Physics 3401, Chapter 5 quiz

1. Your task is to determine energies for particles in a particle beam to investigate a crystal with lattice
spacing 0.1 nm, so to see diffraction effects you want to have a wavelength of at least twice the lattice
spacing, 0.2 nm.

a. What kinetic energy would be required for electrons?
This is in non-relativistic range.
pc=hc/wavelength=1240 eV nm/0.2nm=   6200 eV
E=p^2/2m = (pc)^2/2*mc^2 =(6200 eV)^2/2*511000 eV =    37.61 eV

b. What kinetic energy would be required for neutrons?
E=p^2/2m = (pc)^2/2*mc^2 =(6200 eV)^2/2*940E6 eV =    0.0204 eV

2. A given nucleus has a diameter of about 10 fermis (10 x 10-15 m). To resolve any detail about
this nucleus, you would need a wavelength of about half this diameter, or 5 fermis.

a. What energy x-rays would be required?
E=hf=hc/wavelength=1240 eV nm/5E-6 nm=   248 MeV      I don't think so!

b. What electron kinetic energy would give this wavelength?
KE=sqrt((pc)^2+(mc^2)^2)-mc^2
pc=248 MeV KE= 247.490 MeV     This would require an accelerator of some kind, but it is doable.

3. Given that you have only about 10 MeV of energy available, show that you cannot contain an electron
in  a nucleus of diameter 10 fermis. You may use the uncertainty principle in a variety of different ways,
but make sure you state your assumptions clearly.

Try 3D particle in box with dimension equal to diameter.
<E>=9*hbar^2/2*m*L^2= 9*(hbar*c)^2/(2*m*c^2*L^2) = 9*(197 MeV fm)^2/(2*.511 MeV*10^2fm^2)
<E>= 3417.622309 MeV

delta p*c= hbar*c/2*delta x = 197 eV nm/(2*10E-6 nm) =    9850000 eV
assuming delta p*c of same range as pc, KE=sqrt((pc)^2+(mc^2)^2)-mc^2 =     9352245.967 eV

Show that a proton can be contained with that energy.

<E> = 9*(197 MeV fm)^2/(2*939 MeV*10^2fm^2)
<E>= 1.85985623 MeV

Using uncertainty principle as above
assuming delta p*c of same range as pc, KE=sqrt((pc)^2+(mc^2)^2)-mc^2 =     51661.25192 eV

4. Calculate the wavelengths of electrons with kinetic energy 10 eV, 10 MeV, and 1 GeV and show which
cases must be treated relativistically by comparing non-relativistic and relativistic results.

a. 10 eV Relativistic wavelength = hc/pc=hc/(sqrt((KE+mc^2)^2-(mc^2)^2)
Non-relativistic wavelength = hc/(sqrt(2*mc^2*KE))
Relativistic wavelength =  0.387877076 nm
Non-relativistic wavelength =  0.387878973 nm non rel OK

a. 10 MeV Relativistic wavelength = hc/pc=hc/(sqrt((KE+mc^2)^2-(mc^2)^2)
Non-relativistic wavelength = hc/(sqrt(2*mc^2*KE))
Relativistic wavelength =  0.000118111 nm
Non-relativistic wavelength =  0.000387879 nm X must use rel

a. 1 GeV Relativistic wavelength = hc/pc=hc/(sqrt((KE+mc^2)^2-(mc^2)^2)
Non-relativistic wavelength = hc/(sqrt(2*mc^2*KE))
Relativistic wavelength =  1.23937E-06 nm
Non-relativistic wavelength =  3.87879E-05 nm X must use rel