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Rutherford’s Gold foil scattering experiment

Introduction:

In physics, Rutherford scattering is a phenomenon that was explained by Ernest

Rutherford in 1909, and led to the development of the Rutherford model (planetary model) of the

atom, and eventually to the Bohr model. The initial discovery was made by Hans Geiger and

Ernest Marsden in 1909 when they performed the gold foil experiment under the direction of

Rutherford, in which they fired a beam of alpha particles (helium nuclei) at layers of gold leafs

only a few atoms thick. The intriguing result showed that around 1 in 8000 alpha particles were

deflected by very large angles (over 90°),

while the rest passed straight through with

little or no deflection. From this,

Rutherford concluded that the majority of

the mass was concentrated in a minute,

positively charged region (the nucleus)

surrounded by electrons. When a (positive)

alpha particle approached sufficiently

close to the nucleus, it was repelled strongly enough to rebound at high angles. The small size of

the nucleus explained the small number of alpha particles that were repelled in this way.

Schematic of the Rutherford scattering is shown above. Most particles travel straight through

the gold foil but about 1 in 8000 was turned through a large angle.

Principle: The relationship between the angle of scattering and the rate of scattering of -

particles by gold foil is examined with a semiconductor detector. This detector has high detection

probability for observing -particles so that the number of pulses agrees exactly with the number

of -particles striking the detector.

Alpha source

very few

some

Many particles

gold foil

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Objective:

1. The particle rates are measured at different angles of scattering between about 20 - 90 degrees.

The measurements are compared with the particle rates calculated by means of the Rutherford

theory for the measurement geometry used.

2. The particle rates are measured in the case of scattering by Aluminum and Gold with identical

angles of scattering in each case. The ratio of the two particle rates is compared with the particle

rate calculated from Rutherford’s scattering equation.

Equipments:

Annular diaphragm w. gold and aluminum foil U-magnet, large

Americium-241 source, 370 kBq Alpha detector

Pre-amplifier, alpha detector Impulse height analyser

Digital counter, Oscilloscope, 30 MHz, 2 channels

Hand held mano-/ barometer and Pressure sensor Diaphragm pump, two-stage

Rubber tubing, vacuum, i.d. 6 mm Hose connector Adapter,

BNC-socket/4 mm plug pair Screened cable, BNC, l = 750 mm

Cable connector BNC, 50 Ω

Figure 2: Experimental set-up for Rutherford experiment.

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Theory and evaluation:

Rutherford’s theory on the scattering of -particles is based on the following assumptions:

(a) The mass of the atom is concentrated in a nucleus which can be regarded as a single point

relative to the dimensions of the atom as a whole.

(b) This nucleus carries a positive charge and is therefore surrounded by a coulomb field which

decreases in proportion to 1/ r2.

(c) -particles are repelled in this coulomb field as a result of their likewise positive charge and

subsequently follow hyperbolic paths.

The particle rate n( ) of particles scattered

through the angle in a solid angle d is

governed under the above assumptions by

Rutherford’s scattering equation for -particles:

Figure 3: set-up for Rutherford experiment.

Where, n = particle rate in the foil, N = atomic concentration in the foil

dF = foil thickness Z = nuclear charge of the scattering atoms

E = energy of the -particles e = elementary charge = 1.6021 · 10-19

As

0= electric constant = 8.8524 · 10-12

As/Vm

The solid angle d covered during particle counting is determined by the distance r2 between

foil and detector and by the surface area of the detector AD (Fig. 4):

The scattering equation becomes for the set-up of Fig.3

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With the modified diagram below the r2 will be replaces by r/2. The foil is always arranged

centrally between the source and detector, so that the scattering angle is given by following

expression:

and the counting rate goes as

Fig. 4: Below is the sketch of the experimental set-up of the Rutherford experiment used in this

experiment.

The activity of the -emitter used is 370 kBq. When using an arrangement as shown in Fig. 3

such low counting rates are obtained at this source strength, that the measurements necessary to

verification of the scattering equation cannot be performed during the time available for the

experiment.

An arrangement as shown in Fig. 4 is therefore used, corresponding to a slightly modified

version of a proposal put forward by Chadwick in 1920. The foil forming an annular diaphragm

is, like the source, arranged to be movable. The scattering angle is varied by shifting these two

components.

Set-Up and Procedure:

A. Setting of the experiment

1. Set the experiment as shown in the layout below.

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2. The -detector, the annular diaphragm with gold foil and the 241

Am source screwed into the

movable source holder are placed in the container.

3. The pre-amplifier is set for -particles, Bias switch is set to internal and “–”. The output is

connected with the “in” of the pulse height analyzer.

4. The counter is connected via a BNC-cable and an adapter BNC-socket/4 mm plug to the

potential free input of the counter.

5. The foils are moved with the help of the magnet. For further information please refer to the

operating instructions. The pulse height analyzer remains initially switched off (no voltage

applied to the -detector).

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B. Preparation

6. Evacuate the container to about 10-12 hPa (10-12 mbar). Close the vacuum hose with a hose

clamp. Switch off the pump (the vacuum is sufficient when the pressure has fallen clearly

below 2 hPa).

7. Switch on the pulse height analyzer. Select the “Integral” window on the pulse height

analyzer.

8. Release the “Auto/Man” key (for “manual” operation).

9. Set to a voltage of around 0.5 V with the “base” adjusting knob (0.50 scale divisions) in

order to suppress the noise.

C. Measurements

10. The counting rate is determined for different distances l between source and detector. The

distance l may in this case be varied between 22 cm and 4 cm, corresponding to a

scattering angle range of about 20° to 90°. Tabulate the data in Table 2.

11. The measuring period selected should be sufficient to achieve a pulse count of 100.

12. The pulse height analyzer is then switched off, the container ventilated and opened the

gold foil annular diaphragm replaced by the aluminum foil diaphragm.

13. The procedure (# 6-9) described above under “Preparation” is then followed and the

particle rate determined at various scattering angles. Tabulate the data in table 2.

Observations:

A. A typical example of the measurement of scattering in gold at different angles is summarized

in the table below.

Z(gold) = 79

N(gold) = 5.9 1028

m-3

0 = 8.854 10-12

As/Vm

e = 1.602 10-19

As

E = 3 MeV = 4.8 10-13

J

a = 20.16 mm = 0.02016 m

Q = 370 kBq = 2.22 107 min

-1

AF = 6 cm2 = 6 10-4

m2

dF = 1.5 μm = 1.5 10-6

m

AD = 15 mm2 = 15 10

-6 m

2 (detector 09099.00)

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Table 1: Forward count rate measurement for Au foil:

Plot the measured and theoretical counting rates as a function of scattering angle (as

below). Within the statistical error we received a linear dependence in accordance with the

Rutherford scattering equation. Similar table can be made for Al foil.

B. Calculate the value of dF.Z2 for two scattering foils and tabulate as follows and plot as shown

below. It is evident that the measurement points lie, within the limits of statistical error,

l (cm) (degree) n( ) n( ) (min-1

)

measured

n( ) (min-1

)

theory

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8

10

12

14

16

18

20

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exactly on a line passing through the origin of the coordinates. The error because of low

number of incidents is as high as / n(l) > 20%. Still the measurement result is in the

right order of magnitude.

Figure 5: counting rate for gold and aluminum as a function of dF.Z2.

Precautions:

1. Counting errors may occur if serious sources of interference are present during the

measurement (spark gaps etc.); the possibility of such errors is excluded in the example given

here. If no such sources of interference are present and the zero effect of the detector has

been checked (it should not substantially exceed 0.1/min, if the detector is operating

correctly), it may be assumed that the counted pulses correspond exactly to the number of -

particles scattered in the barrier layer of the detector.

Foil dF ( m) Z dF.Z2 (mm)

Gold 79

Aluminium 13

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Rutherford backscattering experiment

Objective:

1. Relative particle rates are measured in a backscattering configuration in dependence on

scattering angles between 110° and 145°. Scattering rates are compared between gold and

aluminium as scattering specimen.

2. Particle rates are measured at different angles of scattering between 20° and 90° in a scattering

geometry that compensates the angle dependence of the scattering rate in case of scattering

probability proportional to the Rutherford scattering formula. Rates are compared to absolute

rate predictions of the Rutherford scattering formula. For one specific angle scattering rates of

gold and aluminium are compared.

Set-Up and Procedure:

The 370 kBq 241Am source is mounted on the 5 mm thread on the inside of the flange cover

beside the detector. The scattering specimen, the gold foil on annular diaphragm, is placed (the

gold-foil side facing the source) inside the vessel. Be careful not to touch the foil. See Fig. 6.

Rest of the procedure is same as forward scattering configuration.

Figure 6: Backscattering configuration and denotations.

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Theory and evaluation:

Follow exactly as in case of forward scattering till the point eq 1. Since the configuration

is different the measurement of angle and the corresponding parameters are different.

Since the solid angle dΩ of the detector decreases with r-2

= (s + d)-2

, it can be assumed

that mainly the particles with the shortest path from source S to detector D contribute to the

counting rate. For the shortest path the angle of incidence equals the angle of reflexion, thus

tan = sx/sz = dx/dz = xSD/r

and

r = s + d = +

with

xSD = dx + sx = 2.3 cm fixed by the set-up.

In a measurement example the source S was seated at scale reading 2.6 cm so sz = z - 2.6

cm, the detector D was at 2.1 cm so dz = z - 2.1 cm with scale reading of the position of the foil

z. Then

= arctan (xSD / (dz+sz)) = arctan (2.3 cm / (2 ·z - 4.7 cm))

r2 = ((dz +sz) / sin ( ))2 = ((2 ·z - 4.7 cm) / sin ( ))

2

/2 = 90° -

The number of incidents n(z) was measured during t minutes yielding counting rates Δn(z) per

minute (see Table 1).

Observations:

Table 1: Gold foil 1.5 m thick.

Z (cm) (o) (

o) r

2 (cm

2) n(z) t (min) n

(z)(min-1

)

1/sin4( /2) n(z)

r2sin

4( /2)

4.0

4.5

5.0

5.5

6.0

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Figure 7 shows the counting rate dependence of backscattering angle.

For checking the Z2-dependancy of the scattering rate an aluminum foil was used in an equal

geometry as the gold foil. The results match the expected rates within the high uncertainty due to

low number of incidents.

Table 2: Comparison of measured rate at z=4.0 cm for gold foil and 1.5 m and aluminum foil,

0.8 m thick.

Z (cm) n(z) t (min) Z Z2 n (z)/Z

2

(min-1

)

4.0 79

4.0 13

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