---
title: Nuclear Masses, Mass Excess, and Separation Energies
module: Nuclear Properties
moduleNumber: 1
lessonNumber: 3
order: 103
summary: >
  The atomic mass unit fixes the scale, and the mass excess collects the small
  binding-driven deviation from the integer mass number. Penning-trap cyclotron
  frequencies now measure masses to parts in a billion, and every decay and reaction
  Q-value is a difference of these masses. One- and two-nucleon separation energies
  read the binding difference between neighbouring nuclides directly, showing the
  even-odd pairing stagger and the sharp drops at magic numbers, and their vanishing
  marks the neutron and proton drip lines that bound the chart of the nuclides.
topics: [Nuclear Properties]
sources:
  - book: Krane
    ref: "Ch. 3 — Nuclear Properties; §3.2 Mass and Abundance of Nuclides, §3.3 Nuclear Binding Energy"
  - book: Wong
    ref: "Ch. 4 — Nuclear masses and the mass surface"
  - book: Tipler & Llewellyn
    ref: "Ch. 11 — Nuclear Physics; §11-2 Ground-State Properties"
draft: false
---

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.[^krane-mass]

## The atomic mass unit and mass excess

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

> **Definition (Atomic mass unit).** One atomic mass unit is one-twelfth the mass of a
> neutral $^{12}\mathrm{C}$ atom,
> $$
> 1\,\mathrm{u} = \frac{1}{12}\,M(^{12}\mathrm{C}) = 1.660539\times10^{-27}\,\mathrm{kg},
> \qquad u\,c^2 = 931.494\ \mathrm{MeV}.
> $$
> Tabulated masses $M(Z, A)$ are **atomic** masses: they include the $Z$ electrons and
> their (keV-scale) binding.

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**:

> **Definition (Mass excess).** For an atom of mass $M$ (in $\mathrm{u}$) and mass number
> $A$,
> $$
> \Delta = \bigl(M - A\bigr)\,u\,c^2,
> $$
> quoted in $\mathrm{MeV}$. By construction $\Delta(^{12}\mathrm{C}) = 0$. The mass excess
> is not the binding energy; it collects the neutron-proton mass difference and the
> binding together, and it is the tabulated quantity from which Q-values are computed.

The neutron and hydrogen mass excesses set the reference points,

$$
\Delta_n = 8.071\ \mathrm{MeV}, \qquad \Delta(^1\mathrm{H}) = 7.289\ \mathrm{MeV},
$$

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**

$$
\omega_c = \frac{qB}{m}.
$$

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

$$
\omega_c^2 = \omega_+^2 + \omega_-^2 + \omega_z^2,
$$

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.

$$
% caption: A Penning trap confines a single ion with a strong axial magnetic field and
% a weak electrostatic quadrupole; the measured cyclotron frequency q B over m gives the
% mass by comparison with a reference ion.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % endcap electrodes (top/bottom)
  \draw[black, fill=black!8] (-1.7,1.5) rectangle (1.7,1.85);
  \draw[black, fill=black!8] (-1.7,-1.85) rectangle (1.7,-1.5);
  \node[black, anchor=south, font=\scriptsize] at (0,1.9) {endcap};
  \node[black, anchor=north, font=\scriptsize] at (0,-1.9) {endcap};
  % ring electrode (two side pieces)
  \draw[black, fill=black!8] (-1.95,-0.75) rectangle (-1.55,0.75);
  \draw[black, fill=black!8] (1.55,-0.75) rectangle (1.95,0.75);
  \node[black, anchor=east, font=\scriptsize] at (-2.0,0) {ring};
  % magnetic field
  \draw[very thick, ->] (2.7,-1.6) -- (2.7,1.6) node[anchor=south, black] {$B$};
  % ion orbit (cyclotron loop)
  \draw[acc, thick] (0,0) ellipse (0.75 and 0.32);
  \fill[acc] (0.75,0) circle (2.2pt);
  \node[acc, anchor=west, font=\scriptsize] at (0.9,0.15) {ion};
  \draw[black, ->] (0.55,0.26) arc (30:150:0.63 and 0.27);
  \node[black!70, anchor=south, font=\scriptsize] at (0,0.5) {cyclotron orbit};
\end{tikzpicture}
$$

## Q-values from mass differences

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

> **Definition (Q-value).** For initial and final states,
> $$
> Q = \Bigl(\sum m_{\text{initial}} - \sum m_{\text{final}}\Bigr)c^2 .
> $$
> $Q > 0$ is exoergic (energy released, spontaneous decay allowed); $Q < 0$ is
> endoergic (a threshold energy is required).

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

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.

> **Definition (Nucleon separation energies).** The neutron and proton separation
> energies are
> $$
> S_n = \bigl[M(Z, A-1) + m_n - M(Z, A)\bigr]c^2 = B(Z, A) - B(Z, A-1),
> $$
> $$
> S_p = \bigl[M(Z-1, A-1) + M(^1\mathrm{H}) - M(Z, A)\bigr]c^2 = B(Z, A) - B(Z-1, A-1).
> $$
> The two-nucleon separation energies $S_{2n}$ and $S_{2p}$ remove a pair.

$S_n$ 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 $\sim 12\,A^{-1/2}\,\mathrm{MeV}$.
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.

$$
% caption: Two interleaved mass parabolas at fixed odd or even mass number; even-even
% nuclides lie on the lower curve and odd-odd on the upper, producing the even-odd
% staggering of the ground-state masses.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {$Z$};
  \draw[->, black] (0,0) -- (0,4.3) node[above, black!70] {mass};
  % lower parabola (even-even)
  \draw[acc, very thick] (0.6,3.6) .. controls (2.6,0.2) and (4.4,0.2) .. (6.9,3.9);
  % upper parabola (odd-odd), shifted up by pairing
  \draw[black, thick, dashed] (0.7,4.15) .. controls (2.7,1.15) and (4.3,1.15) .. (6.8,4.3);
  % staggered points
  \foreach \x/\y in {1.2/2.55,2.4/0.75,3.6/0.55,4.8/1.35,6.0/3.05} \fill[acc] (\x,\y) circle (1.7pt);
  \foreach \x/\y in {1.8/2.15,3.0/1.4,4.2/1.75,5.4/2.7} \draw[black, thick] (\x,\y) circle (1.9pt);
  \node[acc, anchor=west, font=\scriptsize] at (4.5,0.55) {even-even};
  \node[black, anchor=west, font=\scriptsize] at (4.5,2.05) {odd-odd};
\end{tikzpicture}
$$

$$
% caption: The neutron separation energy zigzags with the even-odd pairing energy and
% drops sharply as N crosses a magic number, marking the closure of a neutron shell.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.8,0) node[right, black!70] {$N$};
  \draw[->, black] (0,0) -- (0,4.3) node[above, black!70] {$S_n$ (MeV)};
  % zigzag descending then a sharp drop at magic N, then resume
  \draw[acc, very thick]
    (0.4,3.7) -- (0.9,3.1) -- (1.4,3.5) -- (1.9,2.9) -- (2.4,3.3)
    -- (2.9,2.7) -- (3.4,3.1) -- (3.9,2.5)
    -- (4.1,2.4);
  % drop at magic number
  \draw[acc, very thick] (4.1,2.4) -- (4.35,1.15);
  \draw[acc, very thick]
    (4.35,1.15) -- (4.85,1.55) -- (5.35,1.0) -- (5.85,1.45) -- (6.35,0.9) -- (6.85,1.3);
  % magic-number marker
  \draw[black, dashed] (4.2,0) -- (4.2,2.4);
  \node[black!70, anchor=north, font=\scriptsize] at (4.2,-0.05) {magic $N$};
  \node[acc, anchor=west, font=\scriptsize] at (0.9,3.55) {pairing zigzag};
\end{tikzpicture}
$$

## 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](/nuclear-physics/nuclear-properties/nuclear-constituents-nuclide-chart)
is a thin thread.

$$
% caption: The chart of bound nuclides in the N-Z plane is bounded below by the proton
% drip line (S_p equals zero) and above by the neutron drip line (S_n equals zero), with
% the line of beta stability threading the interior.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {$N$};
  \draw[->, black] (0,0) -- (0,5.0) node[above, black!70] {$Z$};
  % neutron drip line (upper-right boundary, far to neutron rich)
  \draw[black, thick] (0.3,0.5) .. controls (3.5,1.2) and (6.0,1.9) .. (7.0,2.4);
  \node[black, anchor=north west, font=\scriptsize] at (5.4,1.9) {neutron drip};
  % proton drip line (lower boundary, close to stability)
  \draw[black, thick, dashed] (0.3,0.35) .. controls (2.0,1.6) and (3.3,2.9) .. (4.2,4.2);
  \node[black, anchor=south east, font=\scriptsize] at (2.6,3.0) {proton drip};
  % stability line inside
  \draw[acc, very thick] (0.3,0.42) .. controls (2.6,1.4) and (4.5,2.4) .. (5.6,3.3);
  \node[acc, anchor=west, font=\scriptsize] at (4.6,2.55) {stability};
  % shaded bound region hint (dots)
  \foreach \x/\y in {1.2/0.9,2.0/1.3,2.6/1.05,3.2/1.7,3.8/1.5,4.4/2.1} \fill[black!45] (\x,\y) circle (1.1pt);
\end{tikzpicture}
$$

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.

[^krane-mass]: **Krane**, _Introductory Nuclear Physics_, §3.2 (Mass and Abundance of Nuclides) and §3.3 (Nuclear Binding Energy): the atomic mass unit, mass excess, Q-value expressions in atomic masses, and separation energies. Evaluated masses are the Atomic Mass Evaluation maintained through the NNDC, [https://www.nndc.bnl.gov/](https://www.nndc.bnl.gov/), and the IAEA Nuclear Data Services, [https://www-nds.iaea.org/](https://www-nds.iaea.org/). Nucleon masses are CODATA/NIST values, [https://physics.nist.gov/cuu/Constants/](https://physics.nist.gov/cuu/Constants/).