Unique λ at which X-ray and electron have same energy

[catkin]

Homework Statement

Derive an equation to show that there is a unique wavelength at which X-ray photons and electrons have the same energy. Calculate this wavelength and energy.

Homework Equations

Here's what I thought was relevant. There may be others!

For photons: E = hf = h c / λ

For electrons:
λ = h / p (de Broglie)
K.E. = ½ mv^2
p = mv

The Attempt at a Solution

p = √(2mE)
Substituting in de Broglie
λ = h / √(2mE)
E = h^2 / 2mλ^2

Equating energies
(h c / λ)X-ray = (h^2 / 2mλ^2)electron
Gathering the constants (c, h, 2 and m -- the rest mass of an electron)
(λ)X-ray = (h / 2cm) * (λ^2)electron

... which does not have a unique solution :-(

 

[Borek]

Doesn't (λ)X-ray = (λ)electron?

 

[catkin]

Thanks Borek :)

(nice hair!)

That's what the question asks the answerer to show so I hope it's true! I don't think I've shown that it is so.

Writing y for (λ)X-ray, x for (λ)electron and lumping the constants together as k my attempt shows, when the X-ray and electron energies are the same,
y = k x^2

Mmm ... the more I read the question the more ambiguous it becomes. Perhaps it would help if you could translate the question into unambiguous language.

Best

Charles

 

[Borek]

Why do you still use two variables for wavelength, when you should use one?

 

[catkin]

Thanks again, Borek.

I need to show that λ = f_e(E) and λ = f_p(E) intersect before that is legitimate (subscript e for electron, p for photon).

With that requirement now clear ...

For photons:
λ = hc / E
= (hc) * (1 / E)

For electrons:
λ = sqrt(h^2 / 2mE)
= sqrt(h^2 / 2m) * sqrt(1 / E)

Regardless of the constant values, these functions intersect once.

At the intersect the previous equation becomes (thanks!)
λ = (h / 2cm) * (λ^2)
= 2cm / h

and the rest is trivial.