Nuclear Rotation

Experimental evidences for collective modes

Deformed Nuclei, shell model

Indicators for rotational collectivity: moment of inertia, electric quadrupole moment, lifetime measurements

The energy spectra shope different structuresdepending on the number n of nucleons outside the closed shell

shell modelstates:

n = ± 1, ± 2 …middle shell:

simple spectra

Even-even Nuclei: how can one interprete the energies of the 2+ states?

E [M

eV]

A

2 MeV

100 keV

energy forbreaking a pair

Energy of 2+ state decreases with A

0+ (ground state): all nucleons are coupled to spin 0

2+ (first excited state): in middle shells the nucleus realizes anintrinsic configuration energetically

more favoured than the breaking of a pair

even-even Nuclei: how to explain E(4+)/E(2+) and Q(2+) ?E(

4+)/

E(2

+ ) [M

eV]

A

3.3

2

A<150: E(4+)/E(2+) ~ 2150<A<190 e A>230: E(4+)/E(2+) ~ 3.3

Q(2

+ ) [b] A

Q ∝ <r2>

E(4+)/E(2+) ratio

A<150 Q(2+) ~ 0150<A<190 e A>230: Q(2+) grande

electric quadrupole Q(2+)

Two different collective behaviour : A<150 vibration of a stable spherical nucleus

150<A<190 rotation of a stable deformed nucleus

Collective Vibrations and Rotations

Vibrational Nucleus

120Te

I Energy [keV]0+

2+

4+

6+

0

560.6

1161.9

1776.6ωh⋅= nE n

ωh

E(4+)/E(2+) ~ 2

Rotational Nucleus

168Yb

I Energy [keV]087285.8584.5

969.1

1424.5

1935.1

0+2+4+6+

8+

10+

12+1)(

2

2

+ℑ

= IIE h

E(4+)/E(2+) ~ 3.3

Rotational Motion: It can be observed only in nucleiwith stable equilibrium deformation

deformed nuclei

[ ]),(1),( 20 φθβφθ YRR av +=

The shape of the nucleus isrepresented byan ellipsoid of revolution:

deformation parameter β(ellipsoid eccentricity):

3/10

534

ARR

RR

av

av

=

∆=

πβ

50 100 150 2000

50

100

150

0

Number of neutrons N

Num

bero

f pro

tons

P

β0.450.400.350.300.250.200.150.100.05

ground statequadrupole deformation

β>0 prolate β<0 oblate

Nuclear Shapes

the deformed potential givesthe nuclear shape

(Bohr & Mottelson, 1950)

deformed harmonic oscillator

f7/2 [n,l,j,mj]εΩ

appearance of new magic numbers:• superdeformed nuclei (2:1)• hyperdeformed nuclei (3:1)

1:2 2:1 3:1

the energy levels loosethe (2j+1) degeneracy

Nilsson diagram for neutrons in a prolate deformed potentialen

ergy

/hω

β

[N,l,j,mj]π = (-1)N

Ω = mj

±1/2 ±3/2±5/2

±7/2

at ω=0 the energy levels show a 2 fold degeneracy: ±mj

f7/2

New magic numbersNew minima at larger deformations

Rotation removes the time-reversal invariance

ωr

Hω=Ho-hωjx

The Coriolis interaction gives rise to forces of opposite sign,

depending whether a nucleon moves clockwise or anti-clockwise

⇒ splitting of ±mj energy levels

⇒ changes of shell structureswith rotation Appearance of favorite deformed

minima at high spins

la rotazione provoca una rottura nella degenerazione in mj

Nilsson pairingcranking

[N,l,j,±mj][N,l,j,±mj]

[N,l,j,+mj][N,l,j,-mj]

±1/2 ±3/2

±5/2

±7/2

ad ω=0 i livelli hanno degenerazione 2

Yrast

projectilenucleus

targetnucleus

fusionfast

fission10-22 seccompound nucleus

formation

hω ∼ 0.75 MeV∼ 2×1020 Hzrotation

10-19sec

10-15sec

10-9 sec

groundstateI

E* E1

E2

compound nucleusγ−decay

Heavy ion reactionsallow to populatenuclear states

at high angular momenta( ≥40 h)

A

l(h)

Angular momentum limits(liquid drop calculations)

Bf=8 MeV

Bf= 0triaxial

oblate

Eγ1 Eγ2

γ spectrum

1

24

5

36

Eγ2

Eγ1

10-15 sec

γ detector∆t ∼ 10-8 sec

Eγ2

Eγ1

γ cascade

Eγ [keV]200 400 600

rotational energy of a body withmoment of inerzia ℑangular velocity ω = I/ ℑ

Eγ

rotationalband

12+

10+

8+

6+4+2+0+

1)(222

1 222 +

ℑ→

ℑ=ℑ= IIIE hω

even-even nuclei: 0+, 2+, 4+, 6+, …

E(4+)/E(2+) ~ 3.3

γE∆

ℑ=−+=∆

ℑ=−+=

+ℑ

=

2

2

2

4)()2(

42

)()2()(

1)(2

)(

h

h

h

IEIEE

IIEIEIE

IIIE

γγγ

γ

Channel number

The nucleus is NON a rigid body, is NON an irrotational fluid

)31.01(52 2 β+=ℑ avrigid MR

βπ

2

89

avfluid MR=ℑ

rigidexpfluid ℑ<ℑ<ℑ

this is a consequence of the short range nature of the nuclear force:strong forces exist only among close nucleons→ The nucleus does not show the long range structure typical of rigid bodies

Additional evidence for lack of rigididy:

2'0

0 1)(

ωkIIk

+ℑ=ℑ

++ℑ=ℑ

back-bending

ℑ increases with the rotation(as it happens in fluids, but not in rigid bodies)

“centrifugal stretching”

ℑ is NON constant with I

1)(2

)(2

+ℑ

= IIIE h

ℑ changes with I

back-bending takes places whenthe rotational energy exceeds the energyrequired to break a pair of nucleons

Unpaired nucleons go to different orbitsand change the momet of inertia

The nucleus can constract the rotation in two different ways:

)(1)(2

)(2

iERRIE ++ℑ

=h

iiRI +ℑ=+= ω

collective motion single particlemotion

158Er 147Gd

Deformed nucleus quasi-spherical nucleus

Indicator of collectivity of the nuclear system: the quadrupole moment Q0

Q0 measures the deviation from a symmetric distributionof the nuclear charge distribution

[ ][ ]

000

)(3

)(32222

2222

<>=

><+><+><−><=

++−= ∫zyxzZ

dzyxzQ τρ

spherical shape <z2> = <x2>=<y2>prolate: elongation along z <z2> > <x2>=<y2>oblate: flattening along z <z2> < <x2>=<y2>

oblate

z

prolatez

Large electric quadrupole moments indicate a stable deformation :

measured quadrupole moment+−= 2

72

0 forQQ

)16.01(53 2

0 ββπ

+= ZRQ av

intrinsec quadrupole momenti (osservable only in the intrinsic reference frame)

Q0 can be obtained from the B(E2) reduced transition probability: 22

02 0020

165):2( fifi JJQeJJEBπ

=→

220

2

165)02:2( βπ

∝=→ ++ QeEB

225 )1(08156.0

)2( beE

BEB

totατγ

γ

+=

s13103.0−≈

≈

τ

β

s111005.0−≈

≈

τ

β

γ decay probability=

=

tot

Bα

γ

probability for internal conversion

High Spin limit: 0,1 == totB αγ

sEEB

125 10

)2(08156.0 −×=

γ

τ

in W.U.

γEEB )2(

in MeV

423/40594.01 fmeAWU =expected intensity for 1 transitioninvolving only 1 nucleon

Measurement of nuclear lifetimes τ

τ > 10−3 sdirect measure

10−3 < τ < 10−11 sDelayed coincidence technique

teNtN λ−= 0)(

10−10 < τ < 10−12 sDoppler-recoil method

plunger setupthe produced nuclei escape from the taget

with a velocity v/c

they decay emitting γ’s which are Doppler shifted

the nuclei qre finally completely stoppedinto a stopper, therefore decaying with v/c=0

two peaks are observed:shifted → in flight decaystopped → decay at rest

)cos1(' θγγ cvEE +=

γγ EE ='

Tipical values:v/c ~ 0.1, τ ~ 10-12 s → d ~ 0.03 mm

The ratio of the peak intensities depends on the lifetime τ of the state

10−15 s <τ < 10−12 sDoppler-shift attenuation method(DSAM)the produced nuclei immediately penetrate into a solid backing (Pb o Au)

they immediately start to slow down and at the end they stop

0 < v/c < (v/c)max

the γ emission varies continously in energyEγ < Eγ

‘ < Eγ(1+ v/c cos θ)

from the shape of the peak one extractv/c and therefore τ(once the dE/dx energy loss nechanism is known)

vdxdEE

vxt 1

/∆

=∆

=∆

stopping power

inte

nsit

y time

EγEγ+(∆Eγ)max

DSAM technique

400 800 1200 1600E γ (keV)

0.0

0.2

0.4

0.6

0.8

1.0

F(τ) 10

753

13eb

SD YrastRidgesTriaxial

143Eu: lifetime analysis

5.2+0.4-0.5

MAXcvcv

F)/(

)/()( =τ