Upload
others
View
8
Download
0
Embed Size (px)
Citation preview
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?