PHYS490: Nuclear Physics
Transcript of PHYS490: Nuclear Physics
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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)
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The Impact Parameter…
A measure of how “close” the reactants come together
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…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
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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 ]
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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”
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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 !
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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)
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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.
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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
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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
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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)
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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)
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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)
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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.
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Measuring the relative gamma-ray intensities in heavy nuclei (i.e. Coulex cross-sections) gives an insight into electromagnetic matrix elements
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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
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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
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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
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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 !
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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
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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
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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
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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
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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
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Cold Fusion
Superheavy elements (SHE’s) can be formed by low-energy fusion-evaporation reactions in which only one neutron is emitted
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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)
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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
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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)
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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)
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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) …
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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)
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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)