Rutherford Scattering Calculator

What Is the Rutherford Scattering Calculator?

How to use the Rutherford scattering calculator

1. Choose the unknown. The Solve for menu opens on Scattering angle, and the other two choices are Impact parameter and Alpha energy.
2. Type the alpha energy. It takes MeV and opens on 5.0 MeV. The preset menu also holds 4.85, 8.29 and 20 MeV, and the box accepts 0.01 to 100 MeV.
3. Set the target atomic number. Z is a plain count of protons from 1 to 118, with presets for aluminium, copper, silver, gold, lead and uranium. It is the one box that is an input in every mode.
4. Type the impact parameter. It takes fm, pm or m and opens on 20 fm. This is the aim — the perpendicular distance from the nucleus to the alpha's original line — and zero means dead centre.
5. Read the answer. The headline always carries six significant figures, in the unit that box was showing. Set the unit first, point Solve for at the target second and type your figures last — switching the target wipes every box back to its default and only the unit menus survive it.
6. Read the closest-approach chip. It is how close the alpha really gets on this aim, always further out than a head-on hit would reach, and it equals the head-on figure only at dead centre.
7. Read the head-on chip as a bound. It is printed with the energy it belongs to on purpose: it is an upper bound on the nuclear radius at that energy and not a nuclear size.
8. Read the count-rate chip. It is the rate for this angle relative to 90 degrees, and it depends on the angle alone — the energy and the target cancel out of it completely.
9. Try the refusal. Point Solve for at the alpha energy and set the angle to 180 degrees: the page declines and explains itself underneath the widget rather than printing a number.
10. Open Show working. The steps give the relation, the boxes read in base units, the Rutherford length, the closest approach against the head-on bound, the count rate, the alpha speed, the assumptions and every unit conversion.

This page exists for the one step that turns an energy, a charge and an aim into an angle. The nuclear model guide is the prose half of it, where the argument is made and the problems are worked. How the experiment came to be done at all, and what the discovery narrative looked like, belongs to the atomic structure guide instead.

Three mistakes account for most wrong answers here. The first is dropping the factor of two and using the head-on distance where the Rutherford length belongs, which doubles every aim; the second is reading the head-on chip as a nuclear radius rather than as a bound on one. The third is expecting the energy mode to answer a backscatter, which it cannot, because nothing about the energy survives in a straight-back hit.

Worked example: change one thing at a time

The table starts at the defaults, moves the unknown through all three modes, then loads the five real cases, inverts four measured angles, walks every unit menu, and finally reaches the entries the calculator declines. Every answer and chip cell was read out of the running widget rather than worked out by hand. Where a cell and the tool ever part company, believe the tool.

Rows 1 to 3 are one state — a 5 MeV alpha on gold aimed 20 fm out — with the Solve for menu in each of its three positions. All three chip columns are identical down all three rows, because the chips describe the state rather than the mode. That is not free: they are worked out from the answer substituted back, so the mode in which the energy is the unknown still has an energy to take a head-on distance from.

Rows 4 to 8 are the five real cases, and rows 4 and 8 are the cleanest pair on the page. Both are dead centre at gold, and four times the energy tightens the head-on bound from 45.50 fm to 11.38 fm — a factor of exactly four, because that distance is inversely proportional to the energy. Row 7 is the other exact scaling: the same energy and aim at aluminium gives 7.49 fm, which is 45.50 fm scaled by 13 over 79.

Rows 10 to 14 invert measured angles back into aims, and row 10 is the check worth memorising. At 90 degrees the tangent is 1, so the aim is the Rutherford length exactly: 22.7514 fm, which is half of 45.50 fm. Row 14 is the other exact case, where a full backscatter inverts to 0 fm rather than to the 1.39e-15 fm the cotangent would actually return.

Rows 15 and 16 run the energy mode on two of the real cases and both come back to 5 MeV, which is the round trip of rows 7 and 6. Rows 17 to 20 change nothing but the units: 20 fm typed as 0.02 pm or 2e-14 m gives the identical 97.3648 deg, and the same angle reads 1.69934 rad if the angle box was showing radians.

Rows 21 and 22 are the refusal worth understanding, and the page explains itself underneath the widget in both cases. Rows 23 to 28 are the ordinary refusals: a zero angle, a negative aim, a zero energy, an atomic number past 118, an atomic number of zero, and an angle past 180 degrees. In every one of them the answer is words rather than an infinity or a NaN, and the Coulomb's law calculator is the place to go if what you actually wanted was the force at a given separation rather than a deflection.

Formula and symbol reference

One relation does all the work: theta = 2 arctan(alpha / b), with alpha = k z Z e² / (2E) and z = 2. Everything else on this page is that relation rearranged, or a quantity derived from the state it describes.

The constant that makes it a one-line calculation is k e² = 1.439965 MeV·fm, which is why an energy in MeV and a distance in femtometres need no conversion factors at all. The head-on distance is d = k z Z e² / E, and the Rutherford length is half of it. That halving is the step most often dropped, and the electric potential guide is where the energy-to-distance step it comes from is set out.

Reading the relation backwards gives b = alpha / tan(theta / 2) and E = k z Z e² / (2 b tan(theta / 2)). The first always has an answer for an angle between 0 and 180 degrees; the second has none at 180 degrees exactly, which is the single refusal on the page. Getting d at all is an energy-conservation argument, set out in the conservation of energy guide, and the kinetic energy that goes into it is the kinetic energy guide.

The physics: why one ratio decides everything

The alpha and the nucleus repel each other with an inverse-square force, so the path is a hyperbola with the nucleus at the outer focus. All of the geometry of that hyperbola is set by a single length, and that length is alpha = k z Z e² / (2E). Once you have it, the angle follows from the aim alone.

That is why the energy and the aim are interchangeable here. Doubling the energy halves the Rutherford length, and halving the Rutherford length does exactly what doubling the aim does, so the two moves give the identical angle. Doubling the target atomic number is the opposite move, because the length is proportional to Z.

The head-on distance is the quantity the whole experiment was really about. Fire the alpha straight in and all of its kinetic energy turns into electrostatic potential energy at the turning point, giving d = k z Z e² / E — and the fact that the alpha turned round there under Coulomb repulsion alone means every scrap of the positive charge was inside it. The Coulomb's law simulator is where that force law can be pushed around on its own.

Two things follow from d that are easy to confuse. It is an upper bound on the nuclear radius, and it is a bound only for the energy used: at 5 MeV on gold it is 45.50 fm, at 10 MeV it is 22.75 fm, and at 20 MeV it is 11.38 fm. The real gold nucleus has a radius of about 7.3 fm, so even the tightest of those three is around 1.6 times too loose.

The count rate is the other half of the story, and it behaves far more violently than the angle does. It goes as one over the fourth power of the sine of half the angle, so a 10 degree deflection is 4332.7 times as common as a 90 degree one. Normalising to 90 degrees removes the energy and the target from the expression entirely, which is why a single chip covers every state this calculator can reach.

Where the point-nucleus model breaks down

The arithmetic here is exact to the last digit a double can hold. What goes wrong is physical: a nucleus treated as a point that does not move, a target stripped of its electrons, and a force law with nothing in it but electrostatics.

The nucleus is a point, and that is the assumption being tested

Every distance on this page is computed as though all the target's positive charge sat at one mathematical point. That is deliberate rather than careless: the whole argument is about how concentrated the charge is, and the model is the thing under test. It stops being usable once the alpha would reach the nuclear surface, which for gold is at about 32.58 MeV.

There is no nuclear force in the relation at all

Pure Coulomb repulsion is the only thing bending the path. Once an alpha gets close enough to feel the strong interaction the deflections stop following this law, and nothing on this page warns you about that except the energy you typed. The companion lab stops its energy ladder at 20 MeV for exactly this reason, well below the 32.58 MeV where the head-on distance reaches the empirical radius of around 6.98 fm.

The head-on distance is a bound, and only for one energy

This is the single most misread figure on the page, which is why the chip prints the energy beside it. It is not a nuclear radius and it is not a nuclear diameter. A higher energy pushes closer and gives a tighter bound, so quoting one without its energy says nothing at all; at 1 MeV on gold the bound is 227.51 fm, five times looser than the 5 MeV figure.

The target does not recoil

The nucleus is treated as infinitely heavy and perfectly fixed, which is a good approximation for gold and a worse one the lighter the target gets. Aluminium is the weakest case the preset menu offers, and the angle there is the one most affected. Nothing on this page corrects for it, and no figure for the size of that correction is printed or estimated.

No electrons, no screening

The target's electrons are simply absent from the model. That is tolerable because the large-angle events this relation is interesting for happen well inside the electron cloud, where the nucleus dominates. It is not tolerable at very small angles, where an alpha passing far out is being deflected by a screened charge rather than a bare one.

One encounter, not a foil

Everything here describes a single alpha meeting a single nucleus once. A real foil is thousands of atoms thick, so small deflections accumulate and the single-encounter picture stops holding at small angles. Counting what a real foil does is a different calculation and no fraction for it is published here.

Relativity is left out, and the cost is in the speed only

The alpha speed chip uses the non-relativistic expression, which is 0.40 per cent high at 20 MeV and better than that everywhere below. No angle on this page is affected by it, because the angle depends only on the ratio of the Rutherford length to the aim, and that length is built from the kinetic energy rather than from the speed. The speeds involved run from a few per cent of light speed to about a tenth of it.

The paths are classical

The alpha is treated as a particle on a definite trajectory with a definite aim, which is not what a quantum particle does. The relation survives that treatment better than it has any right to, but this page makes no claim about why, and it does not present itself as a quantum calculation. Nothing from the quantum account of the atom is used here or needed here.

A refusal in the energy mode is an answer, not a fault

Ask for the energy behind a 180 degree deflection and there genuinely is not one, because every energy produces that angle at dead centre. The same goes for an aim of zero, which forces 180 degrees. The page declines, names the reason underneath the widget, and never prints an infinity or a NaN — the formula at that angle actually returns a finite figure near 3.48e-16 MeV, which would be far worse than refusing.

Rounding the angle moves the energy answer

The energy mode is the sensitive direction, and the page measures that rather than hiding it. The defaults carry the angle to eight figures as 97.364841 deg precisely so that all three modes close on the same state; type the displayed 97.3648 deg instead and the energy comes back as 5.00000357 MeV, which still prints as 5 but is not the figure the other boxes hold. Keep the digits you have, especially near 180 degrees.

The calculator measures nothing — you supply the numbers

Three quantities go in, one relation is used, and the answer describes the alpha and the target you described. The case names are shorthand for the four figures behind them and are not claims about any particular experiment, apparatus or sample.

Where this relation is actually used

Turning a measured angle into a distance of closest approach

Point Solve for at the impact parameter and the question inverts into the one an experiment can actually answer: you counted alphas at an angle, and you want to know how close they came. At 150 degrees on gold at 5 MeV the aim was 6.09623 fm and the alpha reached 46.31 fm, barely outside the head-on bound. That gap between the aim and the approach is the thing a reader usually has backwards.

Putting an upper bound on a nuclear size from one number

Load aim dead centre at gold and the head-on chip gives 45.50 fm at 5 MeV. That alone establishes that all the positive charge in a gold atom sits inside a region more than three thousand times smaller than the atom. It does not establish the radius, and the honest way to quote it is as a bound with its energy attached.

Deciding which energy a measurement needs

Because the head-on distance is inversely proportional to the energy, the bound tightens in step with it: 227.51 fm at 1 MeV, 45.50 fm at 5 MeV, 11.38 fm at 20 MeV. Typing energies into the first mode is the fastest way to see how much energy a given bound would cost. Past about 32.58 MeV on gold the exercise stops meaning anything, because the alpha arrives at the nuclear surface.

Comparing targets at a fixed energy

Switch the Z preset from gold to aluminium at the same energy and aim and the angle drops from 132.546 deg to 41.0509 deg, while the head-on bound falls from 45.50 fm to 7.49 fm. The second ratio is exactly 13 over 79, because that distance is directly proportional to the target charge. A light nucleus is a poor backscatterer and this is the arithmetic of why.

Judging whether a reported deflection is plausible

Point Solve for at the alpha energy and the page becomes a consistency test. An angle and an aim that came from different states will return an energy that does not match the one claimed, and an exact backscatter will return nothing at all. That last case is the useful one: an alpha source of known energy aimed dead centre tells you about the charge it met, never about its own energy.

Seeing why the large-angle counts were so hard to get

The count-rate chip at 10 degrees reads 4332.67 x the 90 deg rate and at 150 degrees it reads 0.29. Those two figures are a factor of 15086.6 apart, which is why finding the rare hard deflections was a matter of patience rather than cleverness. The ratio is the same at every energy and every target, so it is a property of the law and not of any apparatus.

Recognising the same shape of problem somewhere else

An inverse-square force, a hyperbolic path and one length that sets the whole geometry is a pattern rather than a fact about nuclei. The electric potential calculator is the nearest relative, because the head-on distance here is nothing but a potential-energy balance solved for a separation. Its subject is a voltage rather than a deflection, but the algebra of the turning point is the same algebra.

Where to go next

For the argument this arithmetic supports, the problems worked through and what the counting actually proved, read the nuclear model guide, which this tool is the calculating half of. The atomic structure guide is the wider story of how the picture of the atom was put together, and the Bohr model guide is what came next once the nucleus was established.

Four more posts pick up the pieces this page uses but does not explain. The Coulomb's law guide is the force law underneath everything here, the electric potential guide is where the energy-to-distance step comes from, the conservation of energy guide is the step that gives the head-on distance, and the kinetic energy guide is the energy the alpha arrives with. Where the alpha itself comes from is the alpha decay guide, and the types of radiation guide is where it sits among the others.

Three tools sit next to this one. The Coulomb's law calculator gives the force at a separation, the electric potential calculator gives the potential that sets the turning point, and the alpha decay energy calculator gives the energy a source emits. The Coulomb's law simulator and the alpha decay simulator are the interactive versions of the first and the last.

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