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

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

Clustering in the Universe

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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!!

Bethe and Bacher (1936)

Ernest Rutherford (1899) Alpha particle decay

Wefelmeier (1937) Wheeler (1937) Von Weizsacker (1938)

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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

Hafstad and Teller (1938) Martin Freer (2010)

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8Be (1 bond)

12C (3 bonds)

16O (6 bonds)

20Ne (9 bonds)

24Mg (12 bonds)

28Si (16 bonds)

Experimental evidences for alpha clusters

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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

• 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

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)

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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 >

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

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

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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

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• 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

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New form of EOS in the neighborhood of saturation density

•

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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

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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

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Coulomb Energy

Equals :

and Where

• 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

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<position>

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

Binding energy and number of bonds

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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

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Density distribution

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

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Thank you…

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backups

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II EOS depends on Spin and Isospin, suitable for QMD

•

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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↓)

•

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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

•

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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)

•

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•

30 30

•

31 31

we get

For small spin polarization, the terms containing can be neglected in relation to and finally we can write

•

32 32

Finally we obtain,

•

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Substitute the terms with

•

34 34

Then we have EOS in the neighborhood of point of saturation

•

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Approximations used in calculating the average Hamiltonian

< 𝞥|𝑯|𝞥 >

we can calculate ,

So the symmetry term assumes following form,

𝑒𝑆𝑦𝑚 = 𝑎0 + 𝑎1ρ + 𝑎2ρ2

•

•

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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 ,

•

•

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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

Masses and radii of nuclei in its ground state according to new model

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• 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

•

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

•

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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