Global Properties of Atomic Nuclei Global Properties of Atomic Nuclei Sizes Sizes

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Global Properties of Atomic Nuclei Global Properties of Atomic Nuclei

Sizes Sizes

homogenous charge distribution finite diffuseness

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ρ 0( ) =0.17nucleons/fm³ ρ r( )=ρ₀ 1+expr −R

a

⎛

⎝ ⎜ ⎞

⎠ ⎟

⎡

⎣ ⎢ ⎤

⎦ ⎥

−1

R ≈1.2A^1/3fm, a ≈0.6fm

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Nuclear sizes (2) Nuclear sizes (2)

Calculated and experimental densities

(3)

Binding Binding

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m(N,Z) = 1

c² E(N,Z) =NM_n +ZM_H − 1

c² B(N,Z)

The mass of a nucleus consisting of A nucleons is not entirely determined by the masses of the nucleons. The difference (the “mass defect”) is the binding energy: that energy required to disassemble the nucleus. Note that the mass is defined in terms of atomic masses (it includes the electron masses).

The binding energy contributes significantly (~1%) to the mass of a nucleus. This implies that the constituents of two (or more) nuclei can be rearranged to yield a different and perhaps greater binding energy and thus points towards the existence of nuclear reactions in close analogy with chemical reactions amongst atoms.

B is positive for a bound system!

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Binding (cont.) Binding (cont.)

The sharp rise of B/A for light nuclei comes from increasing the number of nucleonic pairs. Note that the values are larger for the 4n nuclei (-particle clusters!). For those nuclei, the difference

divided by the number of alpha-particle pairs, n(n-1)/2, is roughly constant (around 2 (MeV). This is nice example of the saturation of nuclear force.

The associated symmetry is known as SU(4), or Wigner supermultiplet symmetry.

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ΔE ≡B(N,Z)−nB(2,2)

4He

(5)

• For most nuclei, the binding energy per nucleon is about 8MeV.

• Binding is less for light nuclei (these are mostly surface) but there are peaks for A in multiples of 4. (But note that the peak for ⁸Be is slightly lower than that for ⁴He.

• The most stable nuclei are in the A~60 mass region

• Light nuclei can gain binding energy per nucleon by fusing; heavy nuclei by fissioning.

• The decrease in binding energy per nucleon for A>60 can be ascribed to the repulsion between the (charged) protons in the nucleus: the

Coulomb energy grows in proportion to the number of possible pairs of protons in the nucleus Z(Z-1)/2

• The binding energy for massive nuclei (A>60) thus grows roughly as A;

if the nuclear force were long range, one would expect a variation in proportion to the number of possible pairs of nucleons, i.e. as A(A-1)/2.

The variation as A suggests that the force is saturated; the effect of the interaction is only felt in a neighborhood of the nucleon.

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The semi-empirical mass formula, based on the liquid drop model, considers five contributions to the binding energy(Bethe-Weizacker 1935/36)

Binding Binding

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B =a_volA−a_surfA^2/3 −a_sym(N −Z)²

A −a_C Z²

A^1/3 −δ(A) 15.68 -18.56 -28.1 -0.717

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δ(A) =

−34A^−3/4 for even- even 0 for even- odd

34A^−3/4 for odd-odd

⎧

⎨ ⎪

⎪

⎩ ⎪

⎪ pairing term

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The semi-empirical mass formula, based on the liquid drop model, compared to the data

stability valley

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∂B

∂N

⎛

⎝ ⎜ ⎞

⎠ ⎟

A=const

=0

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∂B

∂N

⎛

⎝ ⎜ ⎞

⎠ ⎟

Z=const

=0

neutron drip line

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∂B

∂Z

⎛

⎝ ⎜ ⎞

⎠ ⎟

N =const

=0

proton drip line

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Separation Energies Separation Energies

A = 21

(9)

Separation Energies Separation Energies

shell closure subshell deformation

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Odd-even mass difference Odd-even mass difference

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Δ_n=B(N,Z) −B(N +1,Z) +B(N −1,Z) 2

Δ_p =B(N,Z)−B(N,Z+1) +B(N,Z −1) 2

A common phenomenon A common phenomenon in mesoscopic systems!

in mesoscopic systems!

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Finite-range droplet model: Möller, Nix,Myers, Swiatecki, 1994 Finite-range droplet model: Möller, Nix,Myers, Swiatecki, 1994

E(Z, N,shape) = M_HZ +M_nN mss excess of Z hydrogen toms nd N neutrons + −₍ ₁ + J δ ² −¹₂ K ε²₎A volume energy

+ −₍ ₂B₁+ ⁹₄J²δ ²B_s²/ QB₁₎A^{2 / 3} surfce energy +₃A^{1/ 3}B_k curvture energy

+₀A⁰ A⁰ energy +c₁ Z²

A^{1/ 3} B₃ Coulomb energy

−c₂Z²A^{1 / 3}B_r volume redistribution energy

−c₄ Z^{4 / 3}

A^{1 / 3} Coulomb exchnge correction

−c₅Z²B_wB_s/B₁ surfce redistribution energy + f₀ Z²

A proton form-fctor correction to the Coulomb energy

−c_₍N−Z) chrge-symmetry energy +W I + 1/A, Z nd N odd nd equl

0, otherwise

⎧⎨

⎩

⎛

⎝⎜ ⎞

⎠⎟ Wigner energy

+

+ Δ_p + Δ_n −δ_np, Z nd N re odd + Δ_p, Z odd nd N even + Δ_n, Z even nd N odd +0, Z nd N re even

⎧

⎨⎪

⎪

⎩

⎪⎪

verge piring energy

−_elZ^2.39 energy of bound electrons E(Z, N,shape) =

M_HZ+M_nN mss excess of Z hydrogen toms nd N neutrons + −₍ ₁ + J δ ² −¹₂ K ε ²₎A volume energy

+ −₍ ₂B₁+ ⁹₄J²δ ²B_s²/QB₁₎A^{2 / 3} surfce energy +₃A^{1/ 3}B_k curvture energy

+₀A⁰ A⁰ energy +c₁ Z²

A^{1/ 3} B₃ Coulomb energy

−c₂Z²A^{1 / 3}B_r volume redistribution energy

−c₄ Z^{4 / 3}

A^{1 / 3} Coulomb exchnge correction

−c₅Z²B_wB_s/ B₁ surfce redistribution energy + f₀ Z²

A proton form-fctor correction to the Coulomb energy

−c_₍N−Z) chrge-symmetry energy +W I + 1/A, Z nd N odd nd equl

0, otherwise

⎧⎨

⎩

⎛

⎝⎜ ⎞

⎠⎟ Wigner energy

+

+ Δ_p + Δ_n −δ_np, Z nd N re odd + Δ_p, Z odd nd N even + Δ_n, Z even nd N odd +0, Z nd N re even

⎧

⎨⎪

⎪

⎩

⎪⎪

verge piring energy

−_elZ^2.39 energy of bound electrons

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Global Properties of Atomic Nuclei Global Properties of Atomic Nuclei Neutron Star, a bold explanation Neutron Star, a bold explanation

A lone neutron star, as seen by NASA's Hubble Space Telescope.

The Hubble results show the star is very hot (1.2 million degrees Fahrenheit at the surface), and can be no larger than 16.8 miles across.

These results prove that the object must be a neutron star, because no other known type of object can be this hot, small, and dim.

A lone neutron star, as seen by NASA's Hubble Space Telescope.

The Hubble results show the star is very hot (1.2 million degrees Fahrenheit at the surface), and can be no larger than 16.8 miles across.

These results prove that the object must be a neutron star, because no other known type of object can be this hot, small, and dim.

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B =a_volA−a_surfA^2/3 −a_sym(N −Z)²

A −a_C Z²

A^1/3 −δ(A) +3 5

G

r₀A^1/3 M²

Let us consider a giant neutron-rich nucleus. We neglect Coulomb, surface, and pairing energies. Can such an object exist?

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B =a_volA−a_symA+3 5

G

r₀A^1/3(m_nA)² =0 limiting condition

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3 5

G

r₀ m_n²A^2/3=7.5MeV ⇒ A ≅5×10⁵⁵,R ≅4.3km,M ≅0.045M .

More precise calculations give M(min) of about 0.1 solar mass. Must neutron stars have

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R ≅10km,M ≅1.4M .

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experiment experiment

theory theory

discrepancy discrepancy

20 60 100 140

Number of neutrons -10

0 10

-10 0

Shell Energy (MeV)

Nuclei

20 28

50 82 126

0 -1 1

50 100 150 200

-1 0 1

Number of electrons

Shell Energy (eV)

Na Clusters

58 92 138 198

experiment experiment

theory theory

deformed clusters deformed

clusters

spherical clusters spherical

clusters deformed

nuclei deformed

nuclei

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fission and fusion fission and fusion

• All elements heavier than A=110-120 are fission unstable!

• But… the fission process is unimportant for nuclei with A<230. Why?

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The classical droplet stays stable and spherical for x<1.

For x>1, it fissions immediately.

For ²³⁸U, x=0.8.

The classical droplet stays stable and spherical for x<1.

For x>1, it fissions immediately.

For ²³⁸U, x=0.8.

5-10 MeV

E_B

several hundred

MeV

deformation V(s)

The LDM alone cannot produce

stable deformations!

Shell correction!!!

The LDM alone cannot produce

stable deformations!

Shell correction!!!

E_LDM(def ) = E_S( )0 _[B_S₍def_{) −1+ 2}x B₍ _C₍def_{) −1}₎_] B_S(def) = E_S(def)

E_S( )0 , B_C(def) = E_C(def) E_C( )0 x = E_C( )0

2E_S( )0 = Z² / A Z² / A

( )_crit≈ Z² 50A

E_LDM(def ) = E_S( )0 _[B_S₍def_{) −1+ 2}x B₍ _C₍def_{) −1}₎_] B_S(def) = E_S(def)

E_S( )0 , B_C(def) = E_C(def) E_C( )0 x = E_C( )0

2E_S( )0 = Z² / A Z² / A

( )_crit≈ Z² 50 A

fissibility parameter

spontaneous fission spontaneous fission

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spontaneous fission (cont.) spontaneous fission (cont.)

Realistic

calculations

240Pu

1938 - Hahn & Strassmann 1939 Mwitner & Frisch 1939 Bohr & Wheeler 1940 Petrzhak & Flerov

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Global Properties of Atomic Nuclei Global Properties of Atomic Nuclei spontaneous and induced fission (cont.) spontaneous and induced fission (cont.)

Fragment distribution Fragment distribution

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spontaneous fission (cont.) spontaneous fission (cont.)

(19)

spontaneous and induced fission spontaneous and induced fission

neutron multiplicities neutron multiplicities

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92

235U +n→₃₆⁹²Kr+¹⁴²₅₆Ba+2n +180 MeV

235U

238U

Small n implies a small critical mass (the smallest mass for which the chain reaction is self-sustaining)

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Shape deformations Shape deformations

The question of whether nuclei can rotate became an issue already in the very early days of nuclear spectroscopy

•Thibaud, J., Comptes rendus 191, 656 ( 1930)

•Teller, E., and Wheeler, J. A., Phys. Rev. 53, 778 (1938)

•Bohr, N., Nature 137, 344 ( 1936)

•Bohr, N., and Kalckar, F., Mat. Fys. Medd. Dan. Vid. Selsk. 14, no, 10 (1937)

The first evidence for a non-spherical nuclear shape came from the observation of a quadrupole component in the hyperfine structure of optical spectra. The analysis showed that the electric quadrupole moments of the nuclei concerned were more than an order of

magnitude greater than the maximum value that could be attributed to a single proton and suggested a deformation of the nucleus as a

whole.

•Schüler, H., and Schmidt, Th., Z. Physik 94, 457 (1935)

•Casimir, H. B. G., On the Interaction Between Atomic Nuclei and Electrons, Prize Essay, Taylor’s Tweede Genootschap, Haarlem (1936)

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x z

y

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R(θ,ϕ) =c(α)R₀1+ α_λμ^* Y_λμ(θ,ϕ)

μ=−λ

∑

λ λ=1

⎡

∑

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

radius of the sphere with the same volume

deformation parameters For axial shapes m=0 volume conservation

a) l=1 (dipole); m=-1,0,1

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r r

V

∫ ^d

³

^{r =0}

center of mass conservation 3 conditions, they fix _1m

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β

_λ

≡α

_λμ

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Shape deformations (cont.) Shape deformations (cont.) b) l=2 (quadrupole); m=-2,-1,0,1,2

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α

₂₁

=α

₂₋₁

=0, α

₂₂

=α

₂₋₂

Only two deformation parameters left:C 3 conditions, they fix three Euler angles

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α

₂₀

=βcosγ, α

₂₂

= 1

2 β sinγ

(b,g are the so-called Hill- Wheeler coordinates)

c) l=3 (octupole)

d) l=4 (hexadecupole) e) …

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Shape deformations (cont.) Shape deformations (cont.)

Shape Convention Shape Convention

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⇒ 0≤β <∞, 0

⁰

≤γ≤60

⁰

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Shape of a charge distribution in ¹⁵⁴Gd

Theory: Hartree-Fock

experiment: (e,e’) Bates

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Shape deformations (cont.) Shape deformations (cont.)

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140 160 180 200 10¹

10²

neutron energy (eV)

cross section (barn)

neutron resonances neutron resonances

in in ²³³²³³ThTh

Level density Level density

148148

Gd Gd

148148

Gd Gd

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Level density Level density

Fermi energy Fermi

energy

Ground state Excited state

As one increases the excitation energy, many particles can be promoted and the number of ways to create many-body stares increases

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ρ_A(E ) =dN dE

Level density: number of nuclear states per energy bin!

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Level density (cont.) Level density (cont.)

Fermi gas model formula.

Based on independent- particle model of non-

interacting fermions (Bethe 1937)

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ρ

_A

( ) = E 1

12a

^1/4

E

^5/4

e

^{2 aE}

level density parameter

Interaction introduces fluctuation in the spectrum over the smooth form given by the Fermi gas model formula. It alsi introduces a shift in the energy scale by some amount .

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ρ_A( ) =E 1

12a^1/4(E −Δ)^5/4 e^{2 a(E−Δ)} Back-shifted Fermi gas model formula.

56Fe

(29)

Level density (cont.) Level density (cont.)

level density parameter

neutron evaporation spectra, Ag

Global Properties of Atomic Nuclei Global Properties of Atomic Nuclei Sizes Sizes

A = 21

Nuclei

Na Clusters

∑

∑

€

r r

∫ d

r =0

€

β

≡α

€

α

=α

=0, α

=α

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α

=βcosγ, α

= 1

2 β sinγ

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⇒ 0≤β <∞, 0

≤γ≤60

Gd Gd

Gd Gd

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ρ

( ) = E 1

12a

E

e

∫ ^d

^{r =0}