Shapes

Nuclear shapes

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)

Can perfectly spherical nucleus rotate?

Theory: Hartree-Fock

experiment: (e,e’) Bates

Shape of a charge distribution in 154Gd

Nuclear deformation: Jahn-Teller effect. The Jahn–Teller theorem (1937) states that any nonlinear molecule with a spatially degenerate electronic ground state will undergo a geometrical distortion that removes that degeneracy, because the distortion lowers the overall energy of the species.

Cs3C60

http://www.nature.com/ncomms/journal/v3/n6/full/ncomms1910.html

The intrinsic shape of the deuteron by combining the results from experiments at JLab

How to describe nuclear shapes?

x

z

y

€

R(θ,ϕ) = c(α)R0 1+ αλµ* Yλµ(θ,ϕ)

µ =−λ

λ

∑λ=1∑

(

) * *

+

, - -

radius of the sphere with the same volume

deformation parameters For axial shapes µ=0

volume conservation

€

βλ ≡αλµ

a) λ=1 (dipole); µ=-1,0,1 !r

V∫ d3r = 0 center of mass conservation

3 conditions, they fix α1µ

b) λ=2 (quadrupole); µ=-2,-1,0,1,2

€

α21 = α2−1 = 0, α22 =α 2− 2

Only two deformation parameters left (Hill- Wheeler coordinates):

3 conditions, they fix three Euler angles

α20 = β cosγ, α22 =12β sinγ

c) λ=3 (octupole) d) λ=4 (hexadecapole) e) …

~1 s.p.u.

~400 s.p.u.

S. Raman et al., Atomic Data & Nuclear Data Tables 78, 1

0 20 40 60 80 100 120 140 160Neutron Number N

2

5

10-1

2

5

100

2

5

101

2

Energyof2+ 1(MeV)

(a)

LINES CONNECT ISOTOPES

N =8 N =20N =28 N =50 N =82 N =126

0 20 40 60 80 100 120 140 160Neutron Number N

10-3

2

5

10-2

2

5

10-1

2

5

100

2

5

101

2

B(E2)(e2 b2 )

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............................(b)

LINES CONNECT ISOTOPES

N =8

N =20

N =28

N =50 N =82 N =126

Q1 Q

E shape coexistence

Q2

Q0 Q

E fission/fusion exotic decay heavy ion coll.

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

40

42

44

3

5

78

9

11 1010

1213

14

15

15 17

18

19

21

(22 )

2324

25 (24 )27

2829

30

31

32

33

34

35

36

(24 )

(26 )

(28 )

(30 )

(32 )

(34 )

(36 )

(38 )

(40 )

(42 )

(44 )

(46 )

(48 )

(34)

(36)

(38)

(40)

(42)

(44)

(46)

(48)

(50)

(52)

(54)

(36)

(38)

(40)

(42)

(44)

(46)

(48)

(50)

(52)

(54)

(56)

(27 )

(29 )

(31 )

(33 )

(35 )

(37 )

(39 )

(41 )

(43 )

(45 )

(47 )

(49 )

(26 )

(28 )

(30 )

(32 )

(34 )

(36 )

(38 )

(40 )

(42 )

(44 )

(46 )

(48 )

(32 )

(34 )

(36 )

(38 )

(40 )

(42 )

(44 )

(46 )

(48 )

(50 )

(52 )

613.8

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398.0 560.8492.9

758.5

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254.6

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693

636.5 646.8659.5808.6831634

461 610.0

622 742 765.3 984.8

229.9300.1604.5

53.4547

253.6

566525.2

710 770

262.3305 358609670

625

681

382

990.7

490

541.2 483

973754

220.6

304 456825

967.0 1063

147.4

402.3895

890

6131016

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858962

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133 254.2

684.9703851

248668

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613 779.61061

1345

191257

1064

388.6 520

320 386

970

362750

248

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1093

288538

755

(518)

9831086

792

1858

348

1071 121811681192 1362

10051159

763976 10201222

12831912

11421341

1399

1454

602.4

647.5

692.7

738.1

784.0

829.9

876.4

923.2

970.2

1017.4

1064.9

1112.7

826

855.4

895.2

932.2

970

1009

1049.2

1092.6

1137.6

1183.4

1230.6

793.0

839.7

891.2

944.8

998.2

1052.4

1104.9

1156.9

1207.9

1260.4

670

721.0

772.8

825.0

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929.0

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747.0

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1055.0

1105.0

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761.5

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1077.3

1122.7

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152Dy

triaxialband

noncollectivestates

superdeformedbands

Fission

Fission

1938 Hahn & Strassmann 1939 Meitner & Frisch 1939 Bohr & Wheeler 1940 Petrzhak & Flerov

elongation necking

split

N,Z

N2,Z2 N1,Z1

N=N1+N2 Z=Z1+Z2

Fission yields (fragments)

Understanding the fission process is crucial for many areas of science and technology: •  Fission governs the production and existence of many

transuranium elements, including the predicted long-lived super-heavy species.

•  Fission influences the formation of heavy elements in a neutron rich environment.

•  Fission produces reactor antineutrinos •  Improved understanding of the fission process will

enable scientists to enhance the safety and reliability of nuclear reactors.

•  Fission is important for stockpile stewardship

The new phase in fission theory is expected to rely heavily on advanced modeling and simulation capabilities utilizing massively parallel leadership-class computers

© 1939 Nature Publishing Group

1939: Bohr’s paper on fission

© 1939 Nature Publishing Group

© 1939 Nature Publishing Group

ELDM def( ) = ES 0( ) BS def( ) −1+ 2x BC def( ) −1( )[ ]BS def( ) = ES def( )

ES 0( ), BC def( ) = EC def( )

EC 0( )

x =EC 0( )2ES 0( )

=Z 2 / AZ 2 / A( )

crit

≈Z2

50A

Deformed liquid drop (Bohr & Wheeler, 1939)

fission of nuclear droplet

x: fissility parameter

The nuclear droplet stays stable and spherical for x<1. For x>1, it fissions immediately. For 238U, x=0.8

Realistic calculations Nature 409, 785 (2001)

• All elements heavier than A=110-120 are fission unstable! • But… the fission process is fairly unimportant for nuclei with A<230. Why?

240Pu

Realistic calculations

136 140 144 148 152 156 160 16410-9

10-4

101

106

1011

1016

1021

1026

Status: 6.3.2012

N = 162

N = 152

Hs

SgRfNo

Fm

Cf

Cm

Pu

U

T sf /

sNeutron number

238U lives 4.5 billion years 250No fissions after 4.2 µs

EM fission of RNBs at GSI, E*~11 MeV K-H Schmidt et al., NPA 665, 221 (2000)

TKE

Third minimum around 232Th?

Phys. Rev. C 87, 054327 (2013)

2468 (a) (b)

2468 (c) (d)

0 100 200 0 100 200Q20 (b)

E (M

eV) 226Th

228Th230Th232Th

228U230U232U234U

UNEDF1 UNEDF1

SkM* SkM*

Phys. Rev. C 85, 054306 (2012)

deformation

180Tl

QEC

Bf

180Hg

• 2 step process: β+/EC decay of a parent 180Tl nucleus populates an excited state in the 180Hg daughter, which then might fission (in competition with the γ decay to the g.s.)

• Low-energy fission! (E*<QEC=10.8 MeV) • 10 cases know so far (neutron-def. Uranium region)

γ

γ

γ

β+/EC

β+/EC

Curious Fission of 180Hg

Phys. Rev. Lett. 105, 252502 (2010); Rev. Mod. Phys. 85 1541 (2013)

•  Before the ISOLDE experiment: expected SYMMETRIC split in two semi-magic 90Zr

•  The most probable fission fragments are 100Ru (N=56,Z=44) and 80Kr (N=44,Z=36)

http://www.nature.com/news/2010/101201/full/news.2010.642.html

WKB:

multidimensional space of collective parameters

collective inertia (mass parameter)

The action has to be minimized!

Fission half-lives: depend on potential, friction, and inertia terms

Phys. Rev. C 80, 014309 (2009)

Quality  Input  

Numerical  Techniques  

Large-­‐scale  Simula0ons  on  Leadership-­‐class  Computers  

Dynamics  

Low-energy fission: theoretical strategy

PRC 78, 014318 (2008)

PRC 84, 054321(2011)

PRC 80, 014309 (2009) Confronta0on  with  experiment;  predic0ons  

PRC 80, 014309 (2009)

PRC 85, 024304 (2012)

https://people.nscl.msu.edu/~witek/Talks/Fission-JINA.pptx