HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 1

Outline lecture (HL-4)Structure of nuclei

•Shell structure•Deformation

Nuclear reactions•Stripping reaction•DWBA

Literature: PR 16, 17

HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 2

energy of 1. exc. state of even-even nuclei

typ. JP=2+

shell and level structuredetermined bydepth/shape of central potential,spin-orbit force,residual interaction(pairing correlations)

with )(2

1

0)()( with /)()()( 2

rVlE

rVdr

drVslrVrVrV

lsls

centrallslscentral

j=l-1/2j=l+1/2

HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 3

single-particlelevels nlj

shell gapsandintruder states

HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 4

particle- and hole-states and excitations

mirror pair, hole mirror pair, nucleon isospin independence of nuclear force

12/11 p

2/51d

HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 5

deformed nuclei: quadrupole moments

RRZeQ

baZe

xdxxzQ

zyx abbellipsoid

b-a with

5

45

2

3

2

22),,(

322

partially occupied p- and n-shellspolarized the nucleus deformation

prolate

oblate

HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 6

regions of large deformation

half-full shells deformation

Hund rule for atom shells: nl subshell filling: first different orbitals l next substates ml

repulsive Coulomb force favours separate orbitals

HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 7

pairing and polarization attractive NN force favours: pairing: l1 = l2; m1 = -m2; j = 0; JP = 0+

adjacent m-substate fillingoverlapping adjacent m-orbitals deformation

energy levels of deformed nuclei ellipsoidal potential shape

Nilsson Parameters: = lz

= jz

HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 8

extended SPSM: Nilsson model

single-particle motion inanisotropic oscillator field:

slV

ClV

xxxmV

ls

lll

eff

2

1

2

22

21

223

23

quantum numbersN = n3 + n

[Nn3]

deformation - 3

HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 9

deformation parameters: exp. and calc.

= exp. A even = exp. A uneven

= calc. A uneven

HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 10

spectroscopy through nuclear reactions

stripping reaction 16O(d,p)17O

Proton spectrum at 45 and Ed = 25.4 MeV

(n-1d5/2)1

(n-2s1/2)1(n-1d3/2)1

HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 11

single-particlelevels nlj

shell gaps,intruder states

level sequence in sd shell(n-1d3/2)1

(n-2s1/2)1

(n-1d5/2)1

HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 12

stripping cross sectionfor Ed (vd) binding of d and binding of n in A+1 nucleus

A A+1

Fermi´s Golden Rule

ipnffi

dfi

UM

dEv

dppVM

d

d

,

2

3

22

2

2

for weak d-A and p-(A-1) interaction: Born approx.:likely final state with n in outer shell:with shell-model w.f. for n

and short-range stripping interaction:

/exp /exp 1 ppAfdddAi xpixpi

nAA 1

pnU ,

pnpnpn xxUxxU 0, ,

xdxxppixUUM dpdnipnffi3*

0, 0/2/exp

)(2

1,0, pndnp xxxxxx q-transfer to nucleus!

HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 13

DWBA for strippingreaction X(a,b)Y

proton angular distribution for 16O(d,p)17O at Ed = 25.4 MeV

plane waves scattered waves:solutions for complexoptical potential V(r) + iW(r)taking into account absorption by W(r)

is Y=X+(a-b) a single-particle state ??initial-state spin

Y

ba

X JJ jl,

jljl jl

X

Y

jl

d

dS

J

J

d

d

d

d

,, ,

.exp

,

12

12

n transitioparticle singlefor

spectroscopic factors Sl,j measure single-particle character

l=0

l=2

l=2

q=0

excellent agreementfor small q

HL-4 May 2004 Kernfysica: quarks, nucleonen en kernen 14

Summary lecture (HL-4)Nuclear structure•Shell structure and shell gaps appear due to ls term•Deformations in half-filled shells and polarization

Nuclear reactions•Stripping reactions study level structure•DWBA approximation for transfer:

spectroscopic factors Sl,j measure single-particle character