Investigating the Scattering Angle Dependence of …€¦ · Investigating the Scattering Angle...

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WJP X, XXXX.XX (20XX) Wabash Journal of Physics 1 Investigating the Scattering Angle Dependence of the Rutherford Scattering Equation Adam L. Fritsch, Samuel R. Krutz, and Thomas F. Pizarek Department of Physics, Wabash College, Crawfordsville, IN 47933 (Dated: November 7, 2008) The famous Rutherford Scattering Equation predicts a 1/sin 4 (φ/2) dependence for the number of counts of an incident nucleus scattered off of a target nucleus as a function of the scattering angle φ. By scattering α particles incident on gold nuclei, we found a dependence of 1/sin β (φ/2) for β =3.9751 ± 0.0009 (95% CI), which is a reasonable approximation of Rutherford’s value of β = 4. In 1897, J.J. Thomson discovered the electron, prompt- ing scientists to more aggresively attempt to describe the structure of the atom as a whole. Thomson pro- posed a “plum-pudding” atomic model; the negatively charged electrons are evenly spread throughout a posi- tively charged mass like raisins in plum pudding. Lord Ernest Rutherford, however, who worked under Thom- son, knew that Thomson’s model did not agree with ex- perimental data such as spectroscopy [1]. Rutherford proposed a different model for atomic structure. In his model, Rutherford placed all positive charge in the center of the atom as a dense ball he called the nucleus, while the electrons would be arranged much less densely around the nucleus. Rutherford also sug- gested an experiment, now called Rutherford scattering, to be done to test his model. He proposed to bombard a gold foil, which contains large gold atoms, with small positive nuclei, α particles (already known at the time to be ionized helium atoms). According to Rutherford, as the incident α particles pass through the foil, they would be scattered by the like-charged gold atoms. In some cases, an α particle could even be scattered backwards. Rutherford characterized this scattering effect with the Rutherford Scattering Equation [1]: N (φ)= N i nt 16 e 2 4π 0 2 Z 2 1 Z 2 2 r 2 K 2 sin 4 (φ/2) , (1) where N (φ) is the number of incident particles (here, α particles) scattered as a function of scattering angle φ, N i is the total number of incident particles, nt is the number of target nuclei (in this case, gold) per unit area, e is elementary charge, 0 is permittivity of vacuum, Z 1 is the atomic number of the incident particles, Z 2 is the atomic number of the target atoms, r is the radius of the scattering cross section, and K is the kinetic energy of the incident particles. In 1911, Rutherford reported that Thomson’s model was not consistent with data seen in Rutherford scat- tering. Then, in 1913, Hans Geiger and Ernest Mars- den conducted a Rutherford scattering experiment with α particles incident on a gold foil, and their data verified Eq. (1) [1].

Transcript of Investigating the Scattering Angle Dependence of …€¦ · Investigating the Scattering Angle...

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WJP X, XXXX.XX (20XX)Wabash Journal of Physics 1

Investigating the Scattering AngleDependence of the Rutherford Scattering

Equation

Adam L. Fritsch, Samuel R. Krutz,and Thomas F. Pizarek

Department of Physics, Wabash College,Crawfordsville, IN 47933

(Dated: November 7, 2008)

The famous Rutherford Scattering Equation predicts a 1/sin4(φ/2) dependence for the numberof counts of an incident nucleus scattered off of a target nucleus as a function of the scatteringangle φ. By scattering α particles incident on gold nuclei, we found a dependence of 1/sinβ(φ/2) forβ = 3.9751± 0.0009 (95% CI), which is a reasonable approximation of Rutherford’s value of β = 4.

In 1897, J.J. Thomson discovered the electron, prompt-ing scientists to more aggresively attempt to describethe structure of the atom as a whole. Thomson pro-posed a “plum-pudding” atomic model; the negativelycharged electrons are evenly spread throughout a posi-tively charged mass like raisins in plum pudding. LordErnest Rutherford, however, who worked under Thom-son, knew that Thomson’s model did not agree with ex-perimental data such as spectroscopy [1].

Rutherford proposed a different model for atomicstructure. In his model, Rutherford placed all positivecharge in the center of the atom as a dense ball he calledthe nucleus, while the electrons would be arranged muchless densely around the nucleus. Rutherford also sug-gested an experiment, now called Rutherford scattering,to be done to test his model. He proposed to bombarda gold foil, which contains large gold atoms, with smallpositive nuclei, α particles (already known at the time tobe ionized helium atoms). According to Rutherford, asthe incident α particles pass through the foil, they wouldbe scattered by the like-charged gold atoms. In somecases, an α particle could even be scattered backwards.Rutherford characterized this scattering effect with theRutherford Scattering Equation [1]:

N(φ) =Nint

16

(e2

4πε0

)2Z2

1Z22

r2K2sin4(φ/2), (1)

where N(φ) is the number of incident particles (here, αparticles) scattered as a function of scattering angle φ,Ni is the total number of incident particles, nt is thenumber of target nuclei (in this case, gold) per unit area,e is elementary charge, ε0 is permittivity of vacuum, Z1

is the atomic number of the incident particles, Z2 is theatomic number of the target atoms, r is the radius of thescattering cross section, and K is the kinetic energy ofthe incident particles.

In 1911, Rutherford reported that Thomson’s modelwas not consistent with data seen in Rutherford scat-tering. Then, in 1913, Hans Geiger and Ernest Mars-den conducted a Rutherford scattering experiment withα particles incident on a gold foil, and their data verifiedEq. (1) [1].

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Rutherford’s accurate description of the atom openedup physics to a new understanding of how the universe ismade. As such, we believe that recreating the Geiger andMarsden experiment helps undergraduates connect withthe groundbreaking discoveries of the early 20th century.

This paper details one such recreation of Geiger andMarsden. Our experiment tests the 1/sin4(φ/2) depen-dence of Eq. (1) (see Fig. (1)).

φAlpha Particle Path

Gold Nucleus

FIG. 1: This diagram provides a view of the scattering of an αparticle off of a gold nucleus. The angle at which the α particle

is deflected off of its original trajectory is the scattering angle φ,

which Rutherford correctly predicted would give the function of

scattered particles a 1/sin4(φ/2) dependence.

As done in 1913, α particles are scattered off of a goldfoil. The α source is Polonium-210. The source is locatedimmediately before a collimator of two discs with smallholes at their centers that are separated by 41.3 mm. Thegold foil is located after the collimator. Then, 14.3 cm af-ter the gold foil target is a recording film with a diameterof about 9 cm, upon which the scattered α particles areincident. This entire setup is located inside of a vacuumchamber, which is lowered to about 10 torr with a roughvacuum pump connected to a barbed nozzle at the backof the chamber, as seen in Fig. (2). This apparatus, theEN-20 Rutherford Scattering Apparatus, is produced byDaedalon Corporation.

Once the chamber is pumped down, air molecules nolonger impede path of the α particles, which can thentravel the full length of the chamber and reach the record-ing film. The film, EN-21 Alpha Particle Recording Filmmade by Daedalon Corporation, is made of clear plas-tic, but with the side of the film facing the incident αparticles coated with a thin red layer of cellulose nitrate.When struck by α particles, the layer is disturbed.

By bathing the film in a bath of 10% NaOH for 24hours at 313 K, the disturbed areas are etched away, andholes remain in the layer where the α particles hit. A mi-croscope with a magnification of 20x to 50x is then usedto view the holes. A grid with horizontal and verticalline spacings of 0.5 mm is placed beneath the film. Inthe microscope, both the grid and film are seen, allowingfor the counting of holes produced by α particles to bedone such that coordinates can be assigned to each holewith a 0.5 mm resolution.

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Alpha SourceGold Foil

Hose Connection

Film

14.3cm

41.3mm

FIG. 2: Our apparatus consists of an α source (Po-210) located

behind a collimator of length 41.3 mm. Over the end of the colli-

mator was a gold foil, the target used in the experiment. Located

14.3 cm after the target is a recording film. The film is used to

identify the location of incident α particles. The entire apparatus

was vacuum pumped down to about 10 torr. Trigonometric rela-

tions allow for the calculation of the scattering angle φ for each

α particle, allowing us to test Rutherford’s proposed 1/sin4(φ/2)dependence.

In our experiment, the chamber was under vacuum forapproximately 72 hours. In total, about 40,000 holeswere created on the film. Time constraints limited us toscan only half of the film with the microscope. Once wecounted the holes, we binned the totals for every 0.5 mmin the x and y directions and plotted them as a functionof position, as seen in Fig. (3). Also, Poisson statisticswere used to assign uncertainty to the number of datapoints for each region on the graph paper.

A Gaussian fit was done on each plot to determinethe center of the spread, which we made sure to coverduring the scanning of the film under microscope. Oncedetermined, the data was then re-calibrated such thatthe center of the spread is the origin of the coordinatesystem (Fig. (4)).

Setting the center of the origin allows for the data tobe converted to the polar coordinates ρ and θ since

ρ =√x2 + y2 (2)

and

θ = tan−1(yx

). (3)

Once in polar coordinates, the scattering angle φ canthen be determined by the relation

φ = tan−1( ρD

)(4)

for polar coordinate ρ and distance D = 14.3 cm fromthe gold foil to the film as shown in Fig. (5).

Now, we have φ for every scattered α particle. Plottingthe counts of holes as a function of the scattering angle

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FIG. 3: The above plots show how the center of the spread of αparticles on the film was determined. A bin was created for every

0.5 mm, allowing a histogram of the data for x and y. Then, a

Gaussian function was fit to each plot to determine the location of

each peak. Since the peaks correspond to the center of the spread

of α particles, the Gaussian gives us the center of the spread. Note

that the x plot appears to seem incomplete since we did not fully

cover the x axis when counting the film due to time constraints.

binned with increments of ∆φ = 0.0035 rad allows usto fit a curve to the data. As mentioned before, we areinterested in the 1/sin4(φ/2) dependence of the scatter,so we fit the following curve to our graph:

N(φ) =α

sinβ(φ/2)(5)

with the fit parameters α and β. However, since α is just

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!

"

y

x

FIG. 4: This figure shows the center (black dot) of the film as the

origin of the polar coordinate system. Once determined, the center

as the origin allows for the conversion of coordinates from cartesian

(x,y) to polar (ρ,θ).

Recording Film

Gold Foil

!

"

! Particle Path

Non-scattered Path to Origin

D

FIG. 5: This diagram is the same as seen in Fig. (4), but rotated

90◦. Now, one sees that the scattering angle φ can be determined

from the polar coordinate ρ and distance D = 14.3 cm from the

film to the target by the relation φ = tan−1(ρ/D).

a scaling factor, we really just want to know β such that

N(φ) ∝ 1sinβ(φ/2)

. (6)

Fitting this curve to our data and weighting it to theuncertainties in our counts (shown in Fig. (6)), we getβ = 3.9751± 0.0009 (95% CI).

The uncertainty associated with β is underrepresentedas we did not account for uncertainty in physically count-ing the α particles under the microscope, which certainlyadds error to the Poisson statistics. The act of countingthe holes with one’s eyes in a microscope is very strainingand difficult, likely adding error to the counts, especiallyin dense areas near the center of the spread. Another un-certainty is the true location of the center of the spread.More data points would result in a more precisely de-fined center, which would then improve the curve fit seenin Fig. (6).

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-100

0

100

200

300

400

500

600

700

0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13

Tota

l Cou

nts

Scattering Angle (rad)

FIG. 6: This is a plot of counts as a function of the scattering

angle φ. The curve fit is proportional to the equation 1/sinβ(φ/2)for the fit parameter β. The fit gives the value of the parameter

as β = 3.9751± (95% CI), which is in reasonable agreement with

Rutherford’s value of β = 4. Note that some data points near

small values of φ were masked to avoid double counting due to the

nature of the area scanned on the film.

An alternate method we considered using was to usea phosphorescent screen in place of our recording film.The screen would emit light when struck by an α par-ticle. Then, a camera set up behind the screen wouldsee the flash of light. For each flash, the camera wouldrecord a hit with x and y coordinates. Then, we couldautomate the calibration process and conversion of coor-dinates on a computer, greatly improving our efficiencyduring data analysis. However, there are problems withthis method. Our screen was not a true phosphorescentscreen as it did not actually emit light, which caused usto abandon the method. Even if it were to emit light,it is very likely that the camera would not have goodenough resolution to assign the associated data point aposition with a resolution of at least 0.5 mm in the x andy directions.

Perhaps the best method would be to attach a high-resolution camera to a microscope. This is a methodthat we also attempted; however, the camera on our mi-croscope was not quite good enough to allow us to dis-tinguish the holes on the film from the rest of the film. Itis likely that a higher-resolution camera would allow forthis method to be possible. Using photographs from sucha camera would eliminate the strain on the eyes associ-ated with using a microscope when counting the holes,allowing for more efficient counting. Also, being ableto mark the holes on a printed photograph as one countswould improve the accuracy of the counts where the holesare densely populated. Thus, this method would likelygive the most accurate and precise identification of thelocation of the incident α particles.

While our value for β does not agree with Rutherford’svalue of β = 4, it does have a percentage difference from

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4 of only 0.6%. Moreover, considering in the effects ofcounting the holes with the eye in a microscope addsmore uncertainty to the value of β, causing us to concludethat we are in reasonable agreement with the 1/sin4(φ/2)dependence predicted by Rutherford for the scattering ofpositive particles off of nuclei.

[1] Stephen T. Thornton and Andrew Rex, Modern Physics:for Scientists and Engineers, Third Edition, Thomson:Brooks/Cole, Australia: 2006, pp. 128-137.