Pair Production and kinetic energy

[frankR]

Problem:

A photon of energy E strikes an electron at rest and undergoes pair production, producing a positron and another electron.

photon + e^- --------> e⁺ + e^- + e^-

The two electrons and the positron move off with identical linear momentum in the direction of the initial photon. All the electrons are relativistic.

a) Find the kinetic energy of the final three particles.

b) Find the initial energy of the photon.

Express your answers in terms of the rest mass of the electron m_o and the speed of light c.



I set the problem up this way:

For part a)

E is conserved before and after the collision.

E_photon = 3E_e + 3KE_e

hc/&lambda = 3m_oc² + 3KE_e

I then solved for KE_e and found:

KE_e = 1/3hc/&lambda - m_oc²

Is this the correct method to find the kinetic energy of one of the electrons after the collision?


The problem says "state your answer in the rest mass of the electron and the speed of light", however if I don't know the wave length of the photon, how do I eliminate that energy term. Since they say the electrons are moving relativistic, do I just assume their speed is equal to 1/10c and then work backward to find the photons wavelength?

I may be way off, any help is appreciated!



Thanks

Edit: I need to use mc²(&gamma - 1) for KE!

The biggest question I have is linear momentum conserved. I would assume it is, and the problem hints at linear momentum.

 

[quantumdude]

Originally posted by frankR
E_photon = 3E_e + 3KE_e

Not quite. You forgot the rest mass energy of the original electron. Also, what is "E_e" on the right hand side? I presume that's the rest mass energy of each particle.

 

[frankR]

Is this okay?

E_photon + m_oc²= 3m_oc² + 3K_e

E_photon = energy of the photon: hc/&lambda
m_oc² = rest mass energy of electron
K_e = kinetic energy of the electrons in motion (all three with equal linear momentum and kinetic energy) after the photon collides with the initially motionless electron.

 

[quantumdude]

That's a good start, but you have to take it further. The problem said to express the answer in terms of m₀ and c. That means you can't have λ in the answer. So now you need to conserve momentum and get a second equation.

Also, rather than write the total energy of each final particle as m₀c²+K, I would write each one as γm₀c². That will make it easier to simultaneously solve the 2 equations.

 

[frankR]

Is the algebra soposed to be really messy on this problem.[?]

I just want to make sure I'm not going through all this for nothing.

 

[quantumdude]

Originally posted by frankR
Is the algebra soposed to be really messy on this problem.[?]

There is a messy way, and there is a clean way.

First, the 2 equations.

hc/λ+mc²=3γmc² (Conservation of Energy)
h/λ=3γmv (Conservation of Momentum)

The clean way is to multiply the second equation by 'c', and move the mc² to the right side in the first equation. Then, the left hand sides of the two equations are equal, so you can set the right hand sides equal to each other. Solve for v, then get the KE from that. Then use either equation (I'd use the second one because it's shorter) to find the initial energy of the photon.

 

[frankR]

Originally posted by Tom
There is a messy way, and there is a clean way.

The clean way is to multiply the second equation by 'c'

I like your way much better.

I ended up with a &gamma²/[(&gamma +1)(&gamma-1)] = 4/9mc², solving for v would have been fun. I bet I made some mistakes along the way, two pages of algebra and I write small. Plus that I tried to solve for K directly, then subsituted that and tried to go for v. It's ugly.

BTW: I found v to be 2/3c

 

[quantumdude]

It seems you did make a mistake.

Here are your two equations, after the manipulations I described:

hc/λ=(3γ-1)mc² (Conservation of Energy)
hc/λ=3γmcv (Conservation of Momentum)

You can straighforwardly solve for v by equating the right hand sides of each equation. For convenience, I would define β=v/c, and I get the following:

(3γ-1)=3γβ

Also note that γ=1/(1-β²)^1/2. You should be able to solve for β without much algebra.

BTW, I got β=0.8, so v=0.8c.

Edit: fixed math symbol

 

[frankR]

Yeah I found my error. Reworking it right now.

I canceled &gamma, which you can't do.

Since E of initial electron is only mc²

 

[frankR]

Are you sure v isn't 8/9c, not .8c.

 

[quantumdude]

Originally posted by frankR
Are you sure v isn't 8/9c, not .8c.

I got (4/5)c=0.8c.

 

[frankR]

I get the K of electron after collision:

2/3mc^2

and for initial photon energy of:

4mc^2


If I didn't get those right, I'm quiting for the day.

 

[quantumdude]

Those are correct.

 

[Tyger]

The easiest way to solve this one

is backwards. Start in the rest frame of the two electrons and positron, an electron and positron anihilate and the remaining electron absorbs the recoil momentum. Then do a Lorentz transform of the initial and final states.