Download - GR analysis techniques
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.