Two-body Momentum Distributions in 2 A 40 nuclei M. Alvioli - - PowerPoint PPT Presentation

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Two-body Momentum Distributions in 2 A 40 nuclei M. Alvioli National Research Council Research Institute for Geo-Hydrological Hazards Perugia, Italy M. Alvioli MIT 2016 1 CONTENTS 0. Introduction &

slide-1
SLIDE 1

✬ ✫ ✩ ✪

Two-body Momentum Distributions in 2 ≤ A ≤ 40 nuclei

  • M. Alvioli

National Research Council Research Institute for Geo-Hydrological Hazards Perugia, Italy

  • M. Alvioli

1

MIT 2016

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SLIDE 2

✬ ✫ ✩ ✪

CONTENTS

  • 0. Introduction & basic quantities
  • Few-body nuclei, “exact” wave functions
  • Many-body nuclei, cluster expansion
  • Additional slides
  • 1. Two-Body Momentum Distributions:
  • factorization into nrel(krel) and nc.m.(Kc.m.)
  • Scaling to nD(k)
  • Definition of the scaling coefficients Cpn

A

  • Relationship between one- and two-body momentum dis-

tribution

  • Relationship between Kc.m.-integrated and Kc.m.=0 2BMD
  • 2. Comparison with experimental data

2

MIT 2016

slide-3
SLIDE 3

✬ ✫ ✩ ✪

  • 0. Comprehensive review of experimental results

γ cos

  • 1.00
  • 0.98
  • 0.96
  • 0.94
  • 0.92
  • 0.90

Counts

10 20 30 40

γ

p p

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

pn(GeV/c) cosγ 5.9 GeV/c, 98 8.0 GeV/c, 98 9.0 GeV/c, 98 5.9 GeV/c, 94 7.5 GeV/c, 94

  • 1
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55

A(e, e′p)X A(p, 2p) Tang et al., PRL90 (2003) A(p, ppn) Aclander et al., PLB (1999) theory: Ciofi et al., PRC53 (1996) theory: Piasetzky et al., PLB (1999) triple coincidence A(e, e′pN)X: Carbon Helium heavier targets Subedi et al., Science, 320 (2008) Korover et al., PRL 113 (2014) Hen et al., Science, 356 (2014) Shneor et al., PRL99 (2007)

  • M. Alvioli

3

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SLIDE 4

✬ ✫ ✩ ✪

  • 0. Nuclear Hamiltonian
  • The non-relativistic nuclear many-body problem:

ˆ H Ψn = En Ψn , ˆ H = − ℏ2 2 m ∑

i

ˆ ∇2

i + 1

2 ∑

ij

ˆ vij + ...

  • Exact ground-state wave functions obtained by various methods are avail-

able for light nuclei (A ≤ 12); = ⇒ calculations will be shown using 2H, 3He, 4He WFs;

  • Variational wave functions of nuclei can be obtained with approximated

methods; usually difficult to use/generalize = ⇒ we developed an easy-to-use cluster expansion technique for the cal- culation of basic quantities of medium-heavy nuclei, 12C, 16O, 40Ca;

  • SRCs implemented MC generator for nuclear configurations for nuclei from

12C to 238U for the initialization of pA and AA collisions simulations

http://sites.psu.edu/color

  • M. Alvioli

4

MIT 2016

slide-5
SLIDE 5

✬ ✫ ✩ ✪

  • 0. Calculation of basic quantities
  • one- and two-body densities:

ρST

N (r1, r′ 1) =

∫ dr1 ∑ ∫

A

∏

j=3

drjΨo†

A (x1, ..., xA) ˆ

P ST

pN Ψo A(x′ 1, x2, ..., xA)

ρ(2)

pN(r1, r2; r′ 1, r′ 2) =

∑ ∫

A

∏

j=3

drjΨo†

A (x1, ..., xA) ˆ

PpN Ψo

A(x′ 1, x′ 2, x3, ..., xA)

  • one- and two-body momentum distributions:

nST

N (k1) =

1 (2π)3 ∫ dr1dr′

1 e−k1·(r1−r′

1)ρST

N (r1, r′ 1)

n(2)

pN(k1, k2) =

1 (2π)6 ∫ dr1dr′

1dr2dr′ 2 e−k1·(r1−r′

1) e−k2·(r2−r′ 2) ·

· ρ(2)

pN(r1, r2; r′ 1, r′ 2) ←

→ n(2)

pN(krel, KCM)

  • M. Alvioli

5

MIT 2016

slide-6
SLIDE 6

✬ ✫ ✩ ✪

  • 0. Two-Body Momentum Distributions

krel ≡ k = 1 2 ( k1 − k2 ) r = r1 − r2 r′ = r′

1 − r′ 2

KCM ≡ K = k1 + k2 R = 1 2 ( r1 + r2 ) R′ = 1 2 ( r′

1 + r′ 2 )

n(2)(k, K) = n(2)(krel, KCM, Θ) = = 1 (2π)6 ∫ drdr′dRdR′ e−iK·(R−R′) e−ik·(r−r′)ρ(2)(r, r′; R, R′) n(2)(k) = ∫ dK n(k, K) = 1 (2π)3 ∫ drdr′dR e−ik·(r−r′)ρ(2)(r, r′; R, R) n(2)(K) = ∫ dk n(k, K) = 1 (2π)3 ∫ drdRdR′ e−iK·(R−R′)ρ(2)(r, r; R, R′) n(2)(k, K = 0) = 1 (2π)6 ∫ drdr′dRdR′ e−ik·(r−r′)ρ(2)(r, r′; R, R′) KCM = 0 corresponds to k2 = −k1, i.e. back-to-back nucleons

  • M. Alvioli

6

MIT 2016

slide-7
SLIDE 7

✬ ✫ ✩ ✪

  • 0. Using Realistic WFs of large nuclei: Cluster Expansion
  • Cluster Expansion is a technique to reduce the computational effort in

many many-body calculations; we use: Ψo = ˆ FΦo = ∏

ij

∑

n ˆ

f(n)

ij Φo

  • Expectation value over Ψo of any one- or two-body operator ˆ

Q: ⟨Ψo| ˆ Q|Ψo⟩ ⟨Ψo|Ψo⟩ = ⟨ ˆ F † ˆ Q ˆ F⟩ ⟨ ˆ F 2⟩ = ⟨∏ ˆ f† ˆ Q ˆ f⟩ ⟨∏ ˆ f2⟩ = ⟨ ˆ Q ∏ (1 + ˆ η)⟩ ⟨∏ (1 + ˆ η)⟩ = = ⟨ ˆ Q (1 + ∑ ˆ η + ∑ ˆ ηˆ η + . . . )⟩ ⟨(1 + ∑ ˆ η + ∑ ˆ ηˆ η + . . . )⟩ ≃ ⟨ ˆ Q⟩ + ⟨ ˆ Q ∑ ˆ η⟩ 1 + ⟨∑ ˆ η⟩ = ≃ [ ⟨ ˆ Q⟩ + ⟨ ˆ Q ∑ ˆ η⟩ ] ( 1 − ⟨ ∑ ˆ η⟩ + ... ) ≃ ⟨ ˆ Q⟩ + ⟨ ˆ Q ∑ ˆ η⟩L

  • ⟨ˆ

η⟩ = ⟨ [ ˆ f2 − 1 ] ⟩ is the small expansion parameter; ⟨ ˆ Q⟩ ≡ ⟨Φo| ˆ Q|Φo⟩

  • we end up with linked clusters; up to 4b diagrams needed for 2B

density, each involving the square of: ˆ f = ∑

n fn(rij) ˆ

On(ij)

  • M. Alvioli

7

MIT 2016

slide-8
SLIDE 8

✬ ✫ ✩ ✪

  • 0. Using Realistic WFs of large nuclei: Cluster Expansion

correlation functions: Central, Spin-Isospin, Tensor

1 2 3 4 5
  • 0.04
  • 0.02
0.00 0.02 0.04 0.06 0.08 0.10 f(r) 4 - 5 - S 6 - S 16 O - AV8' r [fm] 1 - f c (/10) 2 - 3 -
  • ⟨ˆ

η⟩ = ⟨ [ ˆ f2 − 1 ] ⟩ is the small expansion parameter

  • we end up with linked clusters; up to 4b diagrams needed for 2B

density, each involving the square of: ˆ f = ∑

n fn(rij) ˆ

On(ij)

  • M. Alvioli

8

MIT 2016

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SLIDE 9

✬ ✫ ✩ ✪

  • at first order of the η−expansion, the full correlated one-body mixed

density matrix expression is as follows: ρ(1)(r1, r′

1) = ρ(1)

  • (r1, r′

1) + ρ(1) H (r1, r′ 1) + ρ(1) S (r1, r′ 1) ,

with

ρ(1)

H (r1, r′ 1) =

∫ dr2 [ HD(r12, r1′2) ρ(1)

  • (r1, r′

1) ρo(r2) − HE(r12, r1′2) ρ(1)

  • (r1, r2) ρ(1)
  • (r2, r′

1)

] ρ(1)

S (r1, r′ 1) = −

∫ dr2dr3ρ(1)

  • (r1, r2)

[ HD(r23)ρ(1)

  • (r2, r′

1)ρo(r3) − HE(r23)ρ(1)

  • (r2, r3)ρ(1)
  • (r3, r′

1)

]

and the functions HD and HE are defined as: HD(E)(rij, rkl) =

6

∑

p,q=1

f(p)(rij) f(q)(rkl) C(p,q)

D(E)(rij, rkl) − C(1,1) D(E)(rij, rkl)

with C(p,q)

D(E)(rij, rkl) proper functions arising from spin-isospin traces;

(Alvioli, Ciofi degli Atti, Morita, PRC72 (2005))

  • M. Alvioli

9

MIT 2016

slide-10
SLIDE 10

✬ ✫ ✩ ✪

  • at first order of the η−expansion, the full correlated two-body mixed

density matrix expression is as follows:

ρ(2)(r1, r2; r′

1, r′ 2) = ρ(2) SM(r1, r2; r′ 1, r′ 2) + ρ(2) 2b(r1, r2; r′ 1, r′ 2) + ρ(2) 3b(r1, r2; r′ 1, r′ 2) + ρ(2) 4b(r1, r2; r′ 1, r′ 2)

with:

ρ(2)

SM(r1, r2; r′ 1, r′ 2) = CD ρo(r1, r′ 1) ρo(r2, r′ 2) − CE ρo(r1, r′ 2) ρo(r2, r′ 1)

ρ(2)

2b(r1, r2; r′ 1, r′ 2) = 1

2 ˆ η(r12, r1′2′) ρo(r1, r′

1) ρo(r2, r′ 2) − 1

2 ˆ η(r12, r1′2′) ρo(r1, r′

2) ρo(r2, r′ 1)

ρ(2)

3b(r1, r2; r′ 1, r′ 2) =

∫ dr3 ˆ η(r13, r1′3) [ ρo(r1, r′

1) ρo(r2, r′ 2) ρo(r3, r3) +

− ρo(r1, r′

1) ρo(r2, r3) ρo(r3, r′ 2) +

+ ρo(r1, r3) ρo(r2, r′

1) ρo(r3, r′ 2) +

− ρo(r1, r3) ρo(r2, r′

2) ρo(r3, r′ 1) +

+ ρo(r1, r′

2) ρo(r2, r3) ρo(r3, r′ 1) +

− ρo(r1, r′

2) ρo(r2, r′ 1) ρo(r3, r3) ]

ρ(2)

4b(r1, r2; r′ 1, r′ 2) = 1

4 ∫ dr3dr4 ˆ η(r34) · · ∑

P∈C

(−1)P [ ρo(r1, rP1′) ρo(r2, rP2′) ρo(r3, rP3) ρo(r4, rP4) ]

(Alvioli, Ciofi degli Atti, Morita, PRC72 (2005)) (Alvioli, Ciofi degli Atti, Morita, PRL100 (2008))

  • M. Alvioli

10

MIT 2016

slide-11
SLIDE 11

✬ ✫ ✩ ✪

  • ne-body, non-diagonal

← − ρ(r1, r′

1) diagrams

1 1’

1 1’ 2

2 1’ 1 1’ 1 2 3 3 2 1 1’

two-body, diagonal ρ(2)(r1, r2) diagrams − →

a

  • M. Alvioli

11

slide-12
SLIDE 12

✬ ✫ ✩ ✪

  • 0. One-Body Mom distrs: Few- and Many-Body nuclei
1 2 3 4 5 10
  • 6
10
  • 5
10
  • 4
10
  • 3
10
  • 2
10
  • 1
10 n n(ST) 3 (k) [fm 3 ] k [fm
  • 1
] Full (ST) (10) (00) (01) (11) 3 He AV18 1 2 3 4 5 10
  • 6
10
  • 5
10
  • 4
10
  • 3
10
  • 2
10
  • 1
10 n (ST) 4 (k) [fm 3 ] k [fm
  • 1
] Full (ST) (10) (00) (01) (11) 4 He AV8' 1 2 3 4 5 10
  • 6
10
  • 5
10
  • 4
10
  • 3
10
  • 2
10
  • 1
Full (ST) (10) (00) (01) (11) n (ST) 16 (k) [fm 3 ] 16 O AV8' k [fm
  • 1
] 1 2 3 4 5 10
  • 6
10
  • 5
10
  • 4
10
  • 3
10
  • 2
10
  • 1
Full (ST) (10) (00) (01) (11) n (ST) 40 (k) [fm 3 ] 40 Ca AV8' k [fm
  • 1
]
  • M. Alvioli et al., PRC87 (2013); IntJModPhys E22 (2013)
  • M. Alvioli

12

MIT 2016

slide-13
SLIDE 13

✬ ✫ ✩ ✪

  • 0. One-Body Mom distrs: Many-Body nuclei
2 4 1E-6 1E-5 1E-4 1E-3 0.01 0.1 n(k) [fm 3 ] k [fm
  • 1
] Shell Model (ST) 10 00 01 11 16 O 1 2 3 4 5 10
  • 6
10
  • 5
10
  • 4
10
  • 3
10
  • 2
10
  • 1
Full (ST) (10) (00) (01) (11) n (ST) 16 (k) [fm 3 ] 16 O AV8' k [fm
  • 1
] 1 2 3 4 5 1 2 3 4 5 6 7 AV8' n 16 (k)/n D (k) n (10) 16 (k)/n D (k) n (01) 16 (k)/n D (k) n (11) 16 (k)/n D (k) n (ST) 16 (k)/n D (k) k [fm
  • 1
] 2 4 6 1E-6 1E-5 1E-4 1E-3 0.01 0.1 40 Ca n(k) [fm 3 ] k [fm
  • 1
] Shell Model (ST) 10 00 01 11 1 2 3 4 5 10
  • 6
10
  • 5
10
  • 4
10
  • 3
10
  • 2
10
  • 1
Full (ST) (10) (00) (01) (11) n (ST) 40 (k) [fm 3 ] 40 Ca AV8' k [fm
  • 1
] 1 2 3 4 5 1 2 3 4 5 6 7 8 AV8' n 40 (k)/n D (k) n (10) 40 (k)/n D (k) n (01) 40 (k)/n D (k) n (11) 40 (k)/n D (k) n (ST) 40 (k)/n D (k) k [fm
  • 1
]
  • M. Alvioli et al., PRC87 (2013); IntJModPhys E22 (2013)
  • M. Alvioli

13

MIT 2016

slide-14
SLIDE 14

✬ ✫ ✩ ✪

  • 0. One-Body Mom distrs: ST pairs contribution

[23] Feldmeier, Horiuchi, Neff, Suzuki - PRC84 (2011) 054003 [40] Forest, Pandharipande, Pieper, Wiringa, Schiavilla, Arriaga PRC54 (1996) 646 [46] Vanhalst, Cosyn, Ryckebusch, PRC84 (2011) 031302

  • M. Alvioli et al., PRC87 (2013); IntJModPhys E22 (2013)
  • M. Alvioli

14

MIT 2016

slide-15
SLIDE 15

✬ ✫ ✩ ✪

  • 1. Two-Body Distributions: a closer look to deuteron scaling
1 2 3 4 5 1 2 3 4 X X X X n pn (k rel ,0) / n D (k rel ) [fm 3 ] k rel [fm
  • 1
] 3 He (pn) 4 He (pn) X
  • M. Alvioli, C.Ciofi degli Atti, L.P. Kaptari, C.B. Mezzetti,
  • H. Morita, S. Scopetta; Phys. Rev. C85 (2012) 021001
  • Should a nucleus’ npn(krel, KCM = 0) scale to 2H’s nD(krel)?
  • M. Alvioli

15

MIT 2016

slide-16
SLIDE 16

✬ ✫ ✩ ✪

  • 1. Two-Body Distributions: a closer look to deuteron scaling
1 2 3 4 5 1 2 3 4 X X X X 3 He (ST)=(10) 4 He (ST)=(10) n pn (k rel ,0) / n D (k rel ) [fm 3 ] k rel [fm
  • 1
] 3 He (pn) 4 He (pn) X
  • M. Alvioli, C.Ciofi degli Atti, L.P. Kaptari, C.B. Mezzetti,
  • H. Morita, S. Scopetta; Phys. Rev. C85 (2012) 021001
  • Should a nucleus’ npn(krel, KCM = 0) scale to 2H’s nD(krel)?
  • Including only pairs with deuteron-like quantum numbers (ST)=(10) we find

exact scaling!

  • M. Alvioli

16

MIT 2016

slide-17
SLIDE 17

✬ ✫ ✩ ✪

  • 1. Two-Body Distributions: a closer look to deuteron scaling
1 2 3 4 5 1 2 3 4 n pn, 4 He CM (K CM =0) X X X X 3 He (ST)=(10) 4 He (ST)=(10) n pn (k rel ,0) / n D (k rel ) [fm 3 ] k rel [fm
  • 1
] 3 He (pn) 4 He (pn) X n pn, 4 He CM (K CM =0)
  • M. Alvioli, C.Ciofi degli Atti, L.P. Kaptari, C.B. Mezzetti,
  • H. Morita, S. Scopetta; Phys. Rev. C85 (2012) 021001
  • Should a nucleus’ npn(krel, KCM = 0) scale to 2H’s nD(krel)?
  • Including only pairs with deuteron-like quantum numbers (ST)=(10) we find

exact scaling!

  • n(krel, 0)/nD(krel) ≃ nD(krel)nCM(0)/nD(krel) = nCM(KCM = 0)!
  • M. Alvioli

17

MIT 2016

slide-18
SLIDE 18

✬ ✫ ✩ ✪

example: many-body contributions in 4He 2BMD npN(krel) = ∫ dKCM npN(krel, KCM)

1 2 3 4 5 10
  • 4
10
  • 3
10
  • 2
10
  • 1
10 10 1 10 2 10 3 10 4 AV18 p-p n pp (k rel ) [fm 3 ] k rel [fm
  • 1
] 4 He - pp Total SM 2b 3b (-4b) 1 2 3 4 5 10
  • 4
10
  • 3
10
  • 2
10
  • 1
10 10 1 10 2 10 3 10 4 AV18 p-n n pn (k rel ) [fm 3 ] k rel [fm
  • 1
] Total SM 2b 3b
  • 4b
4 He - pn

(AV18: Schiavilla at al. PRL98 (2007))

1 2 1 2’ 1 1’ 2 2’ 3 1 2 1 2’ 1 1’ 2 2’ 3 4

Shell Model two-body three-body four-body

  • M. Alvioli

18

MIT 2016

slide-19
SLIDE 19

✬ ✫ ✩ ✪

  • 1. Two-Body momentum Distributions of Few-Body Nuclei

Pisa AV18 +UIX ATMS AV8′

1 2 3 4 5 10
  • 10
10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 1 2 3 4 5 10
  • 10
10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 n pN A (k rel ,K c.m. , ) [fm 6 ] Perp

pn

K c.m . = 0.0 K c.m . = 0.5 K c.m . = 1.0 K c.m . = 1.5 3

He

pp

1 2 3 4 5 10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 1 2 3 4 5 10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 Perp 4

He

n pN (k rel ,K c.m. , ) [fm 6 ]

pn pp

K c.m . = 0.0 K c.m . = 0.5 K c.m . = 1.0

here krel is perpendicular to KCM Alvioli, Ciofi, Kaptari, Mezzetti, Morita, Scopetta; PRC85 (2012) Alvioli, Ciofi, Morita; Phys. Rev. C94 (2016) 044309

  • M. Alvioli

19

MIT 2016

slide-20
SLIDE 20

✬ ✫ ✩ ✪

  • 1. Two-Body momentum Distributions of Few-Body Nuclei

Pisa AV18 +UIX ATMS AV8′

1 2 3 4 5 10
  • 10
10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 1 2 3 4 5 10
  • 10
10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 Para Perp

pn

n pN A (k rel ,K c.m. , ) [fm 6 ] K c.m . = 0.0 K c.m . = 0.5 K c.m . = 1.0 K c.m . = 1.5 3

He

pp

1 2 3 4 5 10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 1 2 3 4 5 10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 Para Perp 4

He

n pN (k rel ,K c.m. , ) [fm 6 ]

pn pp

K c.m . = 0.0 K c.m . = 0.5 K c.m . = 1.0

npN(krel, KCM, Θ) is angle independent for large krel and small KCM Alvioli, Ciofi, Kaptari, Mezzetti, Morita, Scopetta; PRC85 (2012) Alvioli, Ciofi, Morita; Phys. Rev. C94 (2016) 044309

  • M. Alvioli

20

MIT 2016

slide-21
SLIDE 21

✬ ✫ ✩ ✪

  • 1. Two-Body momentum Distributions of Few-Body Nuclei

Pisa AV18 +UIX ATMS AV8′

1 2 3 4 5 10
  • 10
10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 1 2 3 4 5 10
  • 10
10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 Para Perp

pn

n pN A (k rel ,K c.m. , ) [fm 6 ] K c.m . = 0.0 K c.m . = 0.5 K c.m . = 1.0 K c.m . = 1.5 K c.m . = 3.0 3

He

pp

1 2 3 4 5 10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 1 2 3 4 5 10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 Para Perp 4

He

n pN (k rel ,K c.m. , ) [fm 6 ]

pn pp

K c.m . = 0.0 K c.m . = 0.5 K c.m . = 1.0 K c.m . = 3.0

three-body correlations must be in the large KCM region Alvioli, Ciofi, Kaptari, Mezzetti, Morita, Scopetta; PRC85 (2012) Alvioli, Ciofi, Morita; Phys. Rev. C94 (2016) 044309

  • M. Alvioli

21

MIT 2016

slide-22
SLIDE 22

✬ ✫ ✩ ✪

  • 1. Two-Body momentum Distributions of Few-Body Nuclei

3He (Pisa AV18+UIX) 4He (ATMS AV8′)

1 2 3 4 5 10
  • 10
10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 K c.m . = 0.0 K c.m . = 0.5 K c.m . = 1.0 K c.m . = 1.5 K c.m . = 3.0 n pN A (k rel ,K c.m. , ) [fm 6 ] Para Perp

pn

1 2 3 4 5 10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 K c.m. = 0.0 K c.m. = 0.5 K c.m. = 1.0 K c.m. = 3.0 Para Perp n pN (k rel ,K c.m. , ) [fm 6 ]

pn

for the pn case we see clear scaling to the deuteron nD Alvioli, Ciofi, Kaptari, Mezzetti, Morita, Scopetta; PRC85 (2012) Alvioli, Ciofi, Morita; Phys. Rev. C94 (2016) 044309

  • M. Alvioli

22

MIT 2016

slide-23
SLIDE 23

✬ ✫ ✩ ✪

  • 1. Two-Body momentum Distributions of Few-Body Nuclei

3He (Pisa AV18+UIX) 4He (ATMS AV8′)

1 2 3 4 5 10
  • 10
10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 K c.m . = 0.0 K c.m . = 0.5 K c.m . = 1.0 K c.m . = 1.5 K c.m . = 3.0 n pN A (k rel ,K c.m. , ) [fm 6 ] Para Perp n D scaling

pn

1 2 3 4 5 10
  • 8
10
  • 6
10
  • 4
10
  • 2
10 10 2 K c.m. = 0.0 K c.m. = 0.5 K c.m. = 1.0 K c.m. = 3.0 Para Perp n D scaling n pN (k rel ,K c.m. , ) [fm 6 ]

pn

solid curves, the TNC model: rescaling of the deuteron by nA

CM(KCM)!

Alvioli, Ciofi, Kaptari, Mezzetti, Morita, Scopetta; PRC85 (2012) Alvioli, Ciofi, Morita; Phys. Rev. C94 (2016) 044309

  • M. Alvioli

23

MIT 2016

slide-24
SLIDE 24

✬ ✫ ✩ ✪

  • 1. Two-Body momentum Distributions of Many-Body Nuclei

cluster expansion ρ(r1, r2; r′

1, r′ 2), =

⇒ npn(krel, KCM, Θ)

1 2 3 4 5 10
  • 6
10
  • 4
10
  • 2
10 10 2 pn k rel || K c.m. k rel [fm
  • 1
] n pn (k rel ,K c.m. , =0) [fm 6 ] K c.m . =0.0 K c.m . =0.5 K c.m . =1.0 12 C 1 2 3 4 5 10
  • 6
10
  • 4
10
  • 2
10 10 2 K c.m . =0.0 K c.m . =0.5 K c.m . =1.0 k rel [fm
  • 1
] 16 O pn k rel || K c.m 1 2 3 4 5 10
  • 6
10
  • 4
10
  • 2
10 10 2 k rel [fm
  • 1
] n pn (k rel ,K c.m. , =0) [fm 6 ] K c.m . =0.0 K c.m . =0.5 K c.m . =1.0 pn k rel || K c.m. 40 Ca
  • M. Alvioli, C.Ciofi degli Atti, H. Morita, PRL100 (2008) 162503
  • M. Alvioli, C.Ciofi degli Atti, H. Morita; Phys. Rev. C94 (2016) 044309
  • M. Alvioli

24

MIT 2016

slide-25
SLIDE 25

✬ ✫ ✩ ✪

  • 1. Two-Body momentum Distributions of Many-Body Nuclei

cluster expansion ρ(r1, r2; r′

1, r′ 2), =

⇒ npn(krel, KCM, Θ)

1 2 3 4 5 10
  • 6
10
  • 4
10
  • 2
10 10 2 pn k rel || K c.m. k rel [fm
  • 1
] n pn (k rel ,K c.m. , =0) [fm 6 ] K c.m . =0.0 K c.m . =0.5 K c.m . =1.0 12 C 1 2 3 4 5 10
  • 6
10
  • 4
10
  • 2
10 10 2 K c.m . =0.0 K c.m . =0.5 K c.m . =1.0 k rel [fm
  • 1
] 16 O pn k rel || K c.m 1 2 3 4 5 10
  • 6
10
  • 4
10
  • 2
10 10 2 k rel [fm
  • 1
] n pn (k rel ,K c.m. , =0) [fm 6 ] K c.m . =0.0 K c.m . =0.5 K c.m . =1.0 pn k rel || K c.m. 40 Ca
  • M. Alvioli, C.Ciofi degli Atti, H. Morita, PRL100 (2008) 162503
  • M. Alvioli, C.Ciofi degli Atti, H. Morita; Phys. Rev. C94 (2016) 044309
  • solid curves are the deuteron scaled with the corresponding npn

c.m.(Kc.m.)

  • same behaviour & conclusions as in few-body (KCM > 1 not shown)
  • universality of NN correlations
  • we can update the TNC model with many-body quantities
  • M. Alvioli

25

MIT 2016

slide-26
SLIDE 26

✬ ✫ ✩ ✪

  • 1. Factorization in momemtum space for 3<A<40
1 2 3 10
  • 5
10
  • 4
10
  • 3
10
  • 2
10
  • 1
10 10 1 This Work CS 4 He n pn c.m. (K c.m. ) [fm 3 ] K c.m. [fm
  • 1
] This Work CS Argonne This Work CS Argonne 3 He 40 Ca (a) 2 4 6 8 10 1 2 3 4 5 2 4 6 8 10 1 2 3 4 5 4 He K cm =0.0 K cm =0.5 K cm =1.0 Argonne K cm =0.0 K cm =0.0 K cm =0.5 K cm =1.0 k rel [fm
  • 1
] 12 C C pn A n D (k rel )n pn c.m. (K c.m. )/n pn A (k rel ,K cm , ) K cm =0.0 K cm =0.5 K cm =1.0 16 O K cm =0.0 K cm =0.5 K cm =1.0 40 Ca

npn

A (krel, KCM) = npn A (krel, Kc.m., Θ)

≃ npn

A (krel, Kc.m. = 0)

npn

c.m.(Kc.m. = 0)

npn

c.m.(Kc.m.)

≃ Cpn

A nD(krel) npn c.m.(Kc.m.) ,

krel ≳ k−

rel = 1.5 fm−1 + Kc.m.

and Kc.m. ≲ 1.0 − 1.5 fm−1

  • M. Alvioli, C.Ciofi degli Atti, H. Morita; Phys. Rev. C94 (2016) 044309
  • M. Alvioli

26

MIT 2016

slide-27
SLIDE 27

✬ ✫ ✩ ✪

  • 1. Scaling coefficients to the deuteron distribution nD
1 2 3 4 5 10
  • 1
10 10 1 10 2 10 3 10 4 C pn 3 (Argonne) C pn 4 (Argonne) C pn 6 (Argonne) C pn 8 (Argonne) n pn A (k rel ,K cm =0)/n D (k rel )n cm (K cm =0) k rel [fm
  • 1
] C pn 3 C pn 4 C pn 12 C pn 16 C pn 40

lim

krel→k−

rel

npn

A (krel, Kc.m. = 0)

nD(krel) npn

c.m.(Kc.m. = 0) = Cpn A

Cpn

A are completely

specified by many-body calculation, only depend on NN potential, approximations, ..

2H 3He 4He 6Li 8Be 12C 16O 40Ca

VMC 1.0 2.0 ± 0.1 4.0 ± 0.1 – – 20 ± 1.6 24 ± 1.8 60 ± 4.0 PRC94 1.0 (2.0 ± 0.1) (5.0± 0.1) (11.1 ± 1.3) (16.5 ± 1.5) (–) (–) (–)

Cpn

A is a measure of the number of BB deuteron-like pairs

  • M. Alvioli, C.Ciofi degli Atti, H. Morita; Phys. Rev. C94 (2016) 044309
  • M. Alvioli

27

MIT 2016

slide-28
SLIDE 28

✬ ✫ ✩ ✪

  • 1. One- and two-body momentum distributions relationship
1 2 3 4 5 2 4 6 8 R N 1 N 2 /N 1 (k 1 ) k 1 [fm
  • 1
] THIS W ORK (K cm
  • integrated)
THIS W ORK (Back-to-Back) ARGONNE (Back-to-Back) ARGONNE (K cm
  • integrated)
4 He (a) 1 2 3 4 5 2 4 6 8 10 12 14 R N 1 N 2 /N 1 (k 1 ) k 1 [fm
  • 1
] 12 C (b)

RBB

N1N2/N1(k1) =

1 np

A(k1)

[npn

A (krel = k1, Kc.m. = 0)

npn

c.m.(Kc.m.=0)

+ 2npp

A (krel = k1, Kc.m. = 0)

npp

c.m.(Kc.m.=0)

] Rint

N1N2/N1(k1) = npn A (krel = k1) + 2npp A (krel = k1)

np

A(k1)

  • M. Alvioli, C.Ciofi degli Atti, H. Morita; Phys. Rev. C94 (2016) 044309
  • M. Alvioli

28

MIT 2016

slide-29
SLIDE 29

✬ ✫ ✩ ✪

  • 1. Relationship between Kc.m.-integrated and Kc.m. = 0

1 2 3 4 5 10

  • 6

10

  • 4

10

  • 2

10 10

2

n

pn A (krel,Kcm=0) /(C pn A n pn cm(Kcm=0)) [fm 3] 2H 12C 3He 16O 4He 40Ca

k rel [fm
  • 1
] (a)

1 2 3 4 5 10

  • 6

10

  • 4

10

  • 2

10 10

2

(b)

n

pn A (krel)/C pn A [fm 3]

krel [fm

  • 1]

2H 12C 3He 16O 4He 40Ca

1 2 3 4 5 10
  • 6
10
  • 4
10
  • 2
10 10 2 n pn A (k rel )/C pn A [fm 3 ] k rel [fm
  • 1
] 2 H This work Argonne 4 He (c) 1 2 3 4 5 10
  • 6
10
  • 4
10
  • 2
10 10 2 n pn A (k rel )/C pn A [fm 3 ] k rel [fm
  • 1
] 2 H This work Argonne 12 C (d)
  • M. Alvioli, C.Ciofi degli Atti, H. Morita; Phys. Rev. C94 (2016) 044309
  • M. Alvioli

29

MIT 2016

slide-30
SLIDE 30

✬ ✫ ✩ ✪

  • 1. Relationship between Kc.m.-integrated and Kc.m. = 0
10
  • 7
10
  • 6
10
  • 5
10
  • 4
10
  • 3
10
  • 2
10
  • 1
10 10 1 1 2 3 4 5 10
  • 7
10
  • 6
10
  • 5
10
  • 4
10
  • 3
10
  • 2
10
  • 1
10 1 2 3 4 5 1 2 3 4 5 n (2) pn (k rel ) [fm 3 ] and n (2) pn (k rel ,0) [fm 6 ] 2 H 3 He n (2) pn (k re l ,0) 3 He n (2) pn (k re l ) AV18 - norm. 1 2 H 4 He n (2) pn (k re l ,0) 4 He n (2) pn (k re l ) AV18 - norm. 1 2 H 12 C n (2) pn (k re l ,0) 12 C n (2) pn (k re l ) AV8' - norm. 1 2 H 16 O n (2) pn (k re l ,0) 16 O n (2) pn (k re l ) AV8' - norm. 1 k rel [fm
  • 1
] 2 H 40 Ca n (2) pn (k re l ,0) 40 Ca n (2) pn (k re l ) AV8' - norm. 1 n (2) pn (k rel ,0 ) 2 H 3 He 4 He 1 2 C 1 6 O 4 Ca AV18 - norm. 1
  • M. Alvioli

30

MIT 2016

slide-31
SLIDE 31

✬ ✫ ✩ ✪

  • 2. Two-body mom distrs results - comparison with data
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5 10 15 20 25 SRC pp/pn FRACTION [%] P m [fm
  • 1
] Argonne 4 He 4 He 12 C (a)

npp(krel, KCM = 0) / npn(krel, KCM = 0) Alvioli, Ciofi, Morita; Phys. Rev. C94 (2016) 044309

  • M. Alvioli

31

MIT 2016

slide-32
SLIDE 32

✬ ✫ ✩ ✪

  • 2. Two-body mom distrs results - comparison with data
2.0 2.5 3.0 3.5 4.0 10 10 1 10 2
  • Exp. pn/p
  • Exp. pp/pn
  • Exp. pp/p
SRC Pair Fraction (%) p m [fm
  • 1
] This W ork pn/p pp/pn pp/p 4 He Argonne pn/p pp/pn pp/p (a) 1.5 2.0 2.5 3.0 10 10 1 10 2 SRC Pair Fraction (%) p m [fm
  • 1
]
  • Exp. pn/p (BNL)
12 C (b)

npp(krel, KCM = 0) / npn(krel, KCM = 0) npn(krel, KCM = 0) / (npn(krel, KCM = 0) + 2 npp(krel, KCM = 0)) npp(krel, KCM = 0) / (npn(krel, KCM = 0) + 2 npp(krel, KCM = 0)) Alvioli, Ciofi, Morita; Phys. Rev. C94 (2016) 044309

  • M. Alvioli

32

MIT 2016

slide-33
SLIDE 33

✬ ✫ ✩ ✪

  • 2. Two-body mom distrs results - comparison with data
0.0 0.2 0.4 0.6 0.8 1.0 1.2 10 20 30 40 50 0.2 0.4 0.6 0.8 1.0 1.2 10 20 30 40 50
  • 1.00
  • 0.98
  • 0.96
  • 0.94
  • 1.00
  • 0.98
  • 0.96
  • 0.94
  • 0.92
COUNTS cos K c.m. [fm
  • 1
] Ex p This W ork CS Argonne 4

He

12

C

cos COUNTS K c.m. [fm
  • 1
] Ex p This work CS Argonne (b) (a)

npN

c.m.(KCM) (krel-integrated)

and normalized to the first data point Alvioli, Ciofi, Morita; Phys. Rev. C94 (2016) 044309

  • M. Alvioli

33

MIT 2016

slide-34
SLIDE 34

✬ ✫ ✩ ✪

Thank you

  • M. Alvioli

34

MIT 2016