MasayukiYAMAGAMI(Univ.ofAizu)

Shape and independent-particle motion in nuclei; the basic ideas from microscopic collective models

Part 2: Pairing correlations and the rotational spectrum

Version 2.02016/8/25

Rotation of “deformed” statesPairing Quadrupole deformation

Symmetry-broken state BCS state HF stateBroken symmetry Number Spatial rotationDeformation parameter(Order parameter)

Δ = 𝐺 𝐵𝐶𝑆 𝑃( 𝐵𝐶𝑆 𝛽 = 𝐻𝐹 𝑟-𝑌-/ 𝐻𝐹

Classical treatment (Cranking model)

𝐻0 = 𝐻 − 𝜆𝑁 𝐻0 = 𝐻 − 𝜔𝐽

Rotational spectrum 𝑁 = 𝑁/, 𝑁/ ± 2, 𝑁/ ± 4,…

𝐸; − 𝐸;< ≈ℏ-

2ℑ@ABC𝑁-

𝐼 = 0, 2, 4, …𝐸G − 𝐸/ ≈

ℏ-

2ℑ 𝐼(𝐼 + 1)

Reaction for excitation Two-particle transfer Coulomb excitation,…

ΔΔ Δ

Rotation𝜙

Δ = 0 Δ ≠ 0

Fermionic superfluidityCooper pairing:

two correlated fermions act like a boson

Attractive 𝑉@ABC

ExamplesSuperconductivity:

Cooper pairs of electrons

Ultra-cold Fermion gas:6Li, 40K atom pairs

Superfluid 3He:3He atom pairs

Vortex phase in 6Li superfluid fermi gas (Zwierlein et al, 2005)

Nuclear superfluidityNuclear matter (neutron stars)

Nuclei ← Finite system !

Evidences of pairing correlations

n n

Attractive 𝑉@ABC

n nEnergy for breaking a pair

1) Even-Odd mass staggering 2) Moment of inertia (Superfluidity)

𝐸G − 𝐸/ =ℏ-

2ℑ 𝐼(𝐼 + 1)

Additional binding by pair creation

𝐸 ~2Δ

No pairing

with pairing

Why “deformation” appears?

Answer (Mottelson,1960) Competition between two opposing coupling schemes

1) Aligned coupling scheme for deformed equilibrium shape 2) Pair coupling scheme for spherical equilibrium shape

𝑁-particles in a degenerate 𝑗-shell [𝑁: even #] e.g., Two-particles in 𝑑R/--shell (pair degeneracy Ω = -UVW

- = 3 )

+ +

𝐻 = −𝐺𝑃(𝑃, 𝑃( = Y 𝑎U[( 𝑎U[\

(�

[^/

𝑚,𝑚\ = W-, −

W- 𝑚,𝑚\ = a

-, −a- 𝑚,𝑚\ = R

-, −R-

𝐻, 𝑃( = −𝐺 Ω − 𝑛 + 2 𝑃(

Model Hamiltonian

This is easily shown by equations of motion

𝐻𝑃(|0⟩ = −𝐺Ω𝑃(|0⟩, 𝐻 𝑃( -|0⟩ = −2𝐺 Ω − 1 𝑃( -|0⟩, …

Scattering of two particles:𝑗𝑚, 𝑗𝑚\ → 𝑗𝑚0, 𝑗𝑚\′

[𝑚\ is time-reversal state of 𝑚]

𝑛 = Y 𝑎U[( 𝑎U[

�

[^/

+ 𝑎U[\( 𝑎U[\

𝑁-particle state

|𝑁⟩ = 𝑃( ;/-|0⟩

𝑎g(: (deformed) single-particle state𝑎g\(: time-reversal state of 𝑎g

(

𝐻 =Y𝜀g𝑎g(𝑎g

�

g

− 𝐺𝑃(𝑃,

𝑃( = Y 𝑎g(𝑎g\

(�

g^/

For 𝑁-particles, the wave-functions

𝑁-particles in non-degenerate levels [𝑁: even #]

|𝑁⟩ = 𝐴( ;/-|0⟩with𝐴( = Y 𝑐g; 𝑎g

(𝑎g\(

�

g^/

1) 𝑐g; : Variational parameters

2) Particle number 𝑁 is conserved, but not easy to work with

𝜈 �̅��̅�′

𝜈′

BCS state (mean-field approximation)

|Φrst⟩ =u 𝑢g + 𝑣g𝑎g(𝑎g\

( |−⟩�

g^/

𝐴( = Y𝑣g𝑢g𝑎g(𝑎g\

(�

g^/

Particle number 𝑁 is NOT conserved!

BCS wave function

𝑣g- 𝑣g-

𝜀g 𝜀g

Uno

ccup

ied

leve

lsO

ccup

ied

leve

lsHF state BCS state|Φrst⟩ = 𝐶Y

1𝑁/2 !

�

;

|𝑁⟩

𝑣g-: Occupation probability of 𝜈, �̅�

𝑢g- + 𝑣g- = 1

|𝑁⟩ = 𝐴(;-|0⟩ with

If 𝑣B- = 1 ℎ𝑜𝑙𝑒 , 𝑣[- = 0(𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒)

|Φrst⟩ =u𝑎B(𝑎�̅

(|−⟩�

B^/

= |𝐻𝐹⟩

Quasiparticle state

|Φrst⟩ =u 𝑢g + 𝑣g𝑎g(𝑎g\

( |−⟩�

g^/

Quasiparticle state 𝛼g( (independent-particle state)

𝛼g |Φrst⟩ = 0

Determination of 𝑢g, 𝑣g

𝛼g( = 𝑢g𝑎g

( − 𝑣g𝑎g\ , 𝛼g\( = 𝑢g𝑎g\

( + 𝑣g𝑎g

𝛼g , 𝛼�( = 𝛿�g, 𝛼g , 𝛼� = 𝛼g

(, 𝛼�( = 0 ⟹𝑢g- + 𝑣g- = 1

for

Bogoliubov transformation

Here, 𝑢gand𝑣g are real numbers (This sets the gauge angle 𝜙 = 0)

Equations of motion for variation 𝛿𝛼g = 𝛿𝑢g𝑎g − 𝛿𝑣g𝑎g\(

⟹ Gap equation

Φrst 𝛿𝛼g , 𝐻0, 𝛼�( Φrst = 𝐸g Φrst 𝛿𝛼g , 𝛼�

( Φrst

Quasiparticle energy and pairing gap

𝐻0 = 𝐻 − 𝜆𝑁� =Y 𝜀g − 𝜆 𝑎g(𝑎g

�

g

− 𝐺𝑃(𝑃

= Φrst 𝐻′ Φrst +Y𝐸g𝛼g(𝛼g

�

g

+ 𝑉C��B��A�

Quasiparticle energy 𝐸g = 𝜀g − 𝜆 - + Δ�

Pairing gap Δ = 𝐺 Φrst 𝑃( Φrst = 𝐺 ∑ 𝑢g𝑣g�g^/

Occupation probability 𝑣g- =W- 1 + ����

��, 𝑢g- = 1 − ����

��

𝜆 (Chemical potential) for Φrst 𝑁� Φrst = 𝑁 on average

Representation by quasiparticle states 𝛼g(

𝐻rst (mean-field part)

Pairing-rotational spectrum (Sn isotopes)

𝐸 = −𝐵 𝑁 + 8.6𝑁 + 45 ≈ 0.10 𝑁 − 66 - MeV

“Excited state” = the ground states in the neighboring nuclei

Quadrupole deformation

Pairing rotation

Pairing rotation (𝑁�� ⇆ 𝑁 + 2 �� ⇆ 𝑁 + 4 �� …)

Two-neutron transfer(𝑁�� ⇆ 𝑁 + 2 �� ⇆ 𝑁 + 4 �� …)

Rotation in gauge space

𝐻rst : 𝐻rst, 𝑁� ≠ 0 (violation of number conservation!)

BCS theory (mean-field approximation)

Rotated BCS state |Φrst 𝜙 ⟩ (gauge angle 𝜙)

𝐻 = 𝐻rst + 𝑉C��B��A�: 𝐻,𝑁� = 0n n

Φrst 𝜙 𝐻 Φrst 𝜙 = Φrst 𝐻 Φrst

|Φrst 𝜙 ⟩ = 𝑒�B;�-�|Φrst⟩ =u 𝑢g + 𝑒�B�𝑣g𝑎g

(𝑎g\( |−⟩

�

g^/

ΔΔ Δ

Rotation𝜙

Δ = 0 Δ ≠ 0

Continuous symmetry0 ≤ 𝜙 < 2𝜋

Quadrupole deformation and Rotation

𝐻𝐹 Ω 𝐻 𝐻𝐹 Ω = 𝐻𝐹 𝐻 𝐻𝐹

𝐻�� : 𝐻��, 𝐽 ≠ 0 (violates the rotational symmetry)

HF model (mean-field approximation)

Rotated HF state |𝐻𝐹 Ω ⟩ by Euler angle Ω = 𝛼, 𝛽, 𝛾

𝐻 = 𝐻�� + 𝑉C��B��A�: 𝐻, 𝐽 = 0

𝑅 Ω = 𝑒�B£¤¥𝑒�B¦¤§𝑒�B¨¤¥

𝛽 𝛽 𝛽 RotationΩ = 𝛼, 𝛽, 𝛾

Schematic model for “moment of inertia”𝑵-particles in a single 𝒋-shell

PairdegeneracyΩ = 2𝑗 + 1 /2

𝜀g = 0

BCS ground state energy at 𝑁 = Ω (half-filled, 𝑣g- = 1/2)

𝐸rst = Φrst 𝐻 − 𝜆𝑁 Φrst + 𝜆𝑁

= 𝜆𝑁 +𝐺4 𝑁

-

≡ 𝜆𝑁 +ℏ-

2ℑ@ABC𝑁-

𝑣g- =;

-UVW =;-´, 𝑢g- = 1 − ;

-´

Occupation probabilities

Pairing gap

Δ = 𝐺Y𝑢g𝑣g

�

g^/

= 𝐺𝑁2Ω 1 −

𝑁2Ω

�×Ω =

𝐺2 𝑁 2Ω − 𝑁�

Two-particle transfer reaction

Pairing gap Δ = 𝐺 Φrst 𝑃( Φrst 𝑃( = Y 𝑎g(𝑎g\

(�

g^/

Two-particle transfer matrix element

Two-particle transfer cross section (pairing rotational band)

𝜎C·¸~ Φrst 𝑃( Φrst-~

Δ𝐺

-~𝐴4 Δ ≈ W-

¹� MeV, 𝐺 ≈ -R¹ MeV

Two-particle transfer cross section (a two-quasiparticle state)

𝜎-º@~ 𝜈�̅� 𝑃( Φrst- = 𝑢g» ≈1

The ratio𝑅¼ =

𝜎C·¸𝜎-º@

≈𝐴4 ≈ 30 (𝑓𝑜𝑟𝑚𝑎𝑠𝑠𝑛𝑢𝑚𝑏𝑒𝑟𝐴 ≈ 120)

Φrst 𝑁� Φrst = 𝑁/ ⟹ |Φrst⟩ contains |𝑁/⟩,|𝑁/ + 2⟩, |𝑁/ − 2⟩,…

Φrst 𝑃( Φrst ⟹ Average of 𝑁/ + 2 𝑃( 𝑁/ , 𝑁/ 𝑃( 𝑁/ − 2 , …

Number-projected wave function

|Φrst 𝜙 ⟩ =u 𝑢g + 𝑒�B�𝑣g𝑎g(𝑎g\

( |−⟩�

g^/

= u𝑢g

�

g^/

1 + 𝑒�B�|2⟩ +12! 𝑒

�-B�|4⟩ + ⋯+1

𝑁/2 ! 𝑒�B;-�|𝑁⟩ + ⋯

Here, 𝑁-particle state |𝑁⟩ = 𝐴( ;/-|−⟩with𝐴( = ∑ Á���𝑎g(𝑎g\

(�g^/

Number-projection operator 𝑃;

|𝑁⟩ = 𝑃;|Φrst 𝜙 ⟩~Â𝑑𝜙2𝜋

-Ã

/𝑒VB

;-�|Φrst 𝜙 ⟩

Pairing rotational band 𝑁 ⟶ 𝑁 ± 2⟶ 𝑁 ± 4⟶ ⋯(Two-particle transfer reaction !)

Pairing rotation and vibration in Sn isotopes

D. M. Brink and R. A. Broglia, Nuclear Superfluidity: Pairing in Finite Systems (Cambridge University Press, 2005)

Pairing rotation𝐸 ≈ 0.10 𝑁 − 66 - MeV

𝐸=−𝐵𝑁

+8.6𝑁

+45

+𝑬 𝒗

𝒊𝒃M

eVExperiment: 𝐼Ã = 0V states in two-particle transfer reactions (t,p) and (p,t)

0��V0��V

0��V

0ÁBÉV0��V

0��V 0��V 0��V0��V

0��V

0ÁBÉV 0ÁBÉV

0ÁBÉV0ÁBÉV

𝜎C·¸ 𝑡, 𝑝𝜎C·¸ 𝑝, 𝑡

Normalized to 𝜎C·¸(116Sn(gs)⟶ 118Sn(gs))

Pairing vibration= cross section

Pairing correlations determine the limit of existence

This nuclear chart is taken from Wikimedia Commons, the free media repository

Three-body system11Li=9Li+n+n

Pairing correlation in 11Li n

n9Li

Experiment: T. Nakamura, et al., PRL 96, 252502 (2006)

Soft E1 excitation

Two-particle density in 11Li

K.Hagino, H.Sagawa, Phys.Rev. C 72, 044321 (2005)

Di-neutron correlation is suggested !

[ ]fmr

r

𝐵 𝐸1 = 1.42 18 𝑒-𝑓𝑚-(𝐸C�� < 3𝑀𝑒𝑉)

𝜃W- = 48�WÌVW» degreecf. 𝜃W- Í·�ηCC��A¸B·Í = 90 degree

𝜃W-

Divergence of nuclear radius(neutron-gas problem )

J.Dobaczewski, H.Flocard, J.Treiner, Nucl. Phys. A422, 103 (1984)

Pair scattering into continuum states

𝜆� < Δ

𝜌 𝑟 = Y 𝑣B- 𝜑B 𝑟 -�

É·�Í��¸A¸��

+  𝑑𝜀�ÍÉ·�Í��¸A¸��

𝑣�- 𝜑� 𝑟 -

ÒBÁ�C��ÍÎ�!

Breakdown of BCS theory in weakly-bound nuclei

𝜆�

How can we overcome this problem ?

Hartree-Fock-Bogoliubov method Selfconsistency between HF state & pairing correlations

𝑎Ó |𝐻𝐹⟩ = 0, 𝑎Ó( =Y𝐷�Ó

�

�

𝑐�(

𝑎Ó(:HF single-particle state𝑐�(:Basis state (e.g., spherical state)𝐷�Ó: Variational parameters

𝛼Ó |𝐵𝐶𝑆⟩ = 0, 𝛼Ó( = 𝑢Ó𝑎Ó

( − 𝑣Ó𝑎Ó¼𝛼Ó( :Quasiparticlestate𝑎Ó(:HF single-particle state𝑢Ó, 𝑣Ó: Variational parameters

𝛼Ó |𝐻𝐹𝐵⟩ = 0, 𝛼Ó( =Y𝑢�Ó𝑎�

( − 𝑣�Ó𝑎�̅�

�𝛼Ó( :Quasiparticlestate𝑎�(:HF single-particle state𝑢�Ó, 𝑣�Ó: Variational parameters

Equations of motion for variation

ℎ Δ−Δ∗ −ℎ∗

𝑢𝑣 = 𝐸

𝑢𝑣

ℎ�Ó = 𝐻𝐹𝐵 𝑎� , 𝐻0, 𝑎Ó( 𝐻𝐹𝐵

Δ�Ó = − 𝐻𝐹𝐵 𝑎� , 𝐻0, 𝑎Ó 𝐻𝐹𝐵

HFB equation (matrix form)

𝛿𝛼Ó =Y𝛿𝑢�Ó𝑎� − 𝛿𝑣�Ó𝑎�̅(

�

�

HFB quasiparticle state

Two selfconsistent potentials① HF potential② Pairing potential

Pairing anti-halo effectIdea: K. Bennaceur, J. Dobaczewski, M. Ploszajczak, Phys. Lett. 496B, 154 (2000)

Quasiparticle wave function (hole component)

𝑣B 𝑟 C→Ü𝑒𝑥𝑝 −𝛼B𝑟 /𝑟

HFB:𝛼B = ÞßℏÞ

�à��� ≥ Þß

ℏÞâ

� > 0

HF: 𝛼B = −ÞßℏÞ�à

�

�à,�→/0

M. Grasso, N. Sandulescu, Nguyen Van Giai, and R. J. Liotta, Phys. Rev. C64, 064321 (2001) M. Y., Phys. Rev. C72, 064308 (2005)

K. Hagino and H. Sagawa,Phys. Rev. C84, 011303 (2011)

𝑉ä� 𝑟 & Δ@ABC 𝑟

Ni

l=3l=4l=5l=6

Di-neutron condensation

θO

M.Matsuo, K.Mizuyama, Y.Serizawa, Phys.Rev. C 71, 064326 (2005)

Di-neutron correlation

High-l orbits

Strong correlation in θ direction

𝜃~1/𝑙Contribution of non-resonant continuum states

Cooper pair size (2n-correlation density)

Pairing rotation in Sn isotopesSkyrme-HFB+QRPA calculation : H. Shimoyama and M. Matsuo, Phys. Rev. C 88, 054308 (2013)

Pairing rotation Transition strength for 𝐴�� → (𝐴 + 2)��

θO𝜃~1/𝑙

Di-neutron correlations & condensation (Concept!)

2nS

0

BCS

BEC～ ～

Cooperpair

Few-body Mean-field Equation of stateCluster collective moitons (astrophysics)

・・・

・・・

・・・

・・・

Mass number 𝐴 (Number of di-neutrons)

Drip-line

Stable nuclei

Summary of part 2Pairing Quadrupole deformation

Symmetry-broken state BCS state HF stateBroken symmetry Number Spatial rotationDeformation parameter(Order parameter)

Δ = 𝐺 𝐵𝐶𝑆 𝑃( 𝐵𝐶𝑆 𝛽 = 𝐻𝐹 𝑟-𝑌-/ 𝐻𝐹

Classical treatment (Cranking model)

𝐻0 = 𝐻 − 𝜆𝑁 𝐻0 = 𝐻 − 𝜔𝐽

Rotational spectrum 𝑁 = 𝑁/, 𝑁/ ± 2, 𝑁/ ± 4,…

𝐸; − 𝐸;< ≈ℏ-

2ℑ@ABC𝑁-

𝐼 = 0, 2, 4, …𝐸G − 𝐸/ ≈

ℏ-

2ℑ 𝐼(𝐼 + 1)

Reaction for excitation Two-particle transfer Coulomb excitation,…

Core9Li

11Li=9Li+n+n

di-neutronWeakly-bound nuclei

Heavier-mass region

Di-neutron condensation?

Collective rotation?Surface vibration?

Two-neutron transfer?