Nuclear Masses, Mass Excess, and Separation Energies

The mass of a nucleus is its most precisely known property and the source of every decay and reaction energy. Because binding energies are a fraction of a percent of the rest mass, the useful quantity is not the mass itself but its small deviation from the integer mass number, and the machinery of this lesson is the bookkeeping that turns tabulated masses into Q-values and separation energies.¹

The atomic mass unit and mass excess

Masses are quoted on the atomic scale, fixed by carbon:

The mass of a nuclide is close to $A$ in these units because each nucleon contributes roughly $1\mathrm{u}$. The deviation is the mass excess:

The neutron and hydrogen mass excesses set the reference points,

so the binding energy in terms of mass excesses is $B=Z\Delta {(}^1\mathrm{H})+N{\Delta }_n-\Delta (Z,A)$, a form that avoids subtracting large nearly equal masses.

Penning-trap mass spectrometry

A charged ion of mass $m$ and charge $q$ in a uniform magnetic field $B$ circulates at the cyclotron frequency

Measuring ${\omega }_c$ against a reference ion of well-known mass in the same field eliminates $B$ and returns the mass ratio, hence the unknown mass. Confining the ion for a long observation time requires a weak electrostatic quadrupole in addition to $B$; that is the Penning trap, whose three normal modes (a modified cyclotron mode, a magnetron drift, and an axial oscillation) satisfy the invariance relation

so the true cyclotron frequency is recovered even when the trap is slightly misaligned. Modern traps reach fractional precisions $\delta m/m\sim 10^{-9}$ to $10^{-11}$, resolving binding-energy differences of a few keV between neighbouring nuclides. The evaluated masses feeding the chart of the nuclides come from these measurements combined with reaction and decay Q-values in a global least-squares adjustment.

Q-values from mass differences

Any transformation releases (or absorbs) an energy equal to $c^2$ times the mass lost.

Using atomic rather than nuclear masses is convenient because the electron masses cancel in most modes. For the three beta processes and for alpha decay,

• Negatron (${\beta }^{-}$): ${}_Z^{A}X\to {}_{Z+1}^A{X}^{\prime }+e^{-}+\bar{\nu }$, and with atomic masses the $Z$ electrons of the parent and the emitted electron account for the $Z+1$ electrons of the daughter, $$Q_{{\beta }^{-}}=[M(Z,A)-M(Z+1,A)]c^2.$$
• Positron (${\beta }^{+}$): the daughter has one fewer proton, so two electron masses remain, $$Q_{{\beta }^{+}}=[M(Z,A)-M(Z-1,A)-2m_e]c^2.$$
• Electron capture: $Q_{\text{EC}}=[M(Z,A)-M(Z-1,A)]c^2-B_e$, with $B_e$ the binding energy of the captured atomic electron.
• Alpha decay: $Q_{\alpha }=[M(Z,A)-M(Z-2,A-4)-M{(}^4\mathrm{He})]c^2$.

The $1.022\mathrm{MeV}$ ($2m_e{c}^2$) that separates the ${\beta }^{+}$ and electron-capture thresholds is why proton-rich nuclides with small $Q$ decay only by capture.

Separation energies

Removing one nucleon costs an energy equal to the difference of binding energies, and this difference is a far sharper probe of shell structure than $B$ itself.

$S_n$ is the nuclear analog of an atomic ionization energy. Plotted against $N$ it shows two features. First, a zigzag: an even-$N$ nucleus binds its last neutron more tightly than the neighbouring odd-$N$ nucleus because the added neutron pairs with an existing one, so $S_n$ oscillates by the pairing energy $\sim 12A^{-1/2}\mathrm{MeV}$. The two-neutron separation energy $S_{2n}$ removes the staggering and gives a smooth trend that is the better structural probe. Second, a sharp drop just past a magic number: filling a neutron shell at $N=8,20,28,50,82,126$ binds those neutrons tightly, and the first neutron in the next shell is far less bound, so $S_n$ falls by $1$ to $2\mathrm{MeV}$ as $N$ crosses the closed shell. These discontinuities are the mass-measurement signature of the magic numbers.

The drip lines and the nuclear landscape

A nucleus is bound against nucleon emission only while its separation energies stay positive. When $S_n$ reaches zero, the last neutron is unbound and any additional neutron leaks out immediately: this locus in the $(N,Z)$ plane is the neutron drip line. The mirror condition $S_p=0$ defines the proton drip line. Between them lies every particle-bound nuclide; beyond them a nucleon is not confined by the nuclear potential.

The drip lines sit far from stability. Because the asymmetry energy grows only quadratically while pairing and shell effects intervene, the neutron drip line is poorly known and predicted to lie at very large neutron excess for medium-mass elements, whereas the proton drip line is close to stability and experimentally mapped, held in by the Coulomb barrier that briefly traps even a proton-unbound state. The number of bound nuclides between the drip lines is of order several thousand, of which the stable band is a thin thread.

Mass measurements give the depth of binding at each point of this landscape but not its origin. The next lesson builds a formula for that binding from five physical terms, and the resulting mass surface reproduces the parabolas, the valley of stability, and the onset of fission.