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