semi-classical microscopic approach to the liquid drop ... · semi-classical microscopic approach...
TRANSCRIPT
![Page 1: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/1.jpg)
Semi-classical microscopic approach to the liquid drop model and the emergence of alpha
clusters
Jinesh Kallunkathariyil
Outline : • Introduction to the clustering phenomena • Why not clusters in nuclei? Experimental
evidences • Semi-classical microscopic approach to the
Liquid Drop Model • Emergence of alpha clusters • Conclusions
1 20th Nuclear Physics Workshop "Marie & Pierre Curie“ Kazimierz 2013
![Page 2: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/2.jpg)
2
Nuclear Structure Theories
• Nuclei as we know now is roughly a homogeneous distribution of protons and neutrons
• Experimental evidences: Binding energy curve and charge radius
• Liquid drop model
• Shell model
![Page 3: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/3.jpg)
Clustering in the Universe
3
Galaxy Clusters
Virgo Cluster
Star Clusters
Milky Way
Planets
Solar system
Quark Clusters
Protons and Neutrons
Atom Clusters
Crystal
Proton-Neutron Clusters
Alpha particle
Alpha Clusters
Nucleus?
If not, surprising!!
![Page 4: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/4.jpg)
Bethe and Bacher (1936)
Ernest Rutherford (1899) Alpha particle decay
Wefelmeier (1937) Wheeler (1937) Von Weizsacker (1938)
4
History
“Just because you see alpha particles coming out of nucleus, you should not necessarily conclude that inside they exist as such ”
4 n nuclei
![Page 5: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/5.jpg)
Hafstad and Teller (1938) Martin Freer (2010)
5
8Be (1 bond)
12C (3 bonds)
16O (6 bonds)
20Ne (9 bonds)
24Mg (12 bonds)
28Si (16 bonds)
![Page 6: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/6.jpg)
Experimental evidences for alpha clusters
6
Ikeda diagram
Hoyle state – Triple alpha reaction is important to synthesize heavier nuclei in stars
Alpha decay in 8Be
Alpha cluster structure in Ni-56 H Akimune et al 2013 J. Phys.: Conf. Ser. 436 012010
Alpha-cluster structure in the ground state of 40Ca displayed in a (p,pα) knockout reaction A A Cowley 2013 J. Phys.: Conf. Ser. 436 012011
• 4He(56Ni,4He) inelastic scattering experiment • Even without a dedicated detector • Could not explained by any statistical decay models
![Page 7: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/7.jpg)
• Antisymmetrized Molecular Dynamics (AMD) • Fermionic Molecular Dynamics (FMD • Resonating Group Method (RGM) and Generator Coordinate Method
(GCM) • Bloch–Brink cluster model • Local potential cluster models • Semi-classical, microscopic approach to the liquid drop model
Liquid Drop Model (LDM):
<E> = BV + B
surf + V
coul + …
Zbigniew Sosin (http://arxiv.org/abs/1304.2846)
Models to calculate the structural properties of nuclei
7
![Page 8: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/8.jpg)
in present description we consider the system as the superposition
of nucleons and every nucleon is considered as the wave packet:
The interaction is given by:
𝑉 𝐫, < 𝐫𝑘 > , 𝞼𝑘 = 𝑉𝑁(𝐫, {< 𝐫𝑘 >}, 𝞼𝑘) + 𝑉𝑆(𝐫, {< 𝐫𝑘 >}, 𝞼𝑘) + 𝑉𝐶𝑜𝑢𝑙(𝐫, {< 𝐫𝑘 >}, 𝞼𝑘)
8
![Page 9: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/9.jpg)
9
Equations of Motion
• The equation of motion of variables < 𝒓𝒌 >,< 𝒑𝒌 > and 𝞼𝒌 are derived using the time-dependent variational principle based on the action minimization.
• For this purpose, we define the action as :
• With the Lagrange function given as :
𝑆 = 𝐿 𝞥,𝞥∗ 𝑑𝑡𝑡2
𝑡1
𝐿 =< 𝞥|𝑖ћ 𝑑
𝑑𝑡−𝐻 𝞥 > = < 𝞥 𝑖ћ
𝑑
𝑑𝑡𝞥 > − < 𝞥 𝐻|𝞥 >
![Page 10: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/10.jpg)
Average value of the kinetic energy:
now the average value of the Hamiltonian is given by:
excitation
internal
Fermionic motion
The ground state
volume energy 10
![Page 11: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/11.jpg)
Equation of state of nuclear matter (EOS)
𝒆 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔 =𝑬
𝑨
Parameters are the thermo dynamical parameters like 𝑻, 𝝆𝟎, 𝜹, …
𝒆 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔 = 𝑬𝑲𝒊𝒏
𝒃𝒂𝒓𝒊𝒐𝒏+
𝑽
𝒃𝒂𝒓𝒊𝒐𝒏
Often, the EOS is written as a sum
𝒆 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔 = 𝒆𝑻 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔′, 𝑻 + 𝒆𝒄 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔′, 𝑻 = 𝟎 + 𝑪𝒐𝒏𝒔𝒕𝒂𝒏𝒕
Where 𝒆𝑻 is associated with Thermal excitation and 𝒆𝒄 is the compression excitation If thermal energy is negligible compared to the compression energy (widely used approximation), then we have
𝒆 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔 ≃ 𝒆𝑪 𝒑𝒂𝒓𝒂𝒎𝒆𝒕𝒆𝒓𝒔,′ 𝑻 = 𝟎
11
![Page 12: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/12.jpg)
Expansion of EOS term around the vicinity of saturation density ρ0 • In the case of spin and isopin balanced matter,
𝐸
𝐴= 𝐸 ρ = 𝑒0 +
𝐾
18 ρ−ρ0
ρ0
2
0 = −16𝑀𝑒𝑉 +𝐾
18
0−ρ0
ρ0
2⇒ 𝐾 = 288≃ 300 MeV
𝑒0 = −16MeV
𝑛𝑢𝑐𝑙𝑒𝑜𝑛/𝑓𝑚3
12
![Page 13: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/13.jpg)
• In the case of spin-balanced and unbalanced isospin matter, we can write:
𝐸
𝐴= 𝐸 ρ = 𝑒00 +
𝐾018 ρ − ρ0ρ0
2
+ δ2𝐸𝑆𝑦𝑚(ρ)
δ =ρ𝑛 − ρ𝑝ρ
Isospin symmetry energy is not well known and great experimental and theoretical effort has to be given for its determination
𝐸𝑆𝑦𝑚 ρ = 𝑒𝐼0 +𝐿𝐼3
ρ − ρ0ρ0
+𝐾𝐼18
ρ − ρ0ρ0
2
+⋯
We want to determine: • Wigner constant eI0 • the slope LI and • curvature KI
13
![Page 14: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/14.jpg)
14
New form of EOS in the neighborhood of saturation density
![Page 15: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/15.jpg)
•
15
New EOS and some theoretical predictions.
Theoretical Predictions
New form of EOS
δ=1 0.8 0.6 0.4 0.2 0
Neutron matter
Symmetric matter
![Page 16: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/16.jpg)
16
Surface Energy
fm
Z.Sosin, International Journal of Modern Physics E Vol. 19, No. 4 (2010) 759-767
Surface energy is defined as a function of nucleon wave packet parameters
![Page 17: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/17.jpg)
17
Coulomb Energy
Equals :
and Where
![Page 18: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/18.jpg)
• In this approach by the minimization of average value of Hamiltonian we determined the ground state wave function
• This wave function for even-even nuclei can be interpreted with some probability as a cluster of alpha particles
18
<position>
![Page 19: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/19.jpg)
19
• In our minimization approach, we obtained many local minima eg:- 12 C
• For heavier nuclei there is more different structures which gives local minima.
• Some subtle effects can decide which structure we can take as a correct one
• In the future we will concentrate on more sophisticated methods to select the exact structure for example charge distribution
• We cannot yet claim the given structures are final
![Page 20: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/20.jpg)
Binding energy and number of bonds
20
![Page 21: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/21.jpg)
First Approximation
Binding energy of alpha particle (4He),
𝑏𝛼 = 28.296𝑀𝑒𝑉
𝐶12 = 3𝛼 ?
𝐵 𝐶12 = 92.163𝑀𝑒𝑉
3𝛼 = 84.888𝑀𝑒𝑉 < 𝐵 𝐶12
𝐵 = 𝑛𝛼 × 𝑏𝛼 + 𝑛𝑏 × 𝑏𝛼−𝛼
𝑏𝛼−𝛼 = 2.425𝑀𝑒𝑉
21
![Page 22: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/22.jpg)
22
Density distribution
![Page 23: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/23.jpg)
Conclusions
• This model is at a preliminary stage
• We are considering the available experimental results and theoretical predictions to go further
• We hope that this model can help to understand the mechanism involved in various nuclear reactions
• The model is still developing
23
![Page 24: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/24.jpg)
Thank you…
24
![Page 25: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/25.jpg)
backups
25
![Page 26: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/26.jpg)
II EOS depends on Spin and Isospin, suitable for QMD
•
26 26
Where , • 𝝆𝒑↑ is the density of proton gas with spin up,
• 𝝆𝒑↓ is the density of proton gas with spin down, • 𝝆𝒏↑ is the density of neutron gas with spin up and • 𝝆𝒏↓ is the density of neutron gas with spin down.
Because of the symmetry of the nuclear interaction the following operation conserves energy: 1. The mutual exchange of protons and neutrons 2. Inversion of the spin of the all nucleons .
𝑒 = 𝑒(ρ𝑝↑, ρ𝑝↓, ρ𝑛↑, ρ𝑛↓)
![Page 27: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/27.jpg)
•
27
27
when we mutually exchange neutrons and protons that is: ρ𝑛 → ρ𝑝 and ρ𝑝 → ρ𝑛 then δ becomes - δ
The same case for η𝑛 and η𝑝 when we change the spin
In order to make use of such a symmetry we introduce new coordinates
![Page 28: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/28.jpg)
•
28 28
Now the EOS can be written as,
We can expand above energy around 𝜉 = 0, 𝛿 = 0, η𝑛 = 0, η𝑝 = 0
Exchange operation associated with previously mentioned symmetry causes the odd order of derivate associated with the expansion terms equals to zero, otherwise the energy of the system will change.
𝑒 = 𝑒(𝜉, 𝛿, η𝑛, η𝑝)
![Page 29: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/29.jpg)
•
29 29
![Page 30: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/30.jpg)
•
30 30
![Page 31: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/31.jpg)
•
31 31
we get
For small spin polarization, the terms containing can be neglected in relation to and finally we can write
![Page 32: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/32.jpg)
•
32 32
Finally we obtain,
![Page 33: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/33.jpg)
•
33 33
Substitute the terms with
![Page 34: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/34.jpg)
•
34 34
Then we have EOS in the neighborhood of point of saturation
![Page 35: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/35.jpg)
•
35 35
Approximations used in calculating the average Hamiltonian
< 𝞥|𝑯|𝞥 >
we can calculate ,
So the symmetry term assumes following form,
𝑒𝑆𝑦𝑚 = 𝑎0 + 𝑎1ρ + 𝑎2ρ2
![Page 36: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/36.jpg)
•
•
36 36
Part of olume energy can be calculated by taking the integral of the expression
δ𝟐 𝒆𝒔𝒚𝒎ρ = δ𝟐( 𝒂𝟎 + 𝒂𝟏ρ + 𝒂𝟐ρ
𝟐)ρ
Which contains an expression of type
δ𝟐𝒂𝟎ρ →𝟏
ρ𝟐𝒂𝟎ρ = 𝒂𝟎 ρ
When the density is expressed as a sum of Gaussian distributions, this term can not be integrated analytically Therefore, in this model we propose the form ,
![Page 37: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/37.jpg)
•
•
37 37
Polynomial coefficients calculated from the condition of equal energy, and its first and second derivative:
For this EOS the energy in the mean volume Hamiltonian is given analytically! This form of EOS is also able to communicates with the basic theoretical calculations
![Page 38: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/38.jpg)
Masses and radii of nuclei in its ground state according to new model
38
• In the first step, we determine 𝒆𝟎𝟎 , 𝝆𝟎 , 𝑲𝟎, 𝐚𝐧𝐝 𝒔𝟎 parameters.
• For the primary estimations assumed
𝝆𝟎 = 𝟎. 𝟏𝟓𝟗 𝒏𝒖𝒄𝒍/𝒇𝒎𝟑 and 𝐊 = 𝟑𝟎𝟎 𝐌𝐞𝐕
• Used the binding energy and RMS radius for 𝑯𝒆𝟒 𝐚𝐧𝐝 𝑪𝟏𝟐 for
determining and found
𝒆𝟎𝟎 = −𝟏𝟐. 𝟗 𝑴𝒆𝑽 𝑎𝑛𝑑 𝒔𝟎 = 𝟎. 𝟎𝟗 𝒇𝒎𝟐
(These nuclei have less effect on spin and isospin parameters of EOS)
We are searching for wave function which minimize the average value of Hamiltonian => search for 𝐫𝑘 , 𝐩𝑘 , σ𝑘
![Page 39: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/39.jpg)
•
39
• Next we used BE and RMS radii of 𝑯𝟐 , 𝑯𝟑 , 𝑯𝟑 𝒆 to obtain an estimated value of isospin and spin polarization parameters. The rough estimation value obtained for isospin parameters ,
𝒆𝑰𝟎 = 𝟑𝟎 𝑴𝒆𝑽, 𝑳𝑰 = 𝟏𝟐𝟑 𝑴𝒆𝑽,𝑲𝑰 = 𝟓𝟎𝟎 𝑴𝒆𝑽 and for independent neutron spin and proton spin interaction, 𝒆𝒊𝒊𝟎 = 𝟔𝟓 𝑴𝒆𝑽, 𝑳𝒊𝒊 = 𝟑𝟎𝟎 𝑴𝒆𝑽,𝑲𝒊𝒊 = 𝟑𝟎𝟎 𝑴𝒆𝑽 and the mutual spin interaction of protons and neutrons described by
𝒆𝒊𝒋𝟎 = −𝟏 𝑴𝒆𝑽, 𝑳𝒊𝒋 = 𝟐𝟎 𝑴𝒆𝑽,𝑲𝒊𝒋 = 𝟑𝟎𝟎 𝑴𝒆𝑽
![Page 40: Semi-classical microscopic approach to the liquid drop ... · Semi-classical microscopic approach to the liquid drop model and the emergence of alpha clusters Jinesh Kallunkathariyil](https://reader034.vdocument.in/reader034/viewer/2022042311/5ed9682af59b0f56f45f6eed/html5/thumbnails/40.jpg)
•
40
First Calculation results Applying the searching procedure for 𝐫𝑘 , 𝐩𝑘 , σ𝑘 one can obtain
It shows that VISA QMD model could reproduce the basic properties of atomic nuclei