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Binding energy, Coulomb energy

  1. Apr 1, 2016 #1
    1. The problem statement, all variables and given/known data
    a) Calculate the difference in binding energy between the mirror nuclide ##^{15}O## and ##^{15}N##.
    b) Calculate the radius of both nuclide’s assuming the difference in binding energy exclusively depends on difference in Coulomb energy.

    2. Relevant equations
    Mean nuclear radius
    ##R = R_0A^{1/3}##.
    Binding energy
    ##B = \left[ Zm(^1H)+Nm_n-m(^AX)\right]c^2##.
    Binding energy due to Coulomb repulsion between protons
    ##E_c = -\frac{3}{5}\frac{e^2}{4\pi \epsilon_0 R_0} Z(Z-1)A^{-1/3}##.
    Constants:
    ##m(^1H) = 1.007825u##
    ##m_n = 1.00866501u##
    ##m(^{15}O) = 15.003065u##
    ##m(^{15}N)=15.000109u##
    ##1 u = 931.502MeV/c^2##
    ##\frac{e^2}{4\pi \epsilon_0} = 1.439976MeV \cdot fm##.
    3. The attempt at a solution
    I think I solved part a) but I'm having trouble with part b).
    The difference in binding energy should be
    ##\Delta B = \left[ m_n-m(^1H)+m(^{15}O)-m(^{15}N\right] = (0.00379601u)/c^2 = 3.536MeV##. (The answer says ##3.532## but I guess I worked with more decimals)

    The difference in Coulomb energy should be (since ##8\cdot 7 - 7\cdot 6 = 14##)
    ##\Delta B= 3/5 \frac{e^2}{4\pi \epsilon_0R_0} 14A^{-1/3}##
    ##R = R_0A^{1/3} = \frac{3\cdot 14 \cdot 1.439976}{5\cdot 3.536}fm = 3.420752fm##
    The answer however says ##R = 3.67fm## so I did something wrong (using 3.532 instead doesn't change anything)
     
    Last edited: Apr 1, 2016
  2. jcsd
  3. Apr 1, 2016 #2
    check the calculation .
    what value of R0 was taken? As i find people taking different values!
     
  4. Apr 1, 2016 #3
    As I understand the question I should compute ##R_0## from knowing the binding energy (actually ##R_0A^{(1/3)}##).

    The typical value for ##R_0## I've seen is ##R_0 \approx 1.2fm## but from my calculation I have a value of ##R_0 =1.38## (and from the answer I would need an even larger ##R_0##. So perhaps I'm doing something wrong as I understand it ##R_0## should be about the same for a nuclide. I'm not entirely sure I'm using the right formulas either, they were simply the ones that seemed to apply to the question.
     
  5. Apr 1, 2016 #4
  6. Apr 1, 2016 #5
    Essentially I'm using the coulomb term out of the that formula. However the version I have is different from the one on wikipedia. My book says
    ##B = a_vA -a_sA^{(2/3)}-a_cZ(Z-1)A^{(-1/3)}-a_{\text{sym}}\frac{(A-2Z)^2}{A}+\delta.##
    Note the difference in the Coulomb term.
    If I use ##Z^2## I indeed get the correct result just correcting my earlier value with ##3.42\cdot \frac{15}{14} = 3.67## (since ##8^2-7^2=15##).
    Now I'm confused why there's ##Z(Z-1)## in the version In my book. The book argue that the term is proportional to ##Z(Z-1)## since each proton repels each of the other ones.
     
  7. Apr 1, 2016 #6
    Nucleus Ro (in fermis)

    B11 1.28

    C13 1.34

    N15 1.31

    O17 1.26

    F19 1.26

    Ne21 1.25

    Na23 1.22

    Mg23 1.23

    Al27 1.20

    ref.
    www.bhojvirtualuniversity.com/ss/sim/physics/msc_f_phy_p3u1.doc
    The energy discrepancy is because of the use of classical principles instead of quantum mechanical principles in calculating the coulomb energy Ec..
     
  8. Apr 1, 2016 #7
    The difference from wiki is that Wiki Formula is for B/A -binding energy per nucleon - but meanwhile i think its R0 value which changes with nucleids may be responsible for the difference.
     
  9. Apr 1, 2016 #8
    actually its Z(Z-1) but approximated to Z^2 so when you use the former you are very correct. the culprit seems to be R0 value.
     
  10. Apr 1, 2016 #9
    Thanks for all the help! I believe the answer was calculated using the ##Z^2## approximation so that should explain everything. As for the ##R_0## value I'm not entirely sure. I don't use the actual value of ##R_0## in my calculations although I could compute it.
    I'm guessing the difference in ##R_0##(if I compute it) may be because the question only cares about the Coulomb term which isn't the entire truth.
     
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