moment of inertia, electric quadrupole moment,

Nuclear Rotation

ƒ Experimental evidences for collective modes

ƒ Deformed Nuclei, shell model

ƒ Indicators for rotational collectivity:

moment of inertia, electric quadrupole moment,

lifetime measurements

The energy spectra shope different structures

depending on the number n of nucleons outside the closed shell

shell model

states:

n = ± 1, ± 2 … middle shell:

simple spectra

Even-even Nuclei:

how can one interprete the energies of the 2

⁺

states?

E [MeV]

A

2 MeV

100 keV

energy for breaking 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 an intrinsic 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+ ) [MeV]

A

3.3

2

A<150: E(4+)/E(2+) ~ 2

150<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⁺) ~ 0 150<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

120

Te

I

Energy [keV]

0⁺ 2⁺ 4⁺ 6⁺

0 560.6 1161.9

1776.6

h ω

⋅ E

_n

= n

hω

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

Rotational Nucleus

168

Yb

I

Energy [keV]285.8087 584.5 969.1 1424.5 1935.1

02⁺⁺ 4⁺ 6⁺ 8⁺ 10⁺ 12⁺ 1)

2 (

2 +

= ℑ I I

E h

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

Rotational Motion ^:

It can be observed only in nuclei with stable equilibrium deformation

deformed nuclei

[ ^{1 ( , )} ]

) ,

( θ φ R β Y

₂₀

θ φ

R =

_av

+

The shape of the nucleus is represented by

an ellipsoid of revolution:

deformation parameter β (ellipsoid eccentricity):

3 / 1 0

5 3

4

A R R

R R

av

av

=

= π ∆ β

50 100 150 200

0 50 100 150

0

Number of neutrons N

Numberof protonsP

β

0.450.40 0.350.30 0.250.20 0.150.10 0.05

ground state

quadrupole deformation

β>0 prolate β<0 oblate

Nuclear Shapes

the deformed potential gives the nuclear shape

(Bohr & Mottelson, 1950)

deformed harmonic oscillator

f_7/2

ε

_Ω ^[n,l,j,m_j^]

appearance of new magic numbers:

• superdeformed nuclei (2:1)

• hyperdeformed nuclei (3:1)

1:2 2:1 3:1

the energy levels loose the (2j+1) degeneracy

Nilsson diagram for neutrons in a prolate deformed potential

energy/hω

β

[N,l,j,m_j] π = (-1)^N Ω = m_j

±1/2 ±3/2

±5/2

±7/2

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

f_7/2

ƒ New magic numbers

ƒ New minima at larger deformations

Rotation removes the time-reversal invariance

ω ^r

H_ω=H_o-hωj_x

The Coriolis interaction gives rise to forces of opposite sign,

depending whether a nucleon moves clockwise or anti-clockwise

⇒ splitting of ±m_j energy levels

⇒ changes of shell structures

with rotation Appearance of favorite deformed minima at high spins

la rotazione provoca una rottura nella degenerazione in m_j

Nilsson pairing

cranking

[N,l,j,±m_j] [N,l,j,±m_j]

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

±1/2 ±3/2

±5/2

±7/2

ad ω=0 i livelli hanno degenerazione 2

Yrast

projectile nucleus

target nucleus

fusion

fast fission 10^-22 sec

compound nucleus formation

hω ∼ 0.75 MeV

∼ 2×10²⁰ Hz rotation

10^-19 sec

10^-15 sec

10^-9 sec

ground state

I

E*

_E1

E2 compound nucleus

γ−decay Heavy ion reactions

allow to populate nuclear states

at high angular momenta

(

≥40 h

)

A l(h)

Angular momentum limits (liquid drop calculations)

B_f=8 MeV B_f= 0

triaxial

oblate

E

_γ1

E

_γ2

γ spectrum

1

4 2 5

3 6

E_γ2 E_γ1

10^-15 sec γ detector

∆t ∼ 10^-8 sec

E

_γ2

E

_γ1

γ cascade

E

_γ

[keV]

200 400 600

rotational energy of a body with moment of inerzia

ℑ

angular velocity

ω = I/ ℑ

E

_γ

rotational band

12+

10+

8+

6+

4+

2+0+

1) 2 (

2 2

1 ₂ ^{2 2}

ℑ + ℑ →

= ℑ

= I I I

E ω h

even-even nuclei

: 0⁺, 2⁺, 4⁺, 6⁺, …

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

Eγ

∆

= ℑ

− +

=

∆

= ℑ

− +

= ℑ +

=

2 2 2

) 4 ( )

2 (

2 4 ) ( )

2 (

) (

1) 2 (

) (

h h h

I E I

E E

I I

E I

E I

E

I I I

E

γ γ

γ γ

Channel number

The nucleus is

NON

a rigid body, is

NON

an irrotational fluid

) 31 . 0 1 5 (

2 ₂ + β

=

ℑ_rigid MR _av

π β

2

8 9

av

fluid = MR

ℑ

rigid exp

fluid

< 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)

ω k

I I k + 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 I I I

E h

ℑ changes with I

back-bending takes places when

the rotational energy exceeds the energy required to break a pair of nucleons

Unpaired nucleons go to different orbits and change the momet of inertia

The nucleus can constract the rotation in two different ways:

) ( 1)

2 ( )

(

2

i E R

R I

E + +

= hℑ

i i

R

I = + = ℑω +

collective motion single particle motion

158

Er

¹⁴⁷

Gd

Deformed nucleus quasi-spherical nucleus

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

₀ Q₀ measures the

deviation

from a symmetric distribution

of the nuclear charge distribution

[ ]

[ ]

0 0 0

) (

3

) (

3

2 2

2 2

2 2

2 2

<

>

=

>

<

+

>

<

+

>

<

−

>

<

=

+ +

−

=

∫

z y

x z

Z

d z y

x z

Q ρ τ

spherical shape <z²>

=

^<x2>=<y2>

prolate: elongation along z <z²>

>

^<x2>=<y2>

oblate: flattening along z <z²>

<

^<x2>=<y2>

oblate z

prolate

z

Large electric quadrupole moments indicate a stable deformation :

measured quadrupole moment

− +

= 2

7 2

0 for

Q Q

) 16 . 0 1 5 (

3 ₂

0 β β

π ⁺

= R Z

Q _av

intrinsec quadrupole momenti

(osservable only in the intrinsic reference frame)

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

2 2 0

2 020 0

16 ) 5

: 2

(E J_i J _f e Q J_i J _f

B ^{→ =} π

2 2

0 2

16 ) 5

0 2

: 2

( β

π ^∝

=

→ ⁺

+ e Q

E B

2 2

5 (1 )

08156 .

) 0 2

( e b

E E B

B

αtot γ τ

γ

= +

13 s 10

3 . 0

≈ −

≈ τ β

11s 10

05 . 0

≈ −

≈ τ

β

γ decay probability

=

=

tot

B α

γ

probability for internal conversion

High Spin limit: ^{Bγ = 1, αtot = 0} E s

E B

12

5 10

) 2 (

08156 .

0 × ₋

=

γ

τ

in W.U.

Eγ

E B( 2)

in MeV

4 2 3 /

0594 4

. 0

1WU = A e fm

expected intensity for 1 transition involving only 1 nucleon

Measurement of nuclear lifetimes τ

τ > 10

⁻³

s

direct measure 10

⁻³

< τ < 10

⁻¹¹

s

Delayed coincidence technique

e t

N t

N ( ) = ₀ ^−λ

10

⁻¹⁰

< τ < 10

⁻¹²

s

Doppler-recoil method

plunger setup

ƒ

the produced nuclei escape from the taget with a velocity v/c

ƒ

they decay emitting γ’s which are Doppler shifted

ƒ

the nuclei qre finally completely stopped

into a stopper, therefore decaying with v/c=0

ƒ

two peaks are observed:

shifted → in flight decay stopped → decay at rest

) cos 1

' _γ ( θ

γ c

E v

E = +

γ

γ E

E ^' =

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

s

Doppler-shift attenuation method (DSAM)

ƒ

the produced nuclei immediately penetrate into a solid backing (Pb o Au)

ƒ

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

0 < v/c < (v/c)_max

ƒ

the γ emission varies continously in energy E_γ < E_γ ^‘ < E_γ(1+ v/c cos θ)

ƒ

from the shape of the peak one extract v/c and therefore

τ

(once the dE/dx energy loss nechanism is known)

v dx dE

E v

t x 1

/

= ∆

= ∆

∆

stopping power

intensity

time

E_γ E_γ+(∆E_γ)_max

DSAM technique

400 800 1200 1600

E

_γ

(keV) 0.0

0.2 0.4 0.6 0.8 1.0

F( τ)

¹⁰

7 5 3 13eb

SD Yrast Ridges Triaxial 143

Eu: lifetime analysis

5.2 ^+0.4 _-0.5

c

MAX

v

c F v

) / (

) / ) (

( τ =