Determination of constraints on the Skyrme energy density...
Transcript of Determination of constraints on the Skyrme energy density...
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Collaborators:
IgalTalmi
Mason Anders
Giacomo Bonasera
Short-range correlations and the proton 3s1/2 wave
function in 206Pb
Shalom ShlomoTexas A&M University
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Outline
1. IntroductionExperimental data: 3s1/2 charge density in 206Pb
2. Determining the corresponding Single-Particlepotential directly from Matter Density: Derivation of method, application to 3s1/2 state in 206Pb.
3. Calculation of short-range correlation effect on single particle density; the 3s1/2 state in 206Pb
4. Conclusions
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In this talk I will consider the well-known experimental data of the
charge distribution of the proton 3s1/2 orbit given by the charge density
difference, 𝛥𝜌𝑐(𝑟), between charge density distributions of the isotones 206Pb – 205Tl, determined by analysis of elastic electron scattering.
The shell model, which is based on the assumption that nucleons in the
atomic nucleus move independently in single particle orbits associated
with a single particle potential, has been very successful in explaining
many features of nuclei.
I will present a novel method, using the single particle Schrodinger
equation with eigen-energy E, to determine the central potential 𝑉 Ԧ𝑟directly from the measured single particle matter density, 𝜌 Ԧ𝑟 and its
first and second derivatives, assuming known for all Ԧ𝑟. Using the
method we obtained the potential for the proton 3s1/2.
Introduction
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The resulting potential can also be used as an additional
experimental constraint in determining a modern energy density
functional (EDF) for more reliable prediction of properties of
nuclei and nuclear matter.
We have carried out calculations of the effects of short-range
correlations on the charge density difference 206Pb – 205Tl, 3s1/2
state, using the Jastrow correlation method. We derive and use a
simple approximation for determining the effect of short range
correlations on the charge density distribution.
Goal: Find out how short range correlations affect the charge
density distribution and whether it is necessary to help explain the
experimental data on the charge density of 3s1/2 orbit
Introduction
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Experimental Data:3s1/2
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r [fm]
ΔρC
[fm
-3]
Solid Line: Adjusted mean field calculation assuming a difference in
occupation probabilities of 0.7 in the 3s shell and 0.3 in the 2d shell
Measured charge density difference
between the isotones 206Pb and 205Tl
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−ћ2
2𝑚𝛥𝛹 + 𝑉𝛹 = 𝐸𝛹
𝑉 Ԧ𝑟 = 𝐸 +ћ2
2𝑚𝑆 Ԧ𝑟 , 𝑆 Ԧ𝑟 =
𝛥𝛹 Ԧ𝑟
𝛹 Ԧ𝑟
Nonsingular V:𝛥𝛹 Ԧ𝑟 = 0 when 𝛹 Ԧ𝑟 = 0
𝜌 Ԧ𝑟 = 𝛹 Ԧ𝑟 2
Single Particle Potential: Formalism
If we know single-particle W.F. we can determine 𝑉 Ԧ𝑟
Experimentally, one measures density:
Single-particle Schrodinger Eq.
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Spherical: 𝛹𝑛𝑙𝑗 Ԧ𝑟 =𝑅𝑛𝑙𝑗(𝑟)
𝑟𝑌𝑙𝑗
𝑉 𝑟 = 𝑉𝑐𝑒𝑛 𝑟 + Ԧ𝑠 ∙ Ԧ𝑙 𝑉𝑠.𝑜.(𝑟) +1
21 − 𝜏𝑧 𝑉𝑐𝑜𝑢𝑙(𝑟)
𝜏𝑧=1 for a neutron and -1 for a proton
𝑉𝑐𝑒𝑛 𝑟 = 𝐸 +ћ2
2𝑚𝑆 𝑟 −
ћ2
2𝑚
𝑙 𝑙 + 1
𝑟2−1
21 − 𝜏𝑧 𝑉𝑐𝑜𝑢𝑙 𝑟 − 𝑐𝑙𝑠 𝑉𝑠.𝑜. 𝑟
𝑆 𝑟 =𝑑2𝑅𝑛𝑙𝑗(𝑟)
𝑑𝑟21
𝑅𝑛𝑙𝑗(𝑟)
𝑐𝑙𝑠 = −(𝑙 + 1) and 𝑙 for 𝑗 = 𝑙 − Τ1 2 and 𝑗 = 𝑙 + Τ1 2, respectively
−ћ2
2𝑚
𝑑2𝑅𝑛𝑙𝑗𝑑𝑟2
+ 𝑉 𝑟 +ћ2
2𝑚
𝑙(𝑙 + 1)
𝑟2𝑅𝑛𝑙𝑗 = 𝐸𝑅𝑛𝑙𝑗
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𝑆(𝑟) =1
2𝑅𝑛𝑙𝑗2
𝑑2(𝑅𝑛𝑙𝑗2 )
𝑑𝑟2−1
2
1
𝑅𝑛𝑙𝑗2
𝑑(𝑅𝑛𝑙𝑗2 )
𝑑𝑟
2
𝑆 𝑟 =𝑑2𝑅𝑛𝑙𝑗(𝑟)
𝑑𝑟21
𝑅𝑛𝑙𝑗(𝑟)
𝑑(𝑅𝑛𝑙𝑗2 )
𝑑𝑟= 2𝑅𝑛𝑙𝑗
𝑑𝑅𝑛𝑙𝑗
𝑑𝑟
𝑑2(𝑅𝑛𝑙𝑗2 )
𝑑𝑟2= 2
𝑑𝑅𝑛𝑙𝑗
𝑑𝑟
2
+ 2𝑅𝑛𝑙𝑗𝑑2𝑅𝑛𝑙𝑗(𝑟)
𝑑𝑟2
𝑑(𝑅𝑛𝑙𝑗2 )
𝑑𝑟= 0 and
𝑑2(𝑅𝑛𝑙𝑗2 )
𝑑𝑟2−
1
2
1
𝑅𝑛𝑙𝑗2
𝑑(𝑅𝑛𝑙𝑗2 )
𝑑𝑟
2
= 0
And using
When 𝑅𝑛𝑙𝑗 = 0:
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𝑅𝑛𝑙𝑗2 𝑟 = 4𝜋𝑟2𝜌𝑛𝑙𝑗(𝑟)
𝑆 𝑟 =1
2𝜌𝑛𝑙𝑗
𝑑2𝜌𝑛𝑙𝑗𝑑𝑟2
+2
𝑟
𝑑𝜌𝑛𝑙𝑗𝑑𝑟
−1
2𝜌𝑛𝑙𝑗
𝑑𝜌𝑛𝑙𝑗𝑑𝑟
2
𝑑𝜌𝑛𝑙𝑗
𝑑𝑟=0 and
𝑑2𝜌𝑛𝑙𝑗
𝑑𝑟2+
2
𝑟
𝑑𝜌𝑛𝑙𝑗
𝑑𝑟−
1
2𝜌𝑛𝑙𝑗
𝑑𝜌𝑛𝑙𝑗
𝑑𝑟
2
=0
We consider the 3s1/2 proton orbit. No S.O. and l=0
Therefore, using the relation:
When 𝜌𝑛𝑙𝑗 = 0:
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𝜌𝑐ℎ Ԧ𝑟 = න𝜌𝑝(𝑟′) 𝜌𝑝𝑓𝑠(Ԧ𝑟 − 𝑟′)𝑑3𝑟′
𝜌𝑝𝑓𝑠 Ԧ𝑟 =1
8𝜋𝑎3𝑒 Τ−𝑟 𝑎
𝑎2 =1
12𝑟𝑝𝑓𝑠2 with 𝑟𝑝𝑓𝑠 = 0.85 fm (rms radius of 𝜌𝑝𝑓𝑠)
𝑟2 𝑐ℎ = 𝑟2𝜌𝑐ℎ Ԧ𝑟 𝑑 Ԧ𝑟 𝜌𝑐ℎ/ Ԧ𝑟 𝑑 Ԧ𝑟
𝑟2 𝑐ℎ = 𝑟2 𝑝 + 𝑟2 𝑝𝑓𝑠
Free proton Charge Distribution
𝜌𝑐ℎ Ԧ𝑟 =1
4𝜋𝑎න0
∞
𝑟′𝑑𝑟′𝜌𝑝(𝑟′) 1 +
𝑟 − 𝑟′
𝑎𝑒− Τ𝑟−𝑟′ 𝑎
− 1 +(𝑟 + 𝑟′)
𝑎𝑒− Τ(𝑟+𝑟′) 𝑎
NOTE Experiment determines charge distribution;
Theory determines point distribution
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𝐹 𝑞 =4𝜋
𝑞න0
∞
sin 𝑞𝑟 𝜌 𝑟 𝑟𝑑𝑟
𝜌 𝑟 =1
2𝜋 3
4𝜋
𝑟න0
∞
sin 𝑞𝑟 𝐹 𝑞 𝑞𝑑𝑞
𝐹𝑝𝑓𝑠 𝑞 = 1 +1
12𝑟𝑝𝑓𝑠2 𝑞2
−2
𝐹𝑐ℎ 𝑞 = 𝐹𝑝𝑓𝑠 𝑞 𝐹𝑝 𝑞
Define Fourier transform of density:
𝜌𝑝 𝑟 =1
2𝜋 3
4𝜋
𝑟න0
∞
sin 𝑞𝑟 𝐹𝑝 𝑞 𝑞𝑑𝑞
We get the point proton density:
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Measured charge density difference
Calculated proton density difference
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Measured charge density difference
Calculated proton radial WF2 difference𝑅2 𝑟 = 4𝜋𝑟2𝜌(𝑟)
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𝛥𝜌𝑐 𝑟 = 𝜌𝑐ℎ(𝑟;206Pb) −𝜌𝑐ℎ(𝑟;
205Tl)
We have therefore determined potentials by fits
to the experimental data.
Due to large uncertainty in the experimental data for
particularly, around the nodes of the 3s1/2 proton
wavefunction, we are not able to extract the corresponding
potential with reasonable accuracy.
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Fitted WS: V0= -167.95 MeV R1= -0.03 fm and a0 = 4.68 fm
Conventional WS: V0= -62.712 MeV R1= 7.087 fm and a0 = 0.65 fm
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Calculation of Short Range correlation effect on 3s1/2 density
We use the Jastrow many-body correlated wave function
with a repulsive two-body correlation factor f12 and
harmonic oscillator single particle wave functions
We derive a simple and accurate (within a few percents)
approximation for the effect of the two-body correlation
factor f12 on matter density and calculate the effect of short
range correlations on charge density distributions of 206Pb
and 205Tl
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The Shell Model and the Jastrow Correlated
many-body wave functions
Two-body correlation factor:
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Shell Model single particle density
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Correlated single particle density
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Two-body Correlation function
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3s1/2
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3s1/2 Proton Point Density
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3s1/2 Proton Charge Density
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Conclusions We have developed and used a new method of determining the
single particle potential directly from the density distributionand applied it to the experimentally determined density distributions of the proton 3s1/2 state in 206Pb and obtaining an acceptable form for the potential.
We carried out a least-squares fit of a potential providing a fit to the density data which is a much better fit than the conventional Woods-Saxon potential especially near r = 0 fm
Clearly more accurate data is needed to better determine the potential and answer the question how well can the data be reproduced by a calculated 3s1/2 single particle wave function
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Conclusions-Cont.
We have calculated the effect of short-range correlations on charge densities of 206Pb and 205Tl, using the Jastrowcorrelated many-body wave function with a repulsive two-body short range correlation function.
We demonstrated that, although correlated 3s1/2 charge density at r=0 is reduced by about 30%, the calculated density disagrees with the experimental data by more than a factor of 2, particularly in the region of r = 2 – 4 fm.
Further investigations are needed using a more realistic two-body correlation function and determining the effect of short range correlations and the corresponding single particle potential.
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Acknowledgements
TAMU and Cyclotron Institute
DOE support
Weston Visiting Professor Award, Weizmann Institute