Probing the atom with light high velocity particles

[jbar18]

Homework Statement

The first part asks to find the necessary momentum of a "relativistic projectile particle" in order to resolve a diameter of 10^-10m (i.e. the atom). The second part asks to find the corresponding energy of the projectile particle, in GeV.

Homework Equations

λ = h/p

E² = p²c²+m₀²c⁴

These are the only relevant equations I can think of.

The Attempt at a Solution

The first part seems easy, I'm assuming I have that one right. It looks as easy as just λ=10^-10m, so p = h/10^-10 ≈ 6.63 * 10^-24kgms^-1 to resolve the atom.

The second part seems less obvious, the only equation I know of that can answer it is the relativistic energy equation, however the problem does not say what the projectile particle is or its mass. E could be approximated to be pc if the mass is very small, but this approximation would only be reasonable if the particle's invariant mass was significantly less than p/c, which in this case is ≈10^-32kg. I don't know of many particles which are this light, so it seems unreasonable to make that approximation. Are there any equations I am forgetting which give the energy of the particle? I'm suspicious that I have forgotten something very basic. Of course, if the first part is wrong, it may mean that I can safely make the approximation.

Thanks for any help.

 

[Dick]

Are you sure about the exponent on that p value? And, sure, they must mean ignore the mass. I can think of some particles that are light enough. Photons and neutrinos would be good. Though neutrinos would have other problems.

 

[jbar18]

This problem is just totally weird. The momentum needed to probe something as large as the atom (not even the nucleus, the whole atom) is not very high. So even if they did want me to ignore the rest mass, the energy E = pc is going to be nowhere near the scale of GeV. Which leads me to think that they want me to include the rest mass, which they have given me absolutely no information about. I'll post the full problem word for word in case I have misread or misinterpreted anything:

a). Estimate the momentum in SI units needed for a relativistic projectile particle
to be used in a scattering experiment to resolve the structure of a target particle of
diameter 10^−10 m (eg. an atom).
b). Find the corresponding energy of the projectile particle in giga-electron-volts
(GeV)

 

[ehild]

The necessary momentum involves the smallest energy of a particle with the given wavelength. It can be even a photon with zero rest mass. You have written the expression of E in terms of the momentum: E² = p²c²+m₀²c⁴, so E≥pc. If the rest mass differs from zero the energy must be higher than that minimum value.

ehild

 

[paranormal]

I would first like to acknowledge that this is a challenging problem and requires careful consideration of the relevant equations and assumptions. Your solution for the first part, finding the necessary momentum, seems reasonable and correct. However, I would like to point out that the value you calculated, 6.63 * 10^-24 kgm/s, is the momentum in SI units. In order to find the corresponding energy in GeV, we need to convert this momentum to units of GeV/c. This can be done by dividing by the speed of light, c, which gives us a value of approximately 6.63 * 10^-15 GeV/c.

Moving on to the second part of the problem, you are correct in using the relativistic energy equation, E^2 = p^2c^2 + m0^2c^4. However, as you mentioned, we do not know the mass of the projectile particle. In order to solve for the energy, we need to make some assumptions about the particle. One possible assumption is that the particle is an electron, which has a mass of approximately 9.11 * 10^-31 kg. Using this value, we can solve for the energy in GeV by substituting the values we have calculated so far into the equation:

E^2 = (6.63 * 10^-15 GeV/c)^2c^2 + (9.11 * 10^-31 kg)^2c^4
E^2 ≈ 4.38 * 10^-28 GeV^2 + 6.45 * 10^-26 GeV^2
E ≈ 6.45 * 10^-13 GeV ≈ 0.645 GeV

Therefore, the corresponding energy of the relativistic projectile particle in GeV is approximately 0.645 GeV. However, it is important to note that this is just one possible solution based on our assumptions. If the problem does not specify the mass of the projectile particle, it is best to state the assumptions made in order to solve for the energy.

In conclusion, probing the atom with light high velocity particles is a challenging problem that requires careful consideration of the relevant equations and assumptions. By using the relativistic energy equation and making some assumptions about the particle, we can find the corresponding energy in GeV. However, it is important to note that this is just one possible solution and other assumptions