Download Advanced Lab: Rutherford Scattering

Rutherford Scattering
Advanced Lab: Rutherford Scattering
Conner Herndon and Nick Lucas
(Dated: 21 April 2015)
Leading up to the 20th century, experiments pointed toward the existence of fundamental constituents known
as atoms. As the atomic theory developed, the plum pudding model stood strong. By shining α-particles on
gold foil, we replicate Rutherford’s famous experiment with modern technology. We predict and observe a
1/ sin4 (θ/2) trend with increasing detection angle. This result is in direct opposition to the plum pudding
model and implies a dense, positive core at the center of the atom.
I.
BACKGROUND
John Dalton, a famous chemist and physicist from
the early 19th century, proposed the existence of a fundamental entity in an effort to describe contemporary
chemistry1 . Dalton argued that if identical chemicals
were formed as aggregates of dissimilar particles, there
would be inconsistent specific gravities of materials. For
this reason Dalton proposed that fundamental particles
known as “atoms” of the same element hold identical
mass, size, and chemical properties.
Although John Dalton captured the fundamental concept of modern atomic theory through his philosophical
and chemical treatment, experimental proof did not arrive for nearly a century. This theory of atoms resounded
strongly with chemists, however, physicists disregarded
atomic theory until the early 20th century, despite Ludwig Boltzmann’s statistical approach to thermodynamics
which assumed atomic theory2 . With Jean Baptiste Perrin’s experiment3 and Albert Einstein’s theoretical explanation of Brownian motion4 , atomic theory finally gained
a foothold in physics.
Even with the existence of atoms, many phenomena
may not be adequately explained by their mere existence
such as emission spectra. As stated by Gottfried and
Yan, “[...] if the atoms are structureless, how can they
display an excitation spectrum – can a structureless piano play a sonata?”5 Before the acceptance of the atom,
a more fundamental particle – the electron – had been
experimentally discovered and studied by J. J. Thomson in 1897 (as detailed by the authors in a previous
report6 ). Understanding the electron in terms of this
newly discovered particle became the forefront goal of
the physical theory of chemistry. J. J. Thomson offered
his plum pudding model which features a positive electric “fluid” within which electrons oscillate7 . According
to Thomson’s model, when the electronic “plums” were
disturbed by gas collisions or electromagnetic radiation,
they would oscillate wildly within the “pudding” and
emit corresponding frequencies of light. Unfortunately,
the model did not accurately predict the frequencies observed.
At the turn of the 20th century, a young Ernest Rutherford and his colleague Paul Villard began work in radioactive decay in an attempt to understand the struc-
FIG. 1. Rutherford experiment apparatus. Americium
(source) shines α-particles at the gold foil. The scattered particles are then detected at an angle θ from the direct path.
ture of the atom.8 By 1907 Rutherford and Thomas
Royds determined that α-radiation is made up of fast
moving helium ions.9 While performing experiments with
α-particle bombardment on very thin gold foil, Rutherford noticed extreme deflections of the α-particle beam.
Rutherford counted the number of scattered α-particles
versus scattering angle through a microscope,10,11 and
found that Thomson’s plum pudding model strikingly
disagreed with the results. He then offered a competing model featuring a dense, positive core surrounded by
quickly moving electrons.12 This paper intends to outline Rutherford’s experiment with a modern apparatus
and prove the necessity of a dense and positive atomic
nucleus.
II.
EXPERIMENT
Our experimental apparatus consists of a scattering
chamber, americium source, gold foil, and detector with
discriminator and counting machinery. The scattering
chamber was evacuated to 650 torr, and within which was
placed the americium source, gold foil, and detector as
shown in figure 1. The detector is free to move an angle θ
from the direct path of the α-particles. We measured the
number of counts per time as a function of the detection
angle. See section IV for resulting data and analysis.
Rutherford Scattering
III.
2
equation (5) to yield
THEORY
In this paper we will treat the Rutherford scattering
problem with a modern approach and take the classical
limit.13 The problem to solve is the scattering of an
α-particle off a much more massive gold nucleus. The
interaction involved is Coulombic repulsion
(1)
where Z1 and Z2 are the atomic numbers of the colliding
atoms, and e is the electronic charge. We will generalize
this potential with an exponential factor that decreases
with radius to account for interactions with the electron
cloud on gold
e−m0 r
r
µ
2π~2
Z
2µZ1 Z2 e2
.
(m20 + q 2 )
~2
We may then return q to our initial variables by
q 2 = (ki − kf )
.
0
0
(8)
and since ki = kf by momentum conservation,
q 2 = 2k 2 (1 − cos θ)
θ
2
2
.
= 4k sin
2
(2)
d3 r0 e−ikf ·r V (r 0 ) eiki ·r ,
2
= ki2 + kf2 + 2ki · kf ,
From the Born approximation, the scattering amplitude
f (θ, ϕ) is given by
f (θ, ϕ) = −
=−
(7)
Z1 Z2 e2
,
V (r) =
r
V (r) = Z1 Z2 e2
Z
Z
Z
0
0
µZ1 Z2 e2 ∞ 0 2π 0 π 0
dθ sin θ0 r0 e−m0 r eiqr cos θ0
dϕ
dr
2π~2
0
0
0
Z
µZ1 Z2 e2 i ∞ −m0 r0 iqr0
−iqr 0
e
e
−
e
=
~2 q
0
f (θ, ϕ) = −
(9)
Thus we may substitute equation (9) into our scattering
amplitude (7) to yield
(3)
f (θ, ϕ) = −
2µZ1 Z2 e2
2
~2 m0 + 4k 2 sin2
θ
2
.
(10)
where ki is the direction of the incident matter wave, kf
is the direction of the resulting matter wave, and µ the
mass of the particle. The scattering amplitude f (θ, ϕ)
relates to the differential cross section by
By equation (4), we then have the differential cross
section
dσ
2
= |f (θ, ϕ)| ,
dΩ
dσ
4µ2 Z12 Z22 e4
h
i.
=
2
dΩ
~4 m20 + 4k 2 sin2 (θ/2)
(4)
the derivative of the total cross section with respect
to solid angle. To obtain our scattering amplitude, we
will then substitute our potential in equation (2) into
equation (3)
µZ1 Z2 e2
2π~2
Z
f (θ, ϕ) = −
=−
µZ1 Z2 e2
2π~2
Z
The α-particles will have a kinetic energy on the order
of 5 MeV. Then
~2 k 2 ' 5 MeV,
(12)
k 2 ' 7.2 · 1055 (m · kg) −2 .
(13)
0
d3 r0 e−ikf ·r
0
d3 r 0
0
e−m0 r iki ·r0
e
r0
e−m0 r iq·r0
e
,
r0
so
(5)
where we have defined q ≡ ki − kf . The inner product
q · r 0 may be evaluated using the angle θ0 between the
detector and initial path
Then we may safely assume
4k 2 sin2 (θ/2) m0 ,
0
(11)
0
0
q · r = qr cos θ ,
(14)
(6)
which we may then substitute into our integral in
since the Coulombic interaction contribution from the
electron cloud is of negligible order in comparison. Then
Rutherford Scattering
3
we may reduce equation (11) to
dσ
µ2 Z 2 Z 2 e4
.
= 4 4 1 42
dΩ
4k ~ sin (θ/2)
(15)
Finally, the energy of a wave is given by E = ~2 k 2 , so
we receive the differential cross section for our system
dσ
=
dΩ
µZ1 Z2 e2
4E sin2 (θ/2)
2
(16)
which gives the number of α-particles deflected into an
angle θ
N (θ) =
1
4π0
2 Z1 Z2 e2
2µv 2
2
1
,
sin4 (θ/2)
(17)
FIG. 2. Counts per second as a function of detector angle.
in SI units and v is the velocity of an α-particle. After
substitution of fundamental constants and Z1 = 79 and
Z2 = 4,
N (θ) '
v2
7.4217
.
sin4 (θ/2)
(18)
The energy of an α-particle according to the lab manual
is E ' 5.48 MeV, so mv 2 /2 = 5.48 MeV means
N (θ) '
2.828 · 1014
.
sin4 (θ/2)
(19)
FIG. 3. Linear regression on counts per second versus
1/ sin4 (angle/2).
IV.
ANALYSIS
We collected data over the period of four weeks ranging from angles of -60 to +65 degrees. Using the fact that
radioactive decay follows a Poisson distribution, the standard deviation grows with the square root of the average
number of counts. These standard deviations are shown
on figure 2. Observe the strong central peak that quickly
decays on either side of 0 angle. To check if our data
agree with the theoretically predicted sin4 (θ/2) trend,
we performed a linear regression on counts per second
versus sin4 (angle/2) as shown in figure 3. This regression yields R2 = 0.885, meaning that the data roughly
agree with the prediction.
only know our angle to a precision of 2.5 degrees. The detector also works as a photon detector and even a vibrations detector at times. For this reason, stray photons (or
vibrations) may have contributed to the α-particle count.
Although we maintained the scattering chamber at 650
torr throughout the experiment, this vacuum is not sufficient for entirely pristine results – stray air molecules
affect the trajectories of the α-particles. The gold foil
used in this experiment is more than a single atom thickness, and there were very likely many scattering events
for each α particle although our analysis describes only
one.
V.
VI.
UNCERTAINTY
For this experiment, we used a dial to alter the detector
angle. The dial has tick marks every 5 degrees, so we can
CONCLUSION
We derived the expected differential cross section for
the Coulomb potential using the Born approximation.
Rutherford Scattering
This prediction was then tested against four weeks of
measurements with angles ranging from -60 to +65 degrees in small steps. From these measurements, we found
a 1/sin4 (θ/2) relationship between α-particle counts per
second and measured angle which agrees with our theoretically predicted trend.
From this experiment we see that the α-particles have
encountered a very dense, positive potential while passing through the gold foil. This potential is not expected
under the Thomson plum pudding model. We then require the atom to have a very dense positive nucleus with
electrons in orbit.
VII.
1 J.
REFERENCES
Dalton, A New System of Chemical Philosophy (William Dalton and Sons, Ltd., 1808) Chap. 2, p. 142.
2 L. Boltzmann, Vorlesungen der Gastheorie (Verlag von Johann
Ambrosia Barth, 1896).
3 J. B. Perrin, Les Principes (Imprimerie Gauthier-Villars, year=).
4 A. Einstein, “On the movement of small particles suspended in
a stationary liquid demanded by the molecular-kinetic theory of
heat,” Annalen der Physik , 649–560 (1905).
4
5 Gottfried
and Yau, Quantum Mechanics: Fundamentals, 2nd ed.
(Springer, 2003).
6 C. Herndon and N. Lucas, “Advanced lab: Wave particle duality,” Advanced Lab (2015).
7 J. J. Thomson, “On the structure of the atom: an investigation of
the stability and periods of oscillation of a number of corpuscles
arranged at equal intervals around the circumference of a circle;
with application of the results to the theory of atomic structure,”
Philosophical Magazine Series , 237–265 (1904).
8 E. Rutherford, “Uranium radiation and the electrical conduction
produced by it,” Philosophical Magazine (1899).
9 Rutherford and Royds, “Spectrum of the radium emanation,”
Nature 78, 220–221 (1908).
10 M. Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, 1966).
11 Gamow and Cleveland, Physics Foundations and Frontiers
(Prentice-Hall, inc., 1960) pp. 365–369.
12 E. Rutherford, “The scattering of α and β-particles by matter
and the structure of the atom,” Philosophical Magazine 21, 669–
688 (1911).
13 J. Townsend, A Modern Approach to Quantum Mechanics, 2nd
ed. (University Science Books, 2012).