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Homework Help: Neutron decay - is my answer unrealistic?

  1. Dec 13, 2009 #1
    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]


    [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


    [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?

  2. jcsd
  3. Dec 13, 2009 #2


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    That's not weird. The electrons rest mass = ~1/1836 of the proton rest mass, so it receives most of the kinetic energy, while the proton has a lot of 'rest' energy. In the problem, 0.78 MeV means the electron is relativistic.

    In reality, the neutrino can take a lot of energy/momentum as well, and in beta decay, there is a continuous spectrum of energy with the most probably energy of the beta being ~ 1/3 of the maximum allowable energy.
  4. Dec 30, 2009 #3

    I would like to know more about the beta decay. Could you give me a good reference book to start with? I would like to calculate the electron energy spectrum resulting from a neutron decaying into a proton, electron and anti-electron neutrino. Any idea where to start? Thanks!
  5. Dec 30, 2009 #4


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    Staff Emeritus
    Science Advisor

    Data for each beta-emitter has been tabulated.

    Here is some general information - http://hyperphysics.phy-astr.gsu.edu/Hbase/nuclear/beta2.html

    Decay of a neutron

    Some notes
    http://www.hep.phys.soton.ac.uk/hepwww/staff/D.Ross/phys3002/beta.pdf [Broken]
    Last edited by a moderator: May 4, 2017
  6. Dec 30, 2009 #5
    Thanks! Do you happen to know where I could find the tabulated energy spectrum for decay electrons in the rest frame of the parent neutron?
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