- #1
jeebs
- 325
- 4
Hi,
here's the problem:
"a neutron at rest decays into a proton with a decay energy of 0.78MeV. What is the maximum kinetic energy of the proton left behind?"
here's what I've tried:
In this decay, I assumed that although it's not mentioned, there would be something negatively charged produced to conserve charge. I went with beta decay, so that an electron and an antineutrino would be produced. I assumed that the neutrino was massless and ignored it (my lecturer said it could be ignored).
In the neutron's rest-frame, i said that the proton and electron would have equal and opposite momenta pp and pe respectively, ie.
[tex]\stackrel{\rightarrow}{p_e} = -\stackrel{\rightarrow}{p_p}[/tex]
ie.
[tex]p_e = p_p = p[/tex] (magnitudes are equal.)
Also, for conservation of energy, i said that neutron energy En = Ep + Ee = (mp2 + p2)1/2 + (me2 + p2)1/2 = mn since the neutron has no momentum.
(Here I have used the expression E2 = p2c2 + m2c4 in c=1 units).
mn - (mp2 + p2)1/2 = (me2 + p2)1/2
(mn - (mp2 + p2)1/2)2 = me2 + p2
mn2 - 2mn(mp2 + p2)1/2 + mp2 + p2 = me2 + p2
mn2 + mp2 - me2 = 2mn(mp2 + p2)1/2
hence
[tex] p = \sqrt{(\frac{m_n^2 + m_p^2 - m_e^2}{2m_n})^2 - m_p^2 } [/tex]
Using wikipedia's data:
mp = 938.272 MeV/c^2
mn = 939.566 MeV/c^2
me = 0.510 MeV/c^2
I get p = 1.188 MeV/c
Again, using E2 = p2c2 + m2c4 I get the proton energy Ep = 938.2727521 MeV and when I subtract the rest energy from this to get the kinetic energy, I am left with 7.52x10-4 MeV.
When I do the same for the electron, I find that Ee is just less than 0.78 MeV.
Is this a reasonable answer? It seems weird to me that the electron should take the vast majority of the energy, especially when I am looking for the maximum PROTON energy?
Thanks.
here's the problem:
"a neutron at rest decays into a proton with a decay energy of 0.78MeV. What is the maximum kinetic energy of the proton left behind?"
here's what I've tried:
In this decay, I assumed that although it's not mentioned, there would be something negatively charged produced to conserve charge. I went with beta decay, so that an electron and an antineutrino would be produced. I assumed that the neutrino was massless and ignored it (my lecturer said it could be ignored).
In the neutron's rest-frame, i said that the proton and electron would have equal and opposite momenta pp and pe respectively, ie.
[tex]\stackrel{\rightarrow}{p_e} = -\stackrel{\rightarrow}{p_p}[/tex]
ie.
[tex]p_e = p_p = p[/tex] (magnitudes are equal.)
Also, for conservation of energy, i said that neutron energy En = Ep + Ee = (mp2 + p2)1/2 + (me2 + p2)1/2 = mn since the neutron has no momentum.
(Here I have used the expression E2 = p2c2 + m2c4 in c=1 units).
mn - (mp2 + p2)1/2 = (me2 + p2)1/2
(mn - (mp2 + p2)1/2)2 = me2 + p2
mn2 - 2mn(mp2 + p2)1/2 + mp2 + p2 = me2 + p2
mn2 + mp2 - me2 = 2mn(mp2 + p2)1/2
hence
[tex] p = \sqrt{(\frac{m_n^2 + m_p^2 - m_e^2}{2m_n})^2 - m_p^2 } [/tex]
Using wikipedia's data:
mp = 938.272 MeV/c^2
mn = 939.566 MeV/c^2
me = 0.510 MeV/c^2
I get p = 1.188 MeV/c
Again, using E2 = p2c2 + m2c4 I get the proton energy Ep = 938.2727521 MeV and when I subtract the rest energy from this to get the kinetic energy, I am left with 7.52x10-4 MeV.
When I do the same for the electron, I find that Ee is just less than 0.78 MeV.
Is this a reasonable answer? It seems weird to me that the electron should take the vast majority of the energy, especially when I am looking for the maximum PROTON energy?
Thanks.