglie wavelength is then:(λ/2π) = ħc/pnc = 197/0.052 = 3.7 x 103 fm

The cross-section then becomes 4.3 x 105 barn

The measured value is only 6% of this estimate !

We must also consider other effects such as the spins of the neutron, target and compound systems


Decay of 116In*

116In* n + 115In 4%

γ + 116In* 96%

For this neutron energy of only 1.46 eV

Γn/Γγ = 0.04, also Γn/Γ ≈ 0.04 (Γ = Γn + Γγ)

This decay fraction can be related to the formation cross-section:

σ = π(λ/2π)2 x Γn/Γ (Γn/Γ ≈ 4%)

Recall the measured formation cross-section was only 6% of the estimate using the de Broglie wavelength !


Proton Capture

For charged-particle capture (and decay) we must consider the Coulomb Barrier which inhibits the formation or decay of a compound system

The proton needs sufficient energy to overcome the Coulomb Barrier (several MeV) and hence its de Brogliewavelength is smaller (than in the case of neutron capture

Consequently, proton-capture cross-sections are ~ 1 barn at maximum


Nuclear Fusion


Compound Nucleus Reactions

Two nuclei coalesce, forming a fused system that lasts for a relatively long time (10-20 to 10-16 s)

De-excitation follows by a combination of particle and/or gamma decay

Compound system has no memory of entrance channel, the cross section of the exit channel is independent

Occurs for central collisions around Coulomb barrier energies

Nuclear Fusion: Coulomb Barrier

Electrostatic repulsion between colliding nuclei



Heavy Ion Fusion Reactions For heavy projectile ions, e.g. 12C or 58Ni, the Coulomb

Barrier is high and the particle enters a continuum of high level densities and overlapping resonances

The excitation of the compound nucleus is also high: 10-80 MeV

Since the neutron binding energy is only ~ 8 MeV, several neutrons are emitted before gamma-ray emission dominates

These Heavy Ion Fusion Evaporation reactions bring large amounts of angular momentum into the compound system


Fusion Evaporation Reactions


David CampbellFlorida State University


Fusion Cross-Section The angular momentum

brought into the compound system depends on the impact parameter b:

ℓ = b p

The partial fusion cross-section is proportional to the angular momentum:

d σfus(ℓ ) ~ ℓ

Total Cross-Section and ℓmax

Summing together the contributions from each partial wave ℓ, the total fusion cross-section is

σR = πλ2∑(2ℓ+1)Tℓ

where λ is the reduced wavelength

Here the transmission coefficients Tℓ are ~1 for ℓ < ℓmax

and ~0 for ℓ > ℓmax and hence

σR ~ πλ2ℓmax

where ℓmax can be considered the maximum angular momentum when the nuclei just touch

See Tutorial 4



Compound Formation And Decay Compound nucleus

formation: 10-20 s Neutron emission:

10-19 s Statistical (cooling)

dipole gamma-ray emission: 10-15 s

Quadrupole (slowing down) gamma-ray emission: 10-12 s

After 10-9 s the nuclear ground state is reached after 1011

rotations

100Mo(36S,4n)132Ce

Beam energy: 4.31 MeV/A


Cold Fusion

Superheavy elements (SHE’s) can be formed by low-energy fusion-evaporation reactions in which only one neutron is emitted

Fusion Reactor



Other Reactions


Transfer Reactions Transfer reactions occur within a timescale comparable

with the transit time of the projectile across the nucleus

Cross sections are a fraction of the nuclear area

The de Broglie wavelength of a 20 MeV incident nucleon is 1 fm and it interacts with individual nucleons at the nuclear surface

The projectile may lose a nucleon (stripping reaction) or gain a nucleon (pick up reaction)


Transfer Reactions


Neutron-Induced Fission

Photo-Nuclear Reactions



Direct Reactions Proceed in a single step, timescale comparable to the

time for the projectile to traverse the target (10-22 s)

Usually only a few bodies involved in the reaction

Excite simple degrees of freedom in nuclei

Mostly surface dominated (peripheral collisions)

Primarily used to study single-particle structure

Examples: elastic and inelastic scattering


Elastic Scattering

Both target and projectile remain in their ground state

a + A a + A

Nuclei can be treated as structureless particles

0 2 4 6 8 1010-5

10-4

10-3

10-2

10-1

100

9Li

11Li core

11Li matter

rm

(r)

[fm

-3]

r [fm]

Example:

Investigation of nuclear matter density distributions in exotic nuclei by elastic p-scattering (inverse kinematics)


Inelastic Scattering Both target and projectile nuclei retain their integrity,

they are only brought to bound excited states

a + A a* + A*

Can excite both single-particle or collective modes of excitation

Example: investigate the GMR by (,’) inelastic scattering, gives access to nuclear incompressibility, key parameter of nuclear EOS

Knm (Z,N) ~ r02 d2(E/A) / dr2 | r0

Example: safe and unsafe Coulomb excitation (below and above Coulomb barrier)


Transfer Reactions One or a few nucleons are transferred between the

projectile and target nuclei

Probes single-particle orbitals to which nucleon(s) is (are) transferred

Characteristics of the entrance channel determines selectivity of the reaction, i.e. alpha particle with T=0leads to states with the same isospin as the ground state, but proton with T=1/2 leads to states with T=T±1

Examples numerous: (d,p) , (p,d) , (t,p) , (t,3He) …


Charge Exchange Reactions

Reactions that exchange a proton for a neutron, or vice versa

Net effect is the same as β+ or β- decay

But not limited by Qβ – can reach higher excited states and giant resonances

Many different probes: (p,n) , (d,2He) , (t,3He) , but also with heavy ions, e.g. (7Li,7Be) or exotic particles, e.g. (π+,π0)


Knockout Reactions One or a few nucleons are ejected from either the

target and/or the projectile nuclei, the rest of the nucleons being spectators

Exit channel is a 3-body state

Becomes dominant at high incident energies

Populates single-hole states, from which spectroscopic information can be derived

Examples: (p,2p) , (p,pn) , (e,e’p) , heavy-ion induced knockout, e.g. (9Be,X)


Types of Nuclear Reaction

Summary

Nuclear interactions

Impact parameter, cross-section

Coulomb excitation

Neutron/proton capture

Fusion - compound nucleus

Transfer/knockout reactions


phys490: nuclear physicsns.ph.liv.ac.uk/phys490/chapter13.pdf · 13. nuclear reactions. a variety...

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