Calculating the Cross Section for Rutherford Scattering

In summary: I used the same units as you did, which is MeV.In summary, the textbook gives the answer of 5.3*104 fm2, while the equation given on hyperphysics gives an answer of 0.022813 fm2.
  • #1
says
594
12

Homework Statement


Calculate the cross section for the scattering of a 10 MeV alpha particle by a gold nucleus (Z=79, A=197) through an angle greater than a) 10 degrees b) 20 degrees c) 30 degrees. Neglect Nuclear recoil.

Homework Equations


http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/crosec.html

The Attempt at a Solution


I've seen a number of equations that relate to rutherford scattering. The one on hyperphysics looks neat, but it doesn't say what the variables in the equation are. I don't know what the small 'ke' means and what the two angles are supposed to be in the equation either. I'm assuming they are both the scattering angle.
 
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  • #2
You can find that by dimension check.
 
  • #3
small ke is Newton / metre? I assume it has something to do with the Coulomb Force / Impact Parameter.
 
  • #4
$$\frac{k\,e^2}{KE}$$
must be on meters.
 
  • #5
kg/s2 on the numerator would give units of barns
 
  • #6
ahhhh! I was thinking in the wrong units. So I found that ke2 = 1.44 MeV*fm

If I plug that into the cross section equation given by hyperphysics I get an answer of 0.022813 fm2

But my textbook says the answer is 5.3*104 fm2
 
  • #7
Try converting the given kinetic energy (KE) into joules. Then you can use standard SI (MKS) units for the rest. k is the Coulomb constant from Coulomb's Law: k = 1/(4πε0). e is the elementary charge (magnitude of charge on electron or proton).
 
Last edited:
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Likes says
  • #8
I still get the incorrect answer.

k= 8.98×109
e = 1.60x10-19

I use those values and plug it into the equation and I don't get 5.3*104 fm2
 
  • #9
says said:
k= 8.98×109
e = 1.60x10-19
What did you use for the KE? (in J)
 
  • #10
1.60218*10-12
 
  • #11
Your input numbers agree with mine, but I do get the textbook's answer (for 10°). Check your arithmetic again, and if you still can't find the error, show exactly how you set it up, with the numbers substituted into the equation.

Make sure your calculator is set to use degrees instead of radians for trig functions.
 
  • #12
π * 22 * ( 8.9*109 * ((1.6*10-19)2) / 1.6*10-12 ) (1+cos(10) / 1-cos(10) )

= 3.3*10-29

Sorry about the formatting, but I've got it in degree mode and I've checked all the brackets etc.
 
  • #13
And my units seem right with dim. analysis

N*m2/C2 * C2 / kg * m2*s-2
 
  • #14
Z is for gold (79), not the alpha particle (2).

Also, you omitted one power of 2. Look at the original equation carefully.

Finally, you may have some discrepancy from roundoff error because you used only two significant figures.
 
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Likes says
  • #15
oh, damn! I should've seen that. I tend to look at equations and hope that someone has wrote, where Z= ..., k=..., because sometimes I either forget things, i.e. like coulomb constant and elementary charge, and I find it easier to understand relationships with equations instead of a slab of text. Thanks for your help! :)
 
  • #16
adding the power of 2 and changing the Z to 79, i get 5.19e-26
 
  • #17
I'm still out by some order of magnitude, but I'm not sure why.
 
  • #18
my answer is in m, and not m2. If I square my answer though it's out by an even greater magnitude...

textbook = 5.3*104 fm2

me = 5.19 *10-26 m = 5.19*10-11 fm
 
  • #19
Figured it out. I replaced the units with MeV and got the correct answer.
 

What is Rutherford scattering?

Rutherford scattering is a physical phenomenon in which a beam of particles (usually alpha particles) is scattered by the positively charged nuclei of atoms.

What did Rutherford's experiment reveal about the structure of the atom?

Rutherford's experiment, known as the gold foil experiment, revealed that atoms have a dense, positively charged nucleus at their center and that most of the atom is empty space. This discovery led to the development of the modern atomic model.

What is the mathematical formula for calculating Rutherford scattering?

The mathematical formula for calculating Rutherford scattering is known as the Rutherford scattering formula, given by: dΩ/dΩ = (Z1Z2e^2)/(4πε0mv^2 sin^4(θ/2)), where Z1 and Z2 are the atomic numbers of the scattering particle and the target nucleus, e is the elementary charge, ε0 is the permittivity of free space, m is the mass of the scattering particle, v is the velocity of the scattering particle, and θ is the scattering angle.

How is Rutherford scattering used in modern research and technology?

Rutherford scattering is used in many areas of modern research and technology, including nuclear physics, particle accelerators, and materials characterization. It is also used in medical imaging techniques such as positron emission tomography (PET) scans.

What are some limitations or assumptions of the Rutherford scattering model?

One limitation of the Rutherford scattering model is that it assumes the target nucleus is stationary, which is not always the case. It also assumes that the scattering particles have a point-like structure, which is not true for many subatomic particles. Additionally, the model does not account for the effects of quantum mechanics and relativistic effects at high energies.

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