Alpha Scattering

Alpha particles fired at gold leaf nearly all go straight through; a few turn sharply, some coming back. Counted angle by angle, they put the atom’s charge in a tiny centre.

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Chapters · 13

01The Atom as Pictured
02One in Eight Thousand
03Counting Flashes in the Dark
04A Single Encounter
05How Close It Gets
06A Count for Every Angle
07The Box of 1913
08Hundreds of Atoms Thick
09Counted Against Computed
10The Charge at the Centre
11What Came Next
12The Box Taken Apart
13Turn the Microscope

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01

The Atom as Pictured

In 1904 J. J. Thomson pictured the atom as a sphere of uniform positive electricity with negative corpuscles, today’s electrons, inside it. The positive charge spread through the whole atom, so nowhere was its field strong.

A fast, heavy alpha particle crossing such an atom would be pushed aside only slightly, and crossing a foil it would add up many small turns. In 1910 Geiger found the most probable turn through one thin gold foil to be only about 0°10′, and through a stack of gold equal to 1 cm of air about 2.1°.

Added up from many small turns like these, the chance of an alpha turning through more than 90° is too small to matter.

Thomson 1904 · Geiger 1910: one thin gold foil, a most probable turn of about 0°10′

02

One in Eight Thousand

In 1909, in Manchester, Geiger and Marsden, then an undergraduate, let alpha particles fall on metal plates. The source was a glass tube drawn to a cone and filled with radium emanation, closed by a mica window; the zinc sulphide screen sat behind a lead plate, out of reach of the direct alphas.

With a plate in place, flashes appeared at once: some alphas came out again on the side they had gone in. To find the fraction they used radium C on a small plate as the source and a platinum plate of about 1 cm²; three determinations all gave about 1 in 8,⁠000 of the incident alphas “reflected” in this way. Under the same conditions a gold plate gave 67 flashes a minute, aluminium 3.4.

It was not a surface effect: about half came from a layer equal to 2 mm of air, and some alphas turned through 90° within 6 × 10⁻⁵ cm of gold.

1909 · platinum · about 1 in 8,⁠000 · gold about twenty times aluminium

03

Counting Flashes in the Dark

An alpha particle striking zinc sulphide makes a tiny flash. In 1908 Rutherford and Geiger checked the flashes against an electrical count and found ratios of 0.96 to 0.99: on a good screen, one alpha is one flash.

The observer sat in a dark room and looked through a low-⁠power microscope at about one square millimetre of the screen, counting each flash. The flashes were faint; in 1908 they counted at night and never for more than two minutes at a stretch, after which the eye tired.

In 1913 they wrote that more than 90 a minute, or fewer than about 5, could not be counted with certainty. That set of experiments counted over 100,⁠000 flashes.

One alpha, one flash · 5 to 90 a minute to count with certainty · over 100,⁠000 counted in 1913

04

A Single Encounter

In 1911 Rutherford proposed another atom: a charge at the centre, packed into a very small volume, surrounded by an equal and opposite charge spread through a sphere the size of the atom. Near the centre the field is intense, and an alpha can turn through a large angle in one go.

Its path is a hyperbola, and the closer it passes, the more it turns: with p the distance of its first line from the centre and b the closest approach of an alpha fired straight at it, cot(φ/2) = 2p/b. At p = 2b it turns 28°; at p = b, 53°; at half of b, 90°; at an eighth, 152°.

His results were the same whether the central charge was positive or negative; he made it positive “for convenience”. The paper calls it the central charge; by 1912 he called it the nucleus.

Rutherford 1911 · a single encounter · a hyperbolic path

05

How Close It Gets

An alpha fired straight at the centre stops at the distance b and is pushed back. Taking a central charge of 100 electron charges and an alpha at 2.09 × 10⁹ cm a second, Rutherford found b to be about 3.4 × 10⁻¹² cm; the atom’s radius, about 10⁻⁸ cm, is some 3,⁠000 times larger.

The central charge can take up no more room than b: b is an upper bound on its size, not its radius.

In today’s numbers, an alpha from radium C (polonium-⁠214) carries 7.69 MeV and gets no closer than about 30 fm (3 × 10⁻¹³ cm) to a gold nucleus head on; the nucleus is about 7 fm in radius, so the alpha is turned back before it touches it. A gold atom’s radius is about 20,⁠000 times its nucleus’s.

Rutherford 1911: b ≈ 3.4 × 10⁻¹² cm (for 100 electron charges) · today: about 30 fm, the gold nucleus about 7 fm

06

A Count for Every Angle

Rutherford worked out the number of alphas falling on unit area of the screen as y = ntb²Q cosec⁴(φ/2) / 16r², with n the atoms in unit volume, t the foil’s thickness, Q the alphas falling on the foil and r the screen’s distance from it.

So the flashes vary with the angle as 1/sin⁴(φ/2); in proportion to the thickness; as the square of the central charge; and inversely as the fourth power of the alphas’ speed.

1/sin⁴(φ/2) changes very fast: from 150° to 15° it grows about 3,⁠000 times. A source that gives a countable rate at 150° gives far too many flashes to count at 15°.

n: atoms in unit volume · t: foil thickness · Q: alphas falling on the foil · r: screen to foil

07

The Box of 1913

To test these predictions Geiger and Marsden built this apparatus: a strong cylindrical metal box holding the source, the scattering foil and a microscope with the zinc sulphide screen fixed to its end. The box was fastened to a graduated circular platform that turned on a conical airtight joint.

Turning the platform carried the box and microscope round while the foil and source stayed put on a tube fastened to the stand. A ground-⁠glass plate closed the box, and the air was pumped out through the tube, so that air would not absorb and scatter the alphas.

The source was a thin-⁠walled glass tube about 1 mm across, holding over 100 millicuries of radium emanation, enclosed in a lead box against the glow the beta rays raised on the screen; an aluminium diaphragm let a pencil of alphas fall square on the foil. In some runs the source was 2.5 cm from the foil and the screen moved on a circle of 1.6 cm radius, from 5° to 150°. The box was lined with paper to cut down alphas scattered from its walls.

Source to foil 2.5 cm · the screen on a circle of 1.6 cm · 5° to 150°

08

Hundreds of Atoms Thick

Gold can be beaten into very thin, even leaf. The gold foil of 1913’s absolute count stopped alphas like 1 mm of air and was 2.1 × 10⁻⁵ cm thick: from gold’s crystal spacing, about eight or nine hundred layers of atoms.

Behind such a foil, a square millimetre of screen 1 cm away at 45° caught only 3.7 × 10⁻⁷ of the incident alphas.

Besides gold they used foils of silver, tin, copper, aluminium and platinum, and sheets of paraffin wax for carbon, to compare the scattering by different atomic weights.

1 mm air equivalent · 2.1 × 10⁻⁵ cm · about 800 to 900 layers of atoms

09

Counted Against Computed

Table II of the 1913 paper gives the flashes N counted at each angle from 150° to 15° for silver and gold, corrected for the decay of the emanation, the natural effect and a change of diaphragm. Gold runs from 33.1 to 132,⁠000, silver from 22.2 to 105,⁠400.

N times sin⁴(φ/2) stays nearly constant: 27.5 to 39.6 for gold, 18.4 to 30.6 for silver, a little larger at small angles. Set against so large a variation, they judged the deviations within the experimental error.

Another run went down to 5°. Their conclusion: from 5° to 150° the numbers varied about 250,⁠000 times, in good agreement with the theory.

Table II, 1913 · N × sin⁴(φ/2): gold 27.5⁠–⁠39.6, silver 18.4⁠–⁠30.6

10

The Charge at the Centre

The same paper tested three more laws: scattering in proportion to the foil’s thickness (one to nine gold foils); per atom, roughly as the square of the atomic weight (carbon to gold); and inversely as the fourth power of the speed (over a tenfold change in the count, the alphas slowed by sheets of mica).

From the fraction 3.7 × 10⁻⁷ they worked out the central charge of the gold atom as about half its atomic weight in elementary charges, correct to perhaps 20 %. Rutherford in 1911 had estimated about 100.

Today that number is the atomic number: 79 for gold. Moseley fixed it from X-⁠ray spectra in 1914; in 1920 James Chadwick measured the nuclear charges of platinum, silver and copper directly by alpha scattering as 77.4, 46.3 and 29.3, against atomic numbers 78, 47 and 29, to within 1 to 1.5 %.

Scattering ∝ thickness · per atom ∝ atomic weight² · ∝ 1/v⁴ · a central charge of about half the atomic weight

11

What Came Next

In 1913 Antonius van den Broek proposed that the charge of the nucleus equals an element’s place in the periodic table. In 1914 Henry Moseley proved it from X-⁠ray spectra, for every element from aluminium to gold.

In 1919 Rutherford broke up nitrogen nuclei with alpha particles; in 1932 Chadwick found the neutron. Rutherford’s Nobel Prize, in chemistry in 1908, was for the disintegration of the elements, three years before the nucleus.

Late in life he recalled Geiger telling him that alphas were coming back: it was almost as if you fired a 15-⁠inch shell at a piece of tissue paper and it came back and hit you. The words come from a lecture of about 1936, printed in 1938 as rewritten by another hand.

1912 nucleus · 1913 van den Broek · 1914 Moseley · 1919 nitrogen · 1932 neutron

12

The Box Taken Apart

The apparatus comes apart into five pieces: the box, the microscope with the zinc sulphide screen on its end, the source in its lead box, the gold foil in its frame, and the graduated platform.

The part under the pointer lifts out alone and is named; left alone, all five carry their names.

Box · microscope · source · gold foil · platform

13

Turn the Microscope

This is the page’s model: source, foil and screen at fixed distances, the flashes a minute from Rutherford’s formula, gold at 150° set to 6 a minute; silver of the same thickness scatters about 0.35 times as much.

The lever turns the microscope. Each press of “Count a minute” counts a fresh minute (sped up) and marks the count on the chart; the pale red band is 5 to 90 a minute, what an eye could count with certainty in 1913. The grey dots are the counts of 1913’s Table II, drawn to the page’s scale.

The page’s model · gold at 150°: 6 a minute · silver about 0.35 times gold

Try thisWith gold, turn the microscope from 150° to 60°: from how many flashes a minute to how many? With silver at 60°, can they still be counted with certainty?

What the Model Leaves Out

The apparatus is this page’s model after fig. 1 of the 1913 paper. The paper gives only two distances, source to foil 2.5 cm and the screen’s circle 1.6 cm; the box’s size and material and the look of the microscope, scale and stand follow the drawing’s proportions, not a surviving instrument, and no original or replica could be found in a museum. The box is cut open on the side facing the camera. The red alpha tracks and the flashes on the screen are a drawing: real alphas are invisible and travel at about 6 % of the speed of light, and neither their number nor their speed is to scale, with far more large turns drawn than happen. The magnified foil is a drawing too: atoms as soft clouds, the nucleus over two thousand times too large, since to scale it is about a twenty-⁠thousandth of the atom; Thomson’s atoms are his picture of 1904. The lab computes the formula only: no slowing of the alphas in the foil, no screen that misses about 15 % of them, no stray alphas from the diaphragm’s edge.

Easily Misread

“About 1 in 8,⁠000” is 1909’s share of alphas coming back out of a thick platinum plate; “about 1 in 20,⁠000” is Rutherford’s 1911 account of one thin gold foil, and that one speaks of an average turn of 90°. Neither means bouncing straight back. Geiger and Marsden did the experiments; Rutherford gave the theory, supplied the emanation and communicated the papers. The 1909 measurements were made in air; the evacuated box came in 1913, which reached 150°, the lead box blocking 150° to 180°. The 1911 paper has no word “nucleus”, does not decide that the central charge is positive, and does not draw electrons as planets round it. 1913 found a central charge of about half the atomic weight; 79 comes from the atomic number. The flashes were counted by eye through a microscope, not with a Geiger counter, and the screens of 1913 showed only about 85 % of the alphas. “Plum pudding” is not Thomson’s phrase; he wrote of corpuscles.

Common questions · 5

What did Geiger and Marsden find in 1909?: With a plate in place, flashes appeared at once: some alphas came out again on the side they had gone in.
02One in Eight Thousand
How were alpha particles counted then?: An alpha particle striking zinc sulphide makes a tiny flash.
03Counting Flashes in the Dark
What atom did Rutherford propose in 1911?: In 1911 Rutherford proposed another atom: a charge at the centre, packed into a very small volume, surrounded by an equal and opposite charge spread through a sphere the size of the atom.
04A Single Encounter
What does the closest approach say about the nucleus’s size?: The central charge can take up no more room than b: b is an upper bound on its size, not its radius.
05How Close It Gets
Did the counts of 1913 fit Rutherford’s formula?: N times sin⁴(φ/2) stays nearly constant: 27.5 to 39.6 for gold, 18.4 to 30.6 for silver, a little larger at small angles.
09Counted Against Computed

Sources

Data

The apparatus, the method and the counts after H. Geiger and E. Marsden (Proc. R. Soc. A 82, 495, 1909; Phil. Mag. 25, 604, 1913), H. Geiger (Proc. R. Soc. A 83, 492, 1910) and E. Rutherford and H. Geiger (Proc. R. Soc. A 81, 141, 1908); the theory after E. Rutherford (Phil. Mag. 21, 669, 1911). The page’s curves and the lab compute Rutherford’s formula against Geiger and Marsden’s Table II (docs/plates/rutherford/computed-⁠scatter.py); the closest approach and the sizes use the LNHB energy of the polonium-⁠214 alpha, CODATA 2022 constants and the IAEA charge radius of gold (computed-⁠theory.py). The apparatus on the board is this page’s own model after fig. 1 of the 1913 paper, not a copy of a surviving instrument. Material presets and the model kit as on the site’s other model plates (public/looks/CREDITS.md).

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