Compton Scattering - determine momentum

[JoeyBob]

Homework Statement

see attached

Relevant Equations

p = h/wavelenghth

So the initial wavelength gives the total momentum, p=h/11.2p. Which is 59.161y.

Then I tried to substract the momentum from the scattered light to get the momentum of the electron.

59.161y-h/13.6p, which ends up being 0.4872 as the final answer, but the answer is supposed to be 0.77?

 

[Steve4Physics]

Homework Statement:: see attached
Relevant Equations:: p = h/wavelenghth

So the initial wavelength gives the total momentum, p=h/11.2p. Which is 59.161y.

Then I tried to substract the momentum from the scattered light to get the momentum of the electron.

59.161y-h/13.6p, which ends up being 0.4872 as the final answer, but the answer is supposed to be 0.77?

Here are a few questions to think about:
1. Are 'p' and 'y' meant to be units?!
2. What do you know about the directions of the electron and the scattered photon after the collision (e.g. include a diagram)?
3. What equation relates the wavelengths of the incident and scattered photons to the the photon scattering angle?
4. Do you know how to add/subtract vectors which have different directions?

 

[jtbell]

So the initial wavelength gives the total momentum, p=h/11.2p. Which is 59.161y.

The problem statement says to "answer in multiples of 10^-22 kg.m/s". Is your "y" supposed to equal 10^-22 kg.m/s? If so, your 59.161y isn't quite right. Check your arithmetic.

Then I tried to substract the momentum from the scattered light to get the momentum of the electron.

The momenta of the incoming and outgoing photons are vectors. They're not in the same direction, so you can't simply subtract the magnitudes. You need to use vector arithmetic.

Hint: draw a vector diagram. You should find that the arithmetic is actually rather simple.

 

[JoeyBob]

The problem statement says to "answer in multiples of 10^-22 kg.m/s". Is your "y" supposed to equal 10^-22 kg.m/s? If so, your 59.161y isn't quite right. Check your arithmetic.

The momenta of the incoming and outgoing photons are vectors. They're not in the same direction, so you can't simply subtract the magnitudes. You need to use vector arithmetic.

Hint: draw a vector diagram. You should find that the arithmetic is actually rather simple.

But the question doesn't give vectors, so how am I suppose to draw a vector diagram or assign vectors if none are known?

y=yocto or 10^-24.

 

[Steve4Physics]

But the question doesn't give vectors, so how am I suppose to draw a vector diagram or assign vectors if none are known?

y=yocto or 10^-24.

You can easily look up the diagram/standard formula for Compton scattering. These should already be in your notes/textbook if you have been taught Compton scattering.
E.g. http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/compton.html

Can you now see how to proceed?

Also note, it is incorrect to use a prefix as a unit. For example for a picometre you should write 'pm' not just 'p'.

 

[jtbell]

y=yocto or 10_-24.

OK, so your number for the incoming photon momentum agrees with mine, which is 5.9161 x 10^-23 kg.m/s, or 0.59161 x 10^-22 kg.m/s using the terms of the problem statement.

how am I suppose to draw a vector diagram or assign vectors if none are known?

You can choose the incoming photon momentum (total momentum of the system) to be in the x-direction. The standard Compton-scattering formula gives you the direction (angle) of the outgoing photon momentum. This is enough information to calculate the outgoing electron momentum, both magnitude (which is what you want) and direction (maybe you can get bonus credit for that ).

 

[JoeyBob]

OK, so your number for the incoming photon momentum agrees with mine, which is 5.9161 x 10^-23 kg.m/s, or 0.59161 x 10^-22 kg.m/s using the terms of the problem statement.

You can choose the incoming photon momentum (total momentum of the system) to be in the x-direction. The standard Compton-scattering formula gives you the direction (angle) of the outgoing photon momentum. This is enough information to calculate the outgoing electron momentum, both magnitude (which is what you want) and direction (maybe you can get bonus credit for that ).

Ok so I used the formula

change in wavelength = (h/mc)(1-cos(angle) to get an angle of 89.422 degrees, but isn't this giving the angle the photon is scattered and not the angle the electron is scattered?

I could understand how I could use this vector to for lights momentum, but i don't see how I can use it for the electron.

 

[jtbell]

Hint: Use conservation of momentum. Total initial (incoming) vector momentum = total final (outgoing) vector momentum.

 

[JoeyBob]

so

0.59161 x 10-22 = cos(89.422) * magnitude light momentum + x component of electron momentum

So I really now just need to find the magnitude of light momentum after scattering occurs.

The scattered light wavelength ss 13.6 x 10-12, p=h/wavelength gives 4.8721 x 10-23.

Using the above equation to calculate x component of electron momentum I get a momentum of 0.5867 x 10-22. But it should be 0.77 x 10-22.

Looking into this, the correct answer doesn't even make sense to me unless the light has a negative component, since the x momentum of the electron is greater than the initial x momentum of the light.

But even if i add the final momentum of the light to the initial, I still only get 0.597 x 10-22.

I could be doing something wrong with converting magnitude to x component, multiplying the magnitude by cos(89.422). I haven't used vectors in a long time, was hoping id never have to see them again.

Or I could have calculated the angle incorrectly. Or something else.

 

[jtbell]

Using the above equation to calculate x component of electron momentum I get a momentum of 0.5867 x 10-22.

OK, that's what I get, too.

But it should be 0.77 x 10-22.

No. that number is what the magnitude of the electron momentum vector (not the x-component) is supposed to turn out to be.

If you know the x-component of a vector, how can you find the magnitude, and what else do you need to know in order to do it? There are actually two ways to do it, but one of them is a bit easier, given the information that you have.

 

[neilparker62]

Alternate Method:

$$E_k=\frac{hf_1-hf_2}{q_e}\;eV$$
$$\gamma=\frac{E_k+E_0}{E_0}=\sec\delta,$$ where $\sin\delta=\frac{v}{c}$ and $E_k,E_0$ are kinetic energy and rest energy of the electron respectively (in eV).
$$p_e=m_ec\tan\delta$$

 

[Steve4Physics]

Alternate Method:

$$E_k=\frac{hf_1-hf_2}{q_e}\;eV$$
$$\gamma=\frac{E_k+E_0}{E_0}=\sec\delta,$$ where $\sin\delta=\frac{v}{c}$ and $E_k,E_0$ are kinetic energy and rest energy of the electron respectively (in eV).
$$p_e=m_ec\tan\delta$$

Your first equation has energy on the left but energy/charge on the right.

 

[jtbell]

Your first equation has energy on the left but energy/charge on the right.

This gives the energy in electron-volts (eV) instead of joules. I actually prefer to use electron-volt units for energy, mass and momentum, because it eliminates a lot of factors of c in relativistic calculations. However, the problem statement clearly calls for SI units, so I strongly suspect the OP has not used eV-units extensively (if at all!) in relativistic calculations.

Starting with conservation of energy, instead of conservation of momentum, is a valid approach. This avoids having to mess with vectors. If you can find the energy of the outgoing electron, you can then find the magnitude of its momentum.

IMHO the rest of the "alternate method" above might be a bit obscure for someone at the introductory level.

 

[neilparker62]

This gives the energy in electron-volts (eV) instead of joules. I actually prefer to use electron-volt units for energy, mass and momentum, because it eliminates a lot of factors of c in relativistic calculations. However, the problem statement clearly calls for SI units, so I strongly suspect the OP has not used eV-units extensively (if at all!) in relativistic calculations.

Starting with conservation of energy, instead of conservation of momentum, is a valid approach. This avoids having to mess with vectors. If you can find the energy of the outgoing electron, you can then find the magnitude of its momentum.

IMHO the rest of the "alternate method" above might be a bit obscure for someone at the introductory level.

Would it perhaps be clearer if I wrote in the last line: $$p_e=m_ec\tan\delta=m_ec\sqrt{\gamma^2-1}$$ ?

 

[Steve4Physics]

This gives the energy in electron-volts (eV) instead of joules.

Yes. It just seems unusual to write $$E_k=\frac{hf_1-hf_2}{q_e}\;eV$$ because it embodies a unit in an otherwise purely symbolic equation. As a result, this means $h,f_1,f_2$ and $q_e$ must be in SI units.

I'm just being pedantic I guess.

Edited to correct formatting.

 

[jtbell]

Indeed, I would write the equation without $q_e$ and units, and use either $h = 6.626 \times 10^{-34}~\rm{J \cdot s}$ or $h = 4.136 \times 10^{-15}~\rm{eV \cdot s}$, according to my desired units. Textbooks list both values of $h$, as does Wikipedia. The numerical value of the electron charge here is just a unit conversion factor: $1~\rm{eV} = 1.602 \times 10^{-19}~\rm{J}$.

 

[jtbell]

$$\gamma=\frac{E_k+E_0}{E_0}=\sec\delta,$$ where $\sin\delta=\frac{v}{c}$ and $E_k,E_0$ are kinetic energy and rest energy of the electron respectively (in eV).
$$p_e=m_ec\tan\delta$$

Would it perhaps be clearer if I wrote in the last line: $$p_e=m_ec\tan\delta=m_ec\sqrt{\gamma^2-1}$$ ?

I don't see any reason to introduce the electron's velocity via $\gamma = 1 / \sqrt{1 - v^2/c^2} = \sec \delta$, when the problem statement doesn't call for finding $v$. One can use the well-known relativistic relationship among (total = kinetic + rest) energy, momentum and mass: $E^2 = (pc)^2 + (mc^2)^2$. If one uses eV-units, this is especially convenient because $E$, $pc$ and $mc^2$ all have units of eV (or keV or MeV, etc.).

 

[neilparker62]

We don't need v to find $\gamma$ or vv since we already have an expression for $\gamma$ in terms of energy. The only reason for introducing v would be to show that $p_e=m_ec\tan\delta=m_ec\sqrt{\gamma^2-1}.$ Personally I find it convenient to set $\sin\delta=\frac{v}{c}$ when it follows that $\gamma=\sec\delta$. Then there are convenient trig expressions for electron KE, electron momentum, electron total energy etc. This provides an easy path from electron KE to electron momentum as indicated previously.

 

[JoeyBob]

OK, that's what I get, too.

No. that number is what the magnitude of the electron momentum vector (not the x-component) is supposed to turn out to be.

If you know the x-component of a vector, how can you find the magnitude, and what else do you need to know in order to do it? There are actually two ways to do it, but one of them is a bit easier, given the information that you have.

So either I need to find the angle the electron is ejected or the y component of its momentum. The y component of the photons momentum is sin(89.422)*4.872 x 10-23 = 4.872 x 10-23. The initial y component of the lights momentum is 0. Therefore the y component of the electrons momentum is -4.872 x 10-23.

Using sqrt(x^2+y^2) to give magnitude, I get 7.63 x 10-23, which is only slightly off probably from rounding.

Attempting the energy method, i have the equation Kinetic E of elec. = (hc/wavelength initial) - (hc/wavelength final) in my notes.

So Ek = 1.988 x 10-25((1/11.2 x 10-12) - (1/13.6 x 10-12)) = 3.132 x 10-15.

Ek=mc^2(1/sqrt(1-(v^2/c^2)))-1

Ill spare the details, I get a velocity of 80630000 m/s or 8.063 x 10^7.

Using p=mv/sqrt(1-(v^2/c^2)) I get a momentum of 7.63 x 10-23. Again, off slightly probably from rounding but overall the right answer.

I would say the second way is arguably easier because vectors are bad for you.

Thanks for all the help

 

[jtbell]

I got 7.623 x 10^-23, three different ways, using an extra couple of digits in c, h, and m_e, and making sure not to round off intermediate results. Slightly different values for those constants probably explain the difference between us two. I think whoever wrote the problem had a bit of roundoff error.

 

[neilparker62]

I got 7.623 x 10^-23, three different ways, using an extra couple of digits in c, h, and m_e, and making sure not to round off intermediate results. Slightly different values for those constants probably explain the difference between us two. I think whoever wrote the problem had a bit of roundoff error.

Agreed - same result per following: