NUCLEAR PHYSICS
Exercise set 12
Turn in your solutions THURSDAY,
Dec. 3rd, BEFORE NOON
Fall term 2015
Period 2
Nuclear structure and nuclear models, liquid-drop model (Bethe-Weizsäcker formula), Fermi-gas model,
single-particle shell model, binding energies. Literature: Bertulani (B); B.A. Brown: Lecture Notes
2011 (BAB), and K. Heyde (H2004). Don’t forget to give the reference you use! For experiments in
nuclear physics, accelerators and detectors, see lectures + Krane.
1. The line of nuclear stability. Neutron and proton drip lines. The liquid-drop model (LDM) is a
useful tool in studying the behaviour of nuclear stability. (Rather good estimates can already be
obtained using the “original” parameter values of Wapstra 1971.)
(a) Line of stability, Zstable = Z(A) is defined by those nuclei that have the largest binding energy
for each A. Show that approximately
Zstable (A) =
A/2
A/2
.
aC 2/3 =
1 + 4aA A
1 + 0.0077A2/3
(b) Test the validity of the above formula in different regions of the nuclear chart: A ≈ 15,
A ≈ 50, A ≈ 140, A ≈ 210, by comparing with the experimental binding energies for the
corresponding isobars1 .
(c) How would your estimate change, if you take into account the total binding energy of electrons, as given approximately by
Bel (Z) = 14.4381Z 2.39 + 1.55468 × 10−6 Z 5.35 . [eV]
drip
drip
(d) Neutron and proton drip lines, Zn (A) and Zp (A), define the borderlines of the nuclear
landscape, where the separation energy of a single neutron or proton becomes negative.
Derive the expressions for LDM-predictions of the drip lines2 , and apply at A ≈ 10 and
A ≈ 100. Are the estimates anywhere near to what has been observed experimentally?
2. Neutron star, a bold extrapolation of the liquid-drop model.
By including the gravitational interaction between the nucleons, as described by
3 GM 2
5 R
we can estimate radius and mass of a neutron star, the most dense form of known matter.
(a) Basic assumptions: Why is it reasonable to include only the volume and asymmetry terms
of LDM (+ gravity of course)?
(b) Stability condition: Assume a spherical shape with a sharp radius R = r0 A1/3 and derive the
condition required to a neutron star to be a bound system.
(c) Plug in the values of the gravitational constant G, neutron mass, and the volume and asymmetry terms (Wapstra 1971) to show that A ≈ 5 × 1055 .
(d) Estimate the radius and mass of such a neutron star.
3. Fermi-gas model (B:5-3, BAB:Ch7)
In the (degenerate) Fermi-gas model of the nucleus, protons and neutrons are assumed to fill
the lowest possible energy states allowed by the Pauli exclusion principle and to move without
mutual interactions (independent particles) as a mixture of two fermion gases.
(a) Expand the total average kinetic energy hT i in terms of D := (N − Z)/A to show that
(N − Z)2
3
hT i = A εF + x
εF
5
A
1 The
2 The
AME2012 mass table: Z, A, and experimental ∆ (mass excess in keV) is here. Please don’t print a hardcopy!
location of the drip lines is an area of very active research in experimental and theoretical nuclear physics.
NUCLEAR PHYSICS
Exercise set 12
Turn in your solutions THURSDAY,
Dec. 3rd, BEFORE NOON
Fall term 2015
Period 2
(b) Determine x in the above formula.
(c) Compare with the liquid-drop model of the binding energy to estimate the depth of the
average mean-field potential hV i:
hV i = V0 A +V1
(N − Z)2
.
A
Determine the constants V0 and V1 .
Hint: Total energy E = hT i + hV i = −B(A, Z). Keep only the volume and symmetry terms in
LDM (aV = 15.49 MeV and aA = 22.6 MeV).
4. Accelerators, detectors and related technologies for nuclear structure research.
The 5-MV electrostatic tandem accelerator
(TAMIA) of the Dept. of Physics can accelerate
gold ions to a maximum energy of about 80 MeV.
The negative atomic ions from the ion source are
preaccelerated with 30 kV voltage and transferred
through the 10 m long low-energy beam transfer
(LEBT) line before injection into the accelerator.
The negative ions are attracted by the high-voltage
terminal (max. UT = 5 MV) where they pass
through a 0.5 m long canal filled with Ar or CO2
gas. The ions are “stripped” to positive charge,
q = n · e, in charge-exchange (CX) reactions with
the gas atoms and get repelled away from the
terminal (tandem principle).
(a) Assuming negligible energy loss in CX, show
that the final (kinetic) energy of the accelerated positive ion is
E = Upreacc. +UT · (1 + n) e.
What is the charge state n required to reach
80 MeV?
(b) For some elements, negative (atomic) ions do not exist and they must be injected as molecular ions that are destroyed in the charge-exchanging collisions at the terminal. Modify the
above formula to the case that a binary molecular ion, Ak Bl , is injected.
(c) The hydrogen depth profiling uses the 1 H(15 N, αγ)12 C resonance reaction at 6.395 MeV.
What terminal voltage-charge state combinations are possible, if the injected molecular ion
is 15 N 1 H−
2?
(d) Some elements form metastable negative ions (mean lifetime τ) that can be injected in the
tandem. If we require at least 60% survival probability at the terminal, how short mean
lifetime can be tolerated? The total length of the accelerator is about 8 m.
5. Nuclear shell model: Will be discussed in the next HW and/or in the lectures.