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

- 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

+ 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 ]c 2

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 HQ=+ exothermicQ=- 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 nucleonBave(A,Z)= Btot(A,Z)/A

• Mass excess ()M(A,Z)-ASee appendix of book for mass tables

Nomenclature

• Packing fraction(M-A)/A

• Separation energy, SSn=[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 =3

5

Z 2e2

R

R =1.2A1/ 3

ECoulomb = 0.72Z 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 qin energy, thus the work involvedis q2=(N-Z)2/4. If we add that=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/ 2K ee

δ = 0K oe,eo

δ = −11A−1/ 2K oo

Relative importance of terms

Values of coefficients

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av =15.56MeV

as =17.23MeV

ac = 0.7MeV

aa = 23.285MeV

Modern version of semi-empirical mass equation (Myers and

Swiatecki)

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Btot (A,Z) = c1A 1− kN − Z

A

⎛

⎝ ⎜

⎞

⎠ ⎟2 ⎡

⎣ ⎢

⎤

⎦ ⎥− c2A

2 / 3 1− kN − Z

A

⎛

⎝ ⎜

⎞

⎠ ⎟2 ⎡

⎣ ⎢

⎤

⎦ ⎥− c3

Z 2

A1/ 3+ c4

Z 2

A+ δ

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c1 =15.677MeV

c2 =18.56MeV

c3 = 0.717MeV

c4 =1.211MeV

k =1.79

δ =11A−1/ 2

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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 +as

A1/ 3+ aa

⎡ ⎣ ⎢

⎤ ⎦ ⎥+ Z M(1H) − M(n) − 4Zaa[ ] + Z 2 ac

A1/ 3+

4aa

A

⎛

⎝ ⎜

⎞

⎠ ⎟

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 =−b

2c=M(1H) − M(n) − 4aa

2ac

A1/ 3+

4aa

A

⎛

⎝ ⎜

⎞

⎠ ⎟

ZA

A≈

1

2

81

80 + 0.6A2 / 3

Valley of Beta Stability

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