Data Analysis: Overview1. Inelastic α scattering is used to study the isoscalar giant resonances

• low background at high excitation energy• Isoscalar giant resonances of all multipoles are excited

2. Differential cross section for inelastic scattering calculated in DWBA using an Optical Model Potential

• cross-section can be related to the form-factor

   

3. Optical Potential () is composed of real ( and imaginary ( components• Real part obtained by single folding effective interaction over density of target nucleus • Imaginary part represented by Woods-Saxon shape • Parameters obtained by fit to elastic scattering data

Data Analysis: Overview4. Target Densities

• Fermi shape for ground-state density • Transition densities to different multipoles obtained by deformation of ground-state

density5. Transition Potentials obtained by single-folding effective interaction over the target nucleus

transition density6. DWBA used to calculate differential cross-section of transition to each multipole

• Due to angular range, difficult to distinguish L>4• Strength of calculated L=0-4 multipoles varied to fit to experimental differential cross-

section• Obtain Energy Weighted Sum Rule (EWSR) for L=0-4 multipoles: sum of transition

possibilities from ground to excited, multiplied by excitation energy 

Transition Densities

Generate transition density by ground-state density deformation or nuclear structure calculation (e.g. RPA)

• Bohr-Mottleson form:

• The transition density for excitation of low-lying vibrational states• Used for GR with

• For GMR transition density, the “breathing mode”:

RPA calculations tend to give TD similar to this form

Transition Densities cont.

The dipole transition density is less transparent. The above form for l=1 corresponds to small displacement of the center of mass without change of shape.

The form used for the dipole, as derived by Harakeh and Dieperink:

where, R is the half-density radius of the Fermi mass distribution, β1 is the coupling collective parameter

Effective Interaction

• N-N interaction is averaged over density distribution of particle, represented by Gaussian with complex strength ()

• Hybrid approach where real and complex parts have different radial shapes (phenomenological W-S for imaginary part)

• Correction to strength by making interaction density dependent

• Dynamic correction to density dependence when applied to inelastic scattering and density becomes deformed. This correction reduces strength in the interior.

𝒗𝒈 (𝒔 )=− (𝒗+𝒊𝒘 )𝒆− 𝒔

𝟐

𝒕𝟐

ℑ𝑈 (𝑟 )=− 𝑊𝑒𝑥+1

, 𝑥=𝑟−𝑅𝑊

𝑎𝑤

When applied to inelastic scattering the density is deformed and this affects the interaction and

Effective Interaction

• N-N interaction is averaged over density distribution of particle, represented by Gaussian with complex strength ()

• Hybrid approach where real and complex parts have different radial shapes (phenomenological W-S for imaginary part)

• Correction to strength by making interaction density dependent

• Dynamic correction to density dependence when applied to inelastic scattering and density becomes deformed. This correction reduces strength in the interior.

𝑣𝑔 (𝑠)=− (𝑣+𝑖𝑤 )𝑒− 𝑠2

𝑡 2

𝐈𝐦𝑼 (𝒓 )=− 𝑾𝒆𝒙+𝟏

, 𝒙=𝒓 −𝑹𝑾

𝒂𝒘

When applied to inelastic scattering the density is deformed and this affects the interaction and

Effective Interactions

• N-N interaction is averaged over density distribution of particle, represented by Gaussian with complex strength ()

• Hybrid approach where real and complex parts have different radial shapes (phenomenological W-S for imaginary part)

• Correction to strength by making interaction density dependent

• Dynamic correction to density dependence when applied to inelastic scattering and density becomes deformed. This correction reduces strength in the interior.

𝑣𝑔 (𝑠)=− (𝑣+𝑖𝑤 )𝑒− 𝑠2

𝑡 2

𝐼𝑚𝑈 (𝑟 )=− 𝑊𝑒𝑥+1

, 𝑥=𝑟 −𝑅𝑊

𝑎𝑤

When applied to inelastic scattering the density is deformed and this affects the interaction and

Effective Interactions

• N-N interaction is averaged over density distribution of particle, represented by Gaussian with complex strength ()

• Hybrid approach where real and complex parts have different radial shapes (phenomenological W-S for imaginary part)

• Correction to strength by making interaction density dependent

• Dynamic correction to density dependence when applied to inelastic scattering and density becomes deformed. This correction reduces strength in the interior.

𝑣𝑔 (𝑠)=− (𝑣+𝑖𝑤 )𝑒− 𝑠2

𝑡 2

𝐼𝑚𝑈 (𝑟 )=− 𝑊𝑒𝑥+1

, 𝑥=𝑟 −𝑅𝑊

𝑎𝑤

When applied to inelastic scattering the density is deformed and this affects the interaction and

Continuum Subtraction

• Each spectrum divided into peak and continuum – straight line at high excitation joined to fermi shape at low excitation

• Results in a distribution which is the weighted average of distributions created using different continuum choices

0 10 20 30 40 50 60 700

2

4

6

8

10

12

Ex (MeV)Co

unts

θAVG = 4.3°

0 10 20 30 40 50 60 700

2

4

6

8

10

12

Ex (MeV)

Coun

ts

θAVG = 1.1° 44Ca

Inelastic α spectra obtained for 44Ca are shown. The lines are examples of continua chosen for analyses.

Fit to data

• Divide peak and continuum cross-sections into bins by excitation energy

• By comparing experimental angular distributions to the DWBA calculation, strengths of isoscalar L=0-4 contributions varied to minimize χ2

• IVGDR contributions are calculated and held fixed in the fits

• Uncertainty determined for each multipole fit by incrementing or decrementing strength of that multipole, adjusting strengths of other multipoles by fitting to the data, continuing until new χ2 is 1 unit larger than the best-fit total χ2

GR peak “sliced” into 300 keV bins for multipole decomposition analysis

Fit to data

• Divide peak and continuum cross-sections into bins by excitation energy

• By comparing experimental angular distributions to the DWBA calculation, strengths of isoscalar L=0-4 contributions varied to minimize χ2

• IVGDR contributions are calculated and held fixed in the fits

• Uncertainty determined for each multipole fit by incrementing or decrementing strength of that multipole, adjusting strengths of other multipoles by fitting to the data, continuing until new χ2 is 1 unit larger than the best-fit total χ2

0 2 4 6 80.1

1

10

100 Cont. 15.9 MeV

0 2 4 6 80.1

1

10

100 Peak 15.9 MeV

dσ/d

Ω(m

b/sr

)

44Ca

L=0L=2

L=1, T=1

0 2 4 6 80.1

1

10

100 Peak 20.2 MeV

dσ/d

Ω(m

b/sr

)

Peak

L=1, T=0

0 2 4 6 80.1

1

10

100 Cont. 20.2 MeV

Cont

0 2 4 6 80.1

1

10 Peak 25.5 MeV

θcm(deg.)

dσ/d

Ω(m

b/sr

)

Peak

L=4

L=3

0 2 4 6 80.1

1

10

100 Cont. 25.5 MeV

θcm(deg.)

Cont

The angular distributions of the 44Ca cross sections for three excitation ranges of the GR peak and the continuum are plotted vs. center-of-mass scattering angle.

Fit to data

• Divide peak and continuum cross-sections into bins by excitation energy

• By comparing experimental angular distributions to the DWBA calculation, strengths of isoscalar L=0-4 contributions varied to minimize χ2

• IVGDR contributions are calculated and held fixed in the fits

• Uncertainty determined for each multipole fit by incrementing or decrementing strength of that multipole, adjusting strengths of other multipoles by fitting to the data, continuing until new χ2 is 1 unit larger than the best-fit total χ2

5 10 15 20 25 30 35 400

0.03

0.06

0.09 E0

Frac

tion

EWSR

/MeV

44Ca

5 10 15 20 25 30 35 400

0.02

0.04

0.06

0.08 E1

5 10 15 20 25 30 35 400

0.05

0.1 E2

Ex (MeV)

Frac

tion

EWSR

/MeV

5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03 E3+E4

Ex (MeV)

Strength distributions obtained for 44Ca are shown by the histograms.