The correlated spectral function of the nucleus e97006 collaboration - - PowerPoint PPT Presentation

The correlated spectral function of the nucleus

e97006 collaboration HallC TJNAF Basel University

Kristoff Normand - Basel - 21 October 2004

Correlated spectral function of the nucleus

Experimental study of Short Range Correlations and reaction mechanism Motivations Processes Setup Hydrogen calibrations Quasielastic carbon Spectral Functions Summary

Kristoff Normand - Basel - 21 October 2004

Nuclear structure

• Nucleus = ensemble of nucleons

⇒ many body problem

• Protons and neutrons
• Interaction between nucleons
• Realistic potentials
• Limits of calculations
• 1932: Magic numbers (Barlett, Elsasser)
• Strong differences with atomic case
• 1949: Goeppert Mayer and Jensen
• Interaction between nucleons neglected
• Independent motion of nucleons

in an average potential

Kristoff Normand - Basel - 21 October 2004

Nuclear structure

• Nucleus = ensemble of nucleons

⇒ many body problem

• Protons and neutrons
• Interaction between nucleons
• Realistic potentials
• Limits of calculations
• 1932: Magic numbers (Barlett, Elsasser)
• Strong differences with atomic case
• 1949: Goeppert Mayer and Jensen
• Interaction between nucleons neglected
• Independent motion of nucleons

in an average potential

?

Kristoff Normand - Basel - 21 October 2004

Shell Model

Independent Particle Shell Model (IPSM) ⇒ Single particle structure of the nucleus

E E1s

1p

• Energy shells
• Magic numbers
• Momentum distribution
• Extensively studied

100 −100 100 −100

k (MeV/c)

Kristoff Normand - Basel - 21 October 2004

Short Range Correlations

101 102 0.0 0.2 0.4 0.6 0.8

target mass S/(2j+1)

7Li 12C 16O 31P 40Ca 48Ca 90Zr 208Pb 1.0

Mean Field Theory

VALENCE PROTONS

Experimental data deviate from 1.

⇒ Short range correlations occur between two nucleons via the repulsive part of the N-N interaction. Various models: Green’s functions, CBF, ...

Kristoff Normand - Basel - 21 October 2004

Spectral function

S(E,− → k ): probability to find a proton in the nucleus with E and − → k .

50 100 150 200 250 300 10 10 10 10 10 10

−5 −4 −3 −2 −1

P(k,E) (fm /MeV) E (MeV)

3

k/k =0.75

F

Nuclear matter spectral function

Theoretical nuclear matter predictions

Momentum distribution: n(k) =

∞

dE · S(E, k) Spectroscopic factor: Z = 4π

∞

dk · k2 · n(k)

Kristoff Normand - Basel - 21 October 2004

Correlated Basis Functions

Separation of the correlated and mean field parts. S(E, k) = SIPSM(E, k) + Scorr(E, k) → Low energy and momentum part

→ IPSM

→ Correlated part

→ 20/22% in the SRC region → Nuclear matter for different densities → Local Density Approximation → Finite nucleus

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100 200 300 400 500 600

Kristoff Normand - Basel - 21 October 2004

Spectral function

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50 100 150 200 250 300 350

⇒ Correlated strength for large E and k. ⇒ Experimental measurement via electron scattering. ⇒ The strength found at large E and k, will be a signature of short range correlations.

Kristoff Normand - Basel - 21 October 2004

Reaction Mechanism

e (E ,p ) e’ (E ,p )

e’ e e e’

A A−1

p(E,k) ( ,q)

ν

p’ (E ,p )

p

Em = Ee − Ee′ − T ′

p − Trec

− → pm = − → q − − → p In PWIA: E = Em − → k = −− → pm dσ dEe′dEpdΩe′dΩp = KσepS(Em, pm)

Kristoff Normand - Basel - 21 October 2004

Final State Interactions

q e e’ p N N’ p’

q e e’ p ∆ π p’

⇒ rescattering process

→ SRC close neighbours → Different nuclei, C, Al, Fe, Au → kinematics choice

⇒ ∆ excitation

→ π threshold → simulate large Em → kinematics choice

Kristoff Normand - Basel - 21 October 2004

Kinematics

q p’

Perpendicular kinematics

k q p’

Parallel kinematics

k

100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700

Em (MeV) Pm (MeV/c)

100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700 100 200 300 400 500 600 700 800 900 100 200 300 400 500 600 700

Em (MeV) Pm (MeV/c)

Kinematical conventions Em Pm region accessible

Kristoff Normand - Basel - 21 October 2004

Continuous electron beam Three experimental halls Hall dedicated energy and current

Kristoff Normand - Basel - 21 October 2004

Kristoff Normand - Basel - 21 October 2004

SETUP e97006

⇒ e beam: 3.2 GeV/c ⇒ Targets:

• Solid targets: C, Al, Fe, Au -
• Cryogenics : H2 -

⇒ SOS: protons

• p: 0.85-1.7 GeV/c -
• angle: 29.0-73.0 deg-

⇒ HMS: electrons

• p: 2.05-2.75 GeV/c -
• angle: 12.5 deg-

Kristoff Normand - Basel - 21 October 2004

Hall C Detectors

xfp zfp Pb-glass Shower Counter S2X S1X S2Y S1Y DC2 DC1 gas Cerenkov

Detector package for both spectrometers Cerenkov Drift chambers Hodoscopes

Kristoff Normand - Basel - 21 October 2004

Analysis

Data reduction ⇒ Kinematical information ⇒ Yield normalisation

• Experimental charge
• Efficiencies
• Background substraction

⇒ Cross section calculation

⇒ De Forest off-shell Cross-section

Monte Carlo Simulation

⇒ Modeling of the two spectrometers ⇒ De Forest off-shell Cross-section ⇒ IPSM spectral function ⇒ Radiative weight (Makins, Weissbach)

Outputs for the analysis

⇒ Phase space ⇒ Radiative corrections

Checks with hydrogen runs Checks with quasielastic carbon runs

Kristoff Normand - Basel - 21 October 2004

Radiative Corrections

q e e’ p p’

q’

⇒ Bremsstrahlung (e and p) ⇒ Radiations emitted by particles

• internal (with nucleus involved)
• external (with other nucleus)

⇒ Two modifications

• cross section of the process
• kinematic of the particle

⇒ Need for corrections

Kristoff Normand - Basel - 21 October 2004

Elastic scattering at H2

Checking of experimental setup → No Final state interactions → Unique kinematics → Cross section known → Radiative corrections Allows various checks: ⇒ Offsets of spectrometers ⇒ Proton transmission ⇒ Cross section parametrisation

Kristoff Normand - Basel - 21 October 2004

Offsets

⇒ Reconstructed quantities Em, Pm, W ⇒ Beam and Spectrometers offsets

Ebeam PHMS θHMS φHMS PSOS θSOS φSOS % % rad rad % rad rad

• 0.1
• 0.234
• 0.001

0.0007 var. 0.0015 0.0007

psos < 1.27 : 0.00225 psos > 1.27 : 0.00225 − 0.033 ∗ (x − 1.27)2

• 0.008
• 0.006
• 0.004
• 0.002

0.002 0.004 0.006 0.8 1 1.2 1.4 1.6 1.8

P SOS (GeV)

• ffset psos

Kristoff Normand - Basel - 21 October 2004

Offsets

⇒ Reconstructed quantities Em, Pm ⇒ Beam and Spectrometers offsets correction applied

Kristoff Normand - Basel - 21 October 2004

H2 Checks

500 1000 1500 2000 2500

• 0.04
• 0.020 0.02

0.04 1000 2000 3000 4000 5000 6000

• 0.05

0.05 1000 2000 3000 4000 5000

• 2

2 1000 2000 3000 4000 5000 6000 7000

• 10 -5

5 10

0.04 −0.04

theta(rad) phi(rad) ytar(cm) p/p(%) ∆

Comparison for spectrometers ⇒ Agreement between simulation and data

0.96 0.98 1 1.02 1.04 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Q (GeV/c) ratio data/simc

2 2

Comparison of cross section ratio Mergell parametrisation (Nucl Phys A596 367)

Kristoff Normand - Basel - 21 October 2004

Carbon quasielastics

2000 4000 6000 8000 10000 12000 14000 16000

• 400 -300 -200 -100

100 200 300 400

10000 200 Pm(MeV/c) 400 −400 −200

500 1000 1500 2000 2500 3000 3500 4000 4500 20 40 60 80 100 500 1000 1500 2000 2500 3000 3500 4000 4500 20 40 60 80 100 500 1000 1500 2000 2500 3000 3500 4000 4500 20 40 60 80 100

Em(MeV)

1s 1p

2000 4000 20 40 60 80 100

Data SIMC

IPSM region: ⇒ low Pm ≤ 300 MeV/c ⇒ low Em ≤ 80 MeV/c data/simc = 0.85 ⇒ missing strength to higher Em and Pm

Kristoff Normand - Basel - 21 October 2004

Carbon

10 2 10 3 10 4 10 5

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• 15
• 10
• 5

5 10 15 20

• ✁

Coincidence time (ns) −20 20 −10 10 10 10 10 10

5 4 3 2

Background evaluation 2 ns beam microstructure

20 40 60 80 100 120 140 160 180

• 0.05

0.05 20 40 60 80 100 120 140 160

• 0.04
• 0.02 0

0.020.04 200 400 600 800 1000 1200

• 2

2 20 40 60 80 100 120 140 160 180

• 10

10

theta(rad) phi(rad) ytar(cm) p/p(%)

∆

Comparison for sos quantities ⇒ Correlation correction included ⇒ Good agreement simulation/data

Kristoff Normand - Basel - 21 October 2004

Transparency

Check of the transparency for 12C ⇒ Fraction of proton that escapes without FSI ⇒ Phase volume with: Em < 80MeV and Pm < 300MeV/c ⇒ PWIA model with radiation (SIMC) ⇒ Taking into account correlations

T =

• V d3pmdEmNexp(Em,pm)

ǫ·

• V d3pmdEmNPWIA(Em,pm)

0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 1

Q (GeV/c) transparency

Data Dutta Garrow 2 2

• D.Dutta, Phys. Rev. C 68 064603, 2003
• K. Garrow, Phys Rev C, 66, 044613, 2002

Kristoff Normand - Basel - 21 October 2004

Transparency

Transparency for the solid targets ⇒ Values from former experiments ⇒ Region of interest Target C Al Fe Au Transparency 0.6 0.5 0.4 0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10

Transparency Q2 (GeV/c)2 D C Fe Au

• D.Dutta, Phys. Rev. C 68 064603, 2003
• K. Garrow, Phys Rev C, 66, 044613, 2002

Kristoff Normand - Basel - 21 October 2004

Analysis

⇒ In the experiment:

dσ dEe′dEpdΩe′dΩp = N(Em,pm)−NB(Em,pm) L·Tp·ǫ·Φ(Em,pm)

Cderad(Em, pm) ⇒ In PWIA:

dσ dEe′dEpdΩe′dΩp = KσepS(Em, pm)

⇒ Deradiation process

• First extracted spectral function
• Fed into simc ⇒Cderad(Em, pm)
• Iteration until agreement

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1 10 10 2 10 3

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• 400
• 200

200 400 600

Kristoff Normand - Basel - 21 October 2004

Carbon

⇒ Extraction of the correlated spectral function

cross section for Carbon par 10

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100 200 300 400 500 600 cross section for carbon perp 10

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Kristoff Normand - Basel - 21 October 2004

Carbon

⇒ Comparison with CBF theory

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100 200 300 400 500

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250 300 350 400 450 500 550 600 650

Momentum distribution: n(pm) =

∆lim

60

dEm · S(Em, pm)

Kristoff Normand - Basel - 21 October 2004

Carbon

Global comparison with CBF theory Optimized working region Correlated strength

Zc = 4π

650

250 dpm · pm2 · n(pm) par. perp calc 0.549 1.265 0.599

• Equivalent strength
• missing protons

250 0.701 0.088 0.831 650 0.005 0.094 0.082 370 60 Pm

(MeV/c) (MeV)

Em 0.117 0.082 Correlated strength distribution

Kristoff Normand - Basel - 21 October 2004

Carbon

Global comparison with CBF theory Optimized working region Correlated strength

Zc = 4π

650

250 dpm · pm2 · n(pm) par. perp calc 0.549 1.265 0.599

• Equivalent strength
• missing protons

250 650 370 60 Pm

(MeV/c) (MeV)

Em 0.091 0.099 Correlated strength distribution

Kristoff Normand - Basel - 21 October 2004

Other targets

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Parallel kinematics for the 4 different targets

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Parallel kinematics for gold target.

Kristoff Normand - Basel - 21 October 2004

Summary

• Shell Model/experiments: missing nucleons

⇒ nucleon-nucleon correlations

• Extraction of correlated spectral functions for 4 nuclei
• Good agreement with expectations:

⇒ FSI role ⇒ kinematics/targets ⇒ theoretical predictions

• First experiment for direct SRC strength measurement

Kristoff Normand - Basel - 21 October 2004