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Nuclear Masses and Binding Energy Lesson 3
Nuclear Masses
• Nuclear masses and atomic masses
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mnuclc2 = Matomicc
2 − [Zmelectronc2 + Belectron (Z)]
Belectron (Z) =15.73Z 7 / 3eV
Because Belectron(Z)is so small, it is neglected in most situations.
Mass Changes in Beta Decay
• β- decay
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14C→14N + β− + ν e
Energy = [(m(14C) + 6melectron ) − (m(14N) + 6melectron ) −m(β
−)]c 2
Energy = [M(14C) −M(14N)]c 2
• β+ decay
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64Cu→64Ni− + β + + ν e
Energy = [(m(64Cu) + 29melectron ) − (m(64Ni) + 28melectron ) −melectron −m(β
+)]c 2
Energy = [M(64Cu) −M(64Ni) − 2melectron ]c2
Mass Changes in Beta Decay
• EC decay
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207Bi+ + e−→207Pb+ ν e
Energy = [(m(207Bi) + 83melectron ) − (m(207Pb) + 82melectron )]c
2
Energy = [M(207Bi) −M(207Pb)]c 2
Conclusion: All calculations can be done with atomic masses
Nomenclature
• Sign convention: Q=(massesreactants-massesproducts)c2
Q has the opposite sign as ΔH Q=+ exothermic Q=- endothermic
Nomenclature
• Total binding energy, Btot(A,Z) Btot(A,Z)=[Z(M(1H))+(A-Z)M(n)-M(A,Z)]c2 • Binding energy per nucleon Bave(A,Z)= Btot(A,Z)/A • Mass excess (Δ) M(A,Z)-A See appendix of book for mass tables
Nomenclature
• Packing fraction (M-A)/A
• Separation energy, S Sn=[M(A-1,Z)+M(n)-M(A,Z)]c2
Sp=[M(A-1,Z-1)+M(1H)-M(A,Z)]c2
Binding energy per nucleon
Separation energy systematics
Abundances
Semi-empirical mass equation
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Btot (A,Z) = avA − asA2 / 3 − ac
Z 2
A1/ 3− aa
(A − 2Z)2
A± δ
Terms
• Volume avA • Surface -asA2/3
• Coulomb -acZ2/A1/3
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ECoulomb =35Z 2e2
RR =1.2A1/ 3
ECoulomb = 0.72 Z 2
A1/ 3
Asymmetry term
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−aa(A − 2Z)2
A= −aa
(N − Z)2
A
To make AZ from Z=N=A/2, need to move q protons qΔ in energy, thus the work involved is q2Δ=(N-Z)2Δ/4. If we add that Δ=1/A, we are done.
Pairing term A Z N # stable e e e 201
o e o 69
o o e 61
e o o 4
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δ = +11A−1/ 2Keeδ = 0Koe,eoδ = −11A−1/ 2Koo
Relative importance of terms
Values of coefficients
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av =15.56MeVas =17.23MeVac = 0.7MeVaa = 23.285MeV
Modern version of semi-empirical mass equation (Myers
and Swiatecki)
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Btot (A,Z) = c1A 1− kN − ZA
⎛
⎝ ⎜
⎞
⎠ ⎟ 2⎡
⎣ ⎢
⎤
⎦ ⎥ − c2A
2 / 3 1− k N − ZA
⎛
⎝ ⎜
⎞
⎠ ⎟ 2⎡
⎣ ⎢
⎤
⎦ ⎥ − c3
Z 2
A1/ 3+ c4
Z 2
A+ δ
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c1 =15.677MeVc2 =18.56MeVc3 = 0.717MeVc4 =1.211MeVk =1.79δ =11A−1/ 2
Mass parabolas and Valley of beta stability
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M(Z,A) = Z •M(1H) + (A − Z)M(n) − Btot (Z,A)
Btot (Z,A) = avA − asA2 / 3 − ac
Z 2
A1/ 3− aa
(A − 2Z)2
A
aa(A − 2Z)2
A= aa
A2 − 4AZ + 4Z 2
A= aa A − 4Z +
4Z 2
A⎛
⎝ ⎜
⎞
⎠ ⎟
M = A M(n) − av +asA1/ 3
+ aa⎡
⎣ ⎢ ⎤
⎦ ⎥ + Z M(1H) −M(n) − 4Zaa[ ] + Z 2 ac
A1/ 3+4aaA
⎛
⎝ ⎜
⎞
⎠ ⎟
This is the equation of a parabola, a+bZ+cZ2
Where is the minimum of the parabolas?
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∂M∂Z⎛
⎝ ⎜
⎞
⎠ ⎟ A
= 0 = b + 2cZA
ZA =−b2c
=M(1H) −M(n) − 4aa2 acA1/ 3
+4aaA
⎛
⎝ ⎜
⎞
⎠ ⎟
ZA
A≈12
8180 + 0.6A2 / 3
Valley of Beta Stability
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