Nuclear Binding Energy Calculator

What Is the Nuclear Binding Energy Calculator?

How to use the nuclear binding energy calculator

1. Choose the unknown. The Solve for menu opens on Binding energy per nucleon, and the other choice is Atomic mass of the nuclide. Whichever you pick stops being a box and becomes the answer.
2. Type the proton number. It is the atomic number of the element and opens on 2 for helium. There is no unit, because it is a count.
3. Type the neutron number. It opens on 2, and it is the neutron count rather than the mass number. Iron-56 goes in as 26 and 30, never as 26 and 56.
4. Type the measured atomic mass. It takes u and opens on 4.00260325413 u. It must be the mass of the neutral atom, electrons included.
5. Read the headline. It carries four significant figures of the binding energy per nucleon. The engine drops a trailing zero, so iron-56 reads 8.79 there and 8.790 in the chip.
6. Read the total binding energy chip. It is the whole nucleus taken apart: 28.30 MeV at the defaults, 492.3 MeV for iron-56, 1784 MeV for uranium-235.
7. Read the mass defect chip. It is the mass that went missing, to nine decimals: 0.030376641 u at the defaults. Multiply it by 931.49410242 and the total binding energy comes back.
8. Read the ceiling chip. It is the distance below 8.8 MeV per nucleon, which nothing in nature passes: 0.009644 for iron-56 against 0.005444 for nickel-62.
9. Solve backwards for the mass. The binding energy box then accepts MeV, keV or J per nucleon, and the answer is an atomic mass — read it from the first chip, not the headline.
10. Open Show working. Three steps: the formula with both constants named, the substitution as far as the mass defect, and the division by the mass number.

This page exists for one step: turning two nucleon counts and a measured mass into the energy that holds that nucleus together. The nuclear binding energy guide is the prose half of it, with the curve set out quantitatively, worked problems of rising difficulty and the misconceptions in full. Where the conversion between mass and energy comes from is a different question and a different page: the guide to E = mc2 is the relation this calculator uses once and never derives.

Three mistakes account for most wrong answers here. Typing the mass number into the neutron box — 56 instead of 30 for iron-56 — describes a nuclide with 82 nucleons and gets refused, because the mass no longer matches the counts. Entering a nuclear mass instead of an atomic one is the next, and it inflates every answer by about Z x 0.511 MeV. Last comes expecting the backwards direction to hand back exactly the mass you started from, which a four-figure binding energy cannot do.

Worked example: change one thing at a time

The table starts at the defaults, climbs the curve from the deuteron to the iron and nickel plateau, falls away again to plutonium, then runs the menu backwards, walks the unit entries and finally reaches the entries the calculator declines. Rows 15 to 19 are the same five nuclides solved in the opposite direction. Every answer and chip cell was read out of the running widget rather than worked out by hand, so where a cell and the tool ever part company, believe the tool.

Rows 1 to 5 are the left-hand side of the curve, and helium-4 is the famous anomaly sitting on it. Two nucleons in the deuteron share 2.225 MeV, which is only 1.112 MeV each, and six nucleons in lithium-6 manage 5.332 — yet the four nucleons of helium-4 are held at 7.074 MeV each, above both. That is why alpha particles leave heavy nuclei as a cluster of four rather than one nucleon at a time.

Rows 6 to 8 are the point of the whole page. Iron-56 gives 8.790 MeV per nucleon, iron-58 8.792 and nickel-62 8.795: three nuclides spanning six nucleons and five thousandths of an MeV. The ceiling chip reads 0.009644 for iron-56 against 0.005444 for nickel-62, and the difference between those two figures, 0.0042 MeV per nucleon, is the entire gap between the nuclide textbooks mark and the one that is actually highest.

Rows 9 to 13 come back down the other side, and the two energy columns stop agreeing with each other. Krypton-86 is 8.712 per nucleon, lead-208 7.867, uranium-235 7.591 and plutonium-239 7.56, falling the whole way; the totals meanwhile climb from 749.2 MeV to 1807 MeV. Plutonium-239 holds more than three and a half times the total binding energy of iron-56 and is the least tightly bound nuclide in rows 4 to 13.

Row 14 is an error the tool cannot catch. Iron-56's mass typed with the two counts swapped describes a nuclide of 30 protons and 26 neutrons, and the calculator answers 8.734 MeV per nucleon quite happily, because that combination is physically meaningful even though it is not iron. Nothing on the page knows which element you meant.

Rows 15 to 19 are the same five nuclides run backwards, and they deliberately do not hand back the masses they started from. Helium-4 returns 4.003 u where the entry was 4.00260325413, and carbon-12 returns 12 u in the headline while the chip reads 12.000001862. A binding energy rounded to four significant figures pins the atomic mass only to about 0.0005 x A / 931.494 u, which is a unit in the seventh figure, so the backwards direction is a check rather than a measurement.

Row 17 shows the same effect in the energy column. Iron-56 forwards gives a total of 492.3 MeV; the same iron backwards from 8.790 gives 492.2 MeV, because 8.790 multiplied by 56 is not quite the unrounded figure. Both are right to the precision a four-figure input carries, and the forward direction is the one to trust when you have a mass.

Rows 20 and 21 change nothing but the unit. 8790 keV per nucleon is the same entry as 8.790 MeV per nucleon and returns the identical strings, which is worth knowing because published tables quote this quantity in keV per nucleon. 1.408e-12 J per nucleon is the same figure again in joules, and it lands one digit away at 55.94 u because four figures in joules is a coarser entry than four figures in MeV.

Rows 22 to 32 sit outside the domain and are declined there rather than answered. A lone proton and a lone neutron both give exactly zero, there being no second nucleon for either one to hold on to, and zero is not a binding energy. No nucleons at all would mean dividing by a mass number of zero; a negative count is not a count; an atomic mass heavier than the free nucleons would mean a nucleus that weighs more than its parts.

The last three are the interesting ones. An atomic mass of 55.9 u on 56 nucleons implies 9.37 MeV per nucleon, which no nuclide reaches, so it is refused as a wrong input rather than published as a discovery. The same guard works in the other direction, declining a typed-in 9 MeV per nucleon, and anything that parses to infinity is refused explicitly — Infinity > 0 is true, so a sign test alone would publish it.

The no answer rows are what the widget actually does with those entries: the result area prints That combination has no valid solution. Check your inputs.

No chips are drawn beside that message, which is why the last two columns of those rows are dashes rather than figures. It is a refusal rather than a crash, and nothing on this page ever returns a NaN or an infinity.

Formula and symbol reference

One line does all the work. Weigh Z hydrogen atoms and N free neutrons, subtract the measured mass of the atom you actually have, and the difference is the mass defect: Z x M(1H) + N x m_n - M. Multiply it by 931.49410242 and it becomes the total binding energy in MeV; divide that by A = Z + N and it becomes the binding energy per nucleon.

The conversion factor is the whole of the mass-energy relation compressed into one number: one unified atomic mass unit is 931.49410242 MeV/c2, so a mass defect in u becomes an energy in MeV by multiplication alone. That is stated here once and derived nowhere — the E = mc2 calculator is the tool that owns the conversion in general, for any mass and any unit.

Why the hydrogen atom and not the proton? Because the mass you enter belongs to a neutral atom and carries Z electrons with it. Subtracting Z hydrogen atoms brings Z electrons of their own into the sum, where they cancel the ones your figure carried; the bare proton brings none, and the answer comes out low by about Z x 0.511 MeV. The alpha decay energy calculator takes atomic masses for exactly the same reason, and the electron binding energies that this cancellation ignores run from a few eV to tens of keV, far below anything this page prints.

The physics: why the curve has a top, and where it really is

The strong force that binds nucleons together is short-ranged: a nucleon feels only its immediate neighbours, not the whole nucleus. Add nucleons to a small nucleus and each one gains more neighbours, so the binding energy per nucleon climbs steeply — from 1.112 MeV in the deuteron to 7.976 in oxygen-16. Once a nucleus is big enough for most nucleons to be surrounded, there is nothing left to gain, and the figure flattens out near 8 MeV.

Past the iron region the headline starts to fall, and the proton box is the reason. Each extra proton pushes electrically on every other one, at unlimited range, so that cost climbs as the square of the proton count while the binding it has to beat grows only in step with the number of nucleons. Type a heavy nuclide in and the cost has already won: lead-208 returns 7.867 MeV per nucleon and plutonium-239 7.56.

Those two trends meeting is what makes a peak, and it is the reason both directions of nuclear energy release exist: light nuclei gain by joining, heavy ones gain by splitting, and both are moving towards the same plateau. The fission and fusion guide is where that consequence is worked out, and this page deliberately does none of the reaction arithmetic.

The top is a plateau rather than a single winner. Press the iron-56 and nickel-62 buttons one after the other and the headline moves from 8.79 to 8.795 — nickel-62 is higher, by 0.0042 MeV per nucleon, with iron-58 at 8.792 in between. The gap is real rather than noise: it is about 4.2 keV per nucleon against a combined uncertainty on the two published figures of 0.0117 keV.

And yet textbooks, diagrams and the interactive lab linked below all mark iron-56, with good reason. Divide each atomic mass by its own nucleon count and iron-56 comes out the lightest of every nuclide there is, which is the ranking that decides what stellar burning can reach, and there is far more iron-56 about than nickel-62. Read the two presets together and the peak is a region a few thousandths of an MeV deep, with first place going to whichever of those two questions you asked.

Where the nuclear binding energy calculator breaks down

The arithmetic here is exact to the last digit a double can hold, and the inputs are measured to parts in a hundred million. What goes wrong is everything the formula cannot see: which element you meant, whether the nuclide exists, and what it will do next.

A mass that does not belong to those counts is still answered

The only masses refused are the ones that break a physical bound: zero or negative, heavier than the free nucleons, or light enough to push past 8.8 MeV per nucleon. Anything in between gets an answer, so iron-56's mass typed against 30 protons and 26 neutrons returns 8.734 MeV per nucleon without complaint. Nothing on the page knows which nuclide you had in mind.

A single nucleon is declined rather than answered with zero

A lone proton or a lone neutron comes out at exactly zero by construction, because the mass you enter and the mass subtracted are the same number. That is the right physics — with no second nucleon there is nothing for the first one to hold on to — but zero is not a binding energy, so the tool declines it instead of publishing it. The smallest real case is the deuteron at 1.112 MeV per nucleon.

Fractional proton and neutron counts are not rejected

The two count boxes step in ones, but they will accept 26.5 and return an answer for a nuclide that cannot exist. Refusing non-integers would also refuse a perfectly good entry of 26.0, so the page answers and leaves the judgement to you. Only a negative count is refused outright.

Nothing here says whether the nuclide is stable

A positive binding energy means a nucleus is bound against falling apart into free nucleons, and almost every nuclide is. It says nothing about alpha or beta decay, which compare one nuclide with another rather than with its own pieces, and nothing about how long it will last. The half-life calculator is the tool for lifetimes, and plutonium-239 at 7.56 MeV per nucleon is firmly bound and still radioactive.

Four significant figures on every energy, and the fifth is a choice not to print

The published uncertainties on these binding energies run from 0.0002 to 0.0377 keV per nucleon, so the data itself would support five or six figures. Four is the ceiling this cluster sets so that every piece of it agrees, and it is still enough to separate nickel-62 from iron-56. The backwards direction is coarser again, because a four-figure energy pins a mass only to about seven figures.

Electron binding energies are neglected, deliberately

Using hydrogen atoms rather than protons cancels the electron masses but not the energy binding those electrons into their atoms, which runs from 13.6 eV in hydrogen to tens of keV in the heaviest elements. That is the standard convention and the error it leaves is thousands of times smaller than the figures printed here. The Bohr model guide and the Bohr model calculator are where atomic binding energies belong.

No reaction energies, in either direction

This page takes one nuclide apart into free nucleons. It does not compute what happens when two nuclei join or one splits, and the familiar figure of about 200 MeV per fission is a different quantity from any binding-energy difference, because the prompt release is only part of it and the fragments carry on decaying long afterwards. Deriving it here from these numbers would be wrong twice over, so it is not done.

No formula for the shape of the curve

The explanations above are qualitative on purpose. There is a standard fitted expression that reproduces the curve from the nucleon count, the surface area and the proton repulsion, but it is an approximation with adjustable coefficients, and this page would rather use a measured mass than a fit. Every figure here comes from a measurement.

The 8.8 ceiling is a sanity bound, not a law

Nothing known exceeds 8.8 MeV per nucleon, and the highest in the reference set behind this page is nickel-62 at 8.7946, so the guard catches wrong inputs rather than unusual nuclides. It is also a deliberately blunt instrument, and how blunt depends on where the nuclide sits. Helium-4 is 1.726 MeV per nucleon below the ceiling, so its mass would have to be 0.0074 u too light before the guard noticed; iron-56 is only 0.009644 below, and 0.00058 u is enough.

The calculator measures nothing — you supply the mass

Two counts and one measured mass go in, one subtraction and one multiplication happen, and the answer describes the nuclide you described. The preset names are shorthand for the three figures behind them, and the masses they carry are quoted from a published ground-state table rather than measured here.

Where the nuclear binding energy calculator is actually used

Settling a homework problem that gives you a mass table

The standard exercise hands you Z, N and an atomic mass and asks for the binding energy and the binding energy per nucleon, and that is exactly the forward direction. The working prints the mass defect as a separate line, which is usually the step that is marked. Helium-4 comes out at 28.30 MeV and 7.074 MeV per nucleon, the pair most textbooks use as their first example.

Checking a figure copied out of a published table

Ground-state tables quote binding energy per nucleon in keV per nucleon, so the unit menu takes that entry directly. Put the published figure in, solve for the mass, and compare it with the mass the same table gives: agreement in the first six figures means you read the right row. Disagreement in the second figure usually means the two rows were different isotopes.

Seeing why the curve has a top at all

Press the five presets in order and watch the two readings separate. The total binding energy climbs the whole way, from 28.30 MeV to 1784 MeV, while the per-nucleon figure rises to the iron plateau and then falls back to 7.591. The fission and fusion guide is where the consequence of that shape is worked through, and the interactive lab linked at the foot of this page draws the curve itself.

Telling iron from nickel with a number instead of an argument

The claim that iron-56 is the most tightly bound nuclide is almost right and worth testing. Two clicks give 8.79 and 8.795 MeV per nucleon, and the ceiling chip puts the gap at 0.0042 MeV per nucleon. That is small enough to explain why diagrams label iron and large enough to be certain about.

Separating two isotopes of the same element

This is the thing the curve on its own cannot do, because the curve is drawn against the mass number. Iron-56 and iron-58 are 8.790 and 8.792 MeV per nucleon, two thousandths apart, and the only way to get that is from their separate measured masses. The same applies to uranium-235 at 7.591 against uranium-238 at 7.57.

Sanity-checking a mass before using it somewhere else

A mass that produces a sensible binding energy per nucleon is probably the mass you meant, and one that produces a refusal is not. Because the result is a subtraction of two nearly equal numbers, a mass wrong in its fourth decimal place already moves the fourth figure of the answer. The beta decay energy calculator and the alpha decay tool depend on the same tables, so the check is worth doing before either.

Looking at the quantity rather than reading about it

The physics lab library has the interactive versions of all of this, including the binding-energy curve itself with the iron peak marked on it. This page is the arithmetic for one nuclide at a time, to the precision a measured mass allows.

Where to go next

For the curve set out quantitatively, worked problems of rising difficulty and the misconceptions in full, read the nuclear binding energy guide, which this tool is the arithmetic half of. The fission and fusion guide is what the shape of the curve is good for, and the guide to E = mc2 is the mass-energy conversion this page applies once.

Three more posts pick up what this page leaves out. The alpha decay guide is why four nucleons leave together rather than one at a time, the atomic structure guide is what the Z electrons in that atomic mass are doing, and the Bohr model guide is the 13.6 eV kind of binding energy rather than the 8.790 MeV kind.

Five tools sit next to this one. The E = mc2 calculator owns the mass-energy conversion; the alpha decay energy calculator and the beta decay energy calculator take the same atomic-mass tables and compute decay energies instead; the half-life calculator and the decay chain calculator answer how long rather than how much.

The full physics lab library and calculator index are open too.