--- title: "What a nuclear mass tells us about binding" description: "Calculate a binding energy from mass excesses, then distinguish average binding from the energy needed to remove the last neutron." canonical_url: "https://sandboxphysics.com/stories/nuclear-bridge/binding-and-scales/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/nuclear-bridge/binding-and-scales/" last_modified: "2026-10-27" --- [Nuclear physics](https://sandboxphysics.com/stories/nuclear-bridge/) L1 · Chapter 1 # What a nuclear mass tells us about binding Sandbox Physics · 2026-10-27 A bound nucleus usually weighs less than its separated protons and neutrons. The missing rest mass corresponds to the energy released when the nucleus forms. A mass measurement can therefore tell us how much energy it takes to pull a nucleus apart, even before we know how its particles move. ## Before you begin Mass, energy and arithmetic. The relation between rest mass and energy is introduced here; quantum mechanics is not yet required. - Calculate deuteron binding from three entries in a mass table. - Use neighboring masses to ask whether the last neutron is weakly bound. ## Choose the separated particles as the reference Write $Z$ for proton number and $N$ for neutron number. Their sum $A=N+Z$ is the mass number. Imagine the free nucleons infinitely far apart, with no mutual interaction. Their total rest energy exceeds the nuclear ground-state energy by the positive binding energy $B$. Nuclear masses are conveniently expressed in unified atomic mass units, $\mathrm u$. The conversion $1\,\mathrm u\,c^2\simeq931.494\,\mathrm{MeV}$ turns a mass difference into an energy; a megaelectronvolt is a million electronvolts. The speed of light $c$ supplies the conversion between rest mass and rest energy. A table of atomic masses includes electrons. Using hydrogen atoms on the separated side and a neutral atom on the bound side cancels the electron rest masses. The deuteron calculation below neglects the small isotope difference between hydrogen and deuterium electronic binding. For heavier nuclei, electronic binding corrections can reach kiloelectronvolts or more and must be assessed against the required precision. $$ B=\left[Zm_p+Nm_n-M_{\mathrm{nuc}}(Z,N)\right]c^2 $$ The nuclear-mass expression uses free proton and neutron masses and the mass of the bare nucleus. Source: Wang et al. ## A worked calculation from mass excesses AME2020 stores a convenient small quantity called mass excess: $\Delta=(M-A\,\mathrm u)c^2$. Its sign refers to an integer multiple of the atomic mass unit. A positive mass excess does not by itself mean that a nucleus is unbound relative to free nucleons. The frozen table gives $8071.31806\,\mathrm{keV}$ for the neutron, $7288.971064\,\mathrm{keV}$ for hydrogen and $13135.722895\,\mathrm{keV}$ for deuterium. Hydrogen plus a neutron has the same total mass number as deuterium, so the integer mass-unit contributions cancel. Add the first two excesses and subtract the third. The result is about $2.224566\,\mathrm{MeV}$. This is the energy required to dissociate a deuteron into a proton and a neutron. In the reverse capture process, radiation and recoil carry away the released energy. As a separate check, twice the tabulated binding per nucleon, $1112.2831\,\mathrm{keV}$, agrees after rounding. $$ B_d\simeq\Delta({}^{1}\mathrm H)+\Delta(n)-\Delta({}^{2}\mathrm H)=2224.566229\,\mathrm{keV} $$ Extra digits show the table arithmetic. The displayed physical answer is rounded, and the small isotope difference in electronic binding is omitted. Source: Wang et al. ## Ask separately about the last neutron Dividing binding energy by mass number makes nuclei of different sizes easier to compare. That average hides the situation of the last neutron. To remove one neutron, compare the original nucleus with its neighbor having the same proton number and one fewer neutron. The difference in their binding energies is the one-neutron separation energy. Consider an illustrative nucleus bound by $160\,\mathrm{MeV}$ and a neighbor bound by $159.5\,\mathrm{MeV}$. Removing the last neutron costs only $0.5\,\mathrm{MeV}$. A strongly bound core can coexist with a weakly bound outer neutron. Chapter 8 turns that small separation energy into a long spatial decay length. Separation energies often alternate between neighboring even and odd neutron numbers because pairing correlations favor pairs. A two-neutron difference reduces that alternation and can reveal a smoother trend. A sudden decrease beyond a closed shell suggests that newly added neutrons must occupy a more costly orbital. $$ S_n(Z,N)=B(Z,N)-B(Z,N-1),\qquad S_{2n}(Z,N)=B(Z,N)-B(Z,N-2) $$ Positive separation energy means removal costs energy. Other open decay channels must be checked separately. Source: Wang et al. ## Masses are the first clue A total binding energy does not specify a unique internal arrangement. Motion in an average field, pairing, vibration and rotation can all contribute. Spectral lines and scattering add information about the motion and spatial structure. We next give one neutron the simplest useful environment: a spherical well of finite depth. Before reading a numerical trend, identify the isotope, energy reference and uncertainty. An evaluation combines evidence into recommended values. AME marks extrapolated masses with a hash sign. Treating those entries as independent measurements would overstate how firmly the trend is known. Source: NNDC ## A different probe reads a different structure Masses reveal energy differences. Electron scattering adds electromagnetic information. The existing form-factor experiment uses a proton target and varies electron energy and detector angle at fixed momentum transfer to separate electric and magnetic responses. It studies the proton itself; its graph is not a three-dimensional density reconstruction of a whole nucleus. The neutron-shell experiment works at a different level. It fills states in a spherical mean field and calculates density from their shared wavefunctions. In Chapter 5 an incident neutron instead probes a calcium nucleus through elastic scattering. Identify the probe, target and observable before deciding what a picture tells you. Source: M. N. Rosenbluth ## Try the reasoning in an experiment ### [Find the next orbital after a shell fills](https://sandboxphysics.com/labs/nuclear-shell-filling/) 1. Predict how the capacity of an orbital depends on angular momentum. 2. Change neutron number and watch which orbital fills next. 3. Explain where a shell gap could produce a drop in separation energy. The scene describes a specified mean-field arrangement. ### [Separate the proton responses](https://sandboxphysics.com/labs/nuclear-form-factor/) 1. Predict how electron energy must change when detector angle changes at fixed momentum transfer. 2. Acquire the existing fixed-transfer scan and compare the electric and magnetic responses. 3. Explain how its proton target differs from the nucleus in the neutron-shell model. ## Exercises and explanations 1. A nucleus has binding $180\,\mathrm{MeV}$. Its one-neutron and two-neutron removal products have bindings $178$ and $174\,\mathrm{MeV}$. Calculate both separation energies. Read the explanation The one-neutron separation energy is $2\,\mathrm{MeV}$ and the two-neutron separation energy is $6\,\mathrm{MeV}$. The second removal changes the system, so twice the first separation energy need not equal the two-neutron value. 2. Does a large binding energy per nucleon guarantee that the last neutron is tightly bound? Read the explanation No. The average uses the whole nucleus. Last-neutron binding is a difference between neighboring total energies, so another mass is needed to answer that question. [Next: How a finite nuclear well selects discrete levels](https://sandboxphysics.com/stories/nuclear-bridge/finite-wells-and-spin-orbit/) ## Sources and reproduced scope Evaluated level energies and published reaction measurements retain their source versions and uncertainties. Model curves, schematic figures and synthetic scattering records are identified separately. The offline benchmark files state which calculations have been reproduced. 1. [M. N. Rosenbluth, High Energy Elastic Scattering of Electrons on Protons, Physical Review 79, 615 (1950)](https://doi.org/10.1103/PhysRev.79.615) The existing proton form-factor Lab uses one-photon Born kinematics and a fixed-momentum-transfer Rosenbluth separation. Its source documentation and numerical model remain owned by Issue 22. 2. [Wang et al., The AME2020 atomic mass evaluation (II), Chinese Physics C 45, 030003 (2021)](https://www-nds.iaea.org/amdc/ame2020/mass_1.mas20.txt) Frozen March 3, 2021 mass-table format, neutron and hydrogen/deuterium rows. Mass excess and binding per nucleon are in keV; # marks extrapolated values. The course uses a three-row numerical excerpt with its original uncertainties. 3. [NNDC, NuDat 3 user guide and terms of use](https://www.nndc.bnl.gov/nudat3/guide/) Contents, units, terms of use and changing evaluations. NuDat reports evaluated recommendations; their quoted errors do not include a model discrepancy.