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Nuclear Masses and Binding Energy Lesson 3

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Page 1: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Nuclear Masses and Binding Energy Lesson 3

Page 2: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Nuclear Masses

•  Nuclear masses and atomic masses

mnuclc2 = Matomicc

2 ! [Zmelectronc2 + Belectron (Z)]

Belectron (Z) =15.73Z 7 / 3eV

Because Belectron(Z)is so small, it is neglected in most situations.

Page 3: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Mass Changes in Beta Decay

•  β- decay

14C!14N + "# + $ e

Energy = [(m(14C) + 6melectron ) # (m(14N) + 6melectron ) #m("

#)]c 2

Energy = [M(14C) #M(14N)]c 2

• β+ decay

64Cu!64Ni" + # + + $ e

Energy = [(m(64Cu) + 29melectron ) " (m(64Ni) + 28melectron ) "melectron "m(#

+)]c 2

Energy = [M(64Cu) "M(64Ni) " 2melectron ]c2

Page 4: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Mass Changes in Beta Decay

•  EC decay

207Bi+ + e!"207Pb+ # 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

Page 5: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Nomenclature

•  Sign convention: Q=(massesreactants-massesproducts)c2

Q has the opposite sign as ΔH Q=+ exothermic Q=- endothermic

Page 6: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

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

Page 7: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

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

Page 8: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Binding energy per nucleon

Page 9: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Separation energy systematics

Page 10: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Abundances

Page 11: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Semi-empirical mass equation

Btot (A,Z) = avA ! asA2 / 3 ! ac

Z 2

A1/ 3! aa

(A ! 2Z)2

A± "

Terms

• Volume avA • Surface -asA2/3

• Coulomb -acZ2/A1/3

ECoulomb = 35Z 2e2

RR =1.2A1/ 3

ECoulomb = 0.72 Z 2

A1/ 3

Page 12: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Asymmetry term

!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.

Page 13: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Pairing term A Z N # stable e e e 201

o e o 69

o o e 61

e o o 4

! = +11A"1/ 2…ee! = 0…oe,eo! = "11A"1/ 2…oo

Page 14: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Relative importance of terms

Page 15: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Values of coefficients

av =15.56MeVas =17.23MeVac = 0.7MeVaa = 23.285MeV

Page 16: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Modern version of semi-empirical mass equation (Myers

and Swiatecki)

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+ .

c1 =15.677MeVc2 =18.56MeVc3 = 0.717MeVc4 =1.211MeVk =1.79! =11A"1/ 2

Page 17: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M
Page 18: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Mass parabolas and Valley of beta stability

M (Z,A) = Z •M (1H )c2 + (A − Z )M (n)c2 − 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)c2 − av +asA1/3

+ aa⎡⎣⎢

⎤⎦⎥+ Z M (1H )c2 −M (n)c2 − 4aa⎡⎣ ⎤⎦ + Z

2 acA1/3

+ 4aaA

⎛⎝⎜

⎞⎠⎟

This is the equation of a parabola, a+bZ+cZ2

Page 19: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M
Page 20: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M
Page 21: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M
Page 22: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Where is the minimum of the parabolas?

!M!Z

" # $

% & ' A

= 0 = b + 2cZA

ZA = (b2c

= M(1H) (M(n) ( 4aa

2 acA1/ 3

+ 4aaA

" # $

% & '

ZA

A) 12

8180 + 0.6A2 / 3

Page 23: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M

Valley of Beta Stability

Page 24: Nuclear Masses and Binding Energy - Oregon State Universityoregonstate.edu/instruct/ch374/ch418518/lecture3-1.pdf · Nuclear Masses • Nuclear masses and atomic masses m nucl c 2=M