1

NUCLEAR CLUSTERING, STEP TO A

SUPERCOMPUTING APPROACH

Yu. M. Tchuvil'sky

Skobeltsyn Institute of Nuclear Physics, Lomonosov

Moscow State University, Moscow, Russia

D. M. Rodkin

Dukhov Research Institute for Automatics, Moscow,

Russia

2

STATEMENT OF THE GENERAL PROBLEM

The presented investigation is in line with the general

strategy formulated in monograph K. Wildermuth and

Y.C. Tang “A unified theory of the Nucleus” Veiweg,

Braunschweig, 1977.

The formula of the strategy is: to build a mathematics

and computational methods which make possible to

study various aspects of shell and cluster nuclear

structure in a unified microscopic approach and thus

throw bridges between the theory of nuclear structure

and the theory of nuclear processes.

3

The new era imposes new requirement – to change

from ordinary “microscopicity” to “ab initio”; and

provides new instrument – supercomputer.

The aim of the current paper is to build up an

algebraic approach adopted for microscopic and ab

initio description of the clustered systems, states with

halo nucleon, one-nucleon and cluster resonance

states .

During last decade a closely related strategy was

progressed and significant advances have been made

in realization of it. The No-Core Shell Model /

Resonating Group Model (NCSM/RGM) and No-Core

Shell Model with Continuum (NCSMC) were created

by P. Navratil, S Quaglioni, R. Roth, G. Hupin S.,

Baroni et al.

4

CONSTRUCTION OF THE BASIS

Two-fragment clustering in bound and resonance states

is considered.

As in the NCSMC the shell-model (polarization) terms

of the basis Ψpol are solutions of the A-nucleon

Schrödinger equation with an arbitrary interaction of ab

initio or an effective type written in the form of Slater-

determinant (SD) superposition.

The cluster-channel terms are built in the form:

1 2

1 ˆ{ ( )} ,JA A A nlm JM

AW

1 2ˆ,A A A A is the antisymmertizer

iA

– translationally-invariant WF of the fragment;

i,

( )nlm

– the oscillator WF of the relative motion.

(1)

5

These terms are determined by the quantum numbers of the

internal cluster states, indexes n,l,m and global QN. The

main problem is to present function (1) as a linear

combination of the SDs containing one-nucleon WFs

1 1/ 2

2

( , ) ( ) ( , ) ( ).l sl s

m m

nl lnlm m

r R r Y

For these purposes function (1) is multiplied by000 ( )R

– the function of zero vibrations of the center of mass. This

operation is commutative with the antisymmetrization. Then

Talmi-Moshinsky transformation for particle of different mass

(Yu.F. Smirnov, Nucl. Phys. 27, 177 (1962)) is performed:

1 2

1 1 1 2 2 2

1 1 1 2 2 2

1 1 1

000 , , 1 , , 2

, , , , , 2 2 2

, ,000( ) ( ) ( ) ( ).

, ,

A A

nlm N L M N L M

N L M N L M

N L MR R R

N L Mnlm

6

Thus WF (1) is expressed in the form

1

1

1 1 1 2 2 2

2

2 2 21 1 1 2

1 1 1

000

, , , , ,

, ,

2

,

2

, 2

2

1

, ,0001( )

, ,

ˆ{ } .)( ()J

A

N L M

A

N L

A

N

A

L M N L

JA MM

M

N L MR

N L Mnlm

A R R

W

A critical procedure is to transform products (in color)into the superposition of SDs.

( )

, , , , ( )( ) .i i

i i i i i i i i

A A k SD

N L M i A N L M A k

k

R X

( )

, , | ( )i

i i i i i i

A k shell

N L M A nl A AX R

Coefficient

is called cluster coefficient (CC). Mathematics of these

objects is in a large measure our know-how. It is well-

developed (M. Ichimura et al. NPA 204 (225 (1973); Yu.F.

Smirnov, YMT, PRC 15, 84 (1977); A. Volya, YMT, PRC

91, 044319 (2015); monograph O. F. Nemetz et al.

Nucleon Clusters in Atomic Nuclei and Multi-NucleonTransfer Reactions (Naukova Dumka, Kiev, 1988 etc.).

(2)

7

Methods elaborated for calculations of CC are many

and varied. The most general one is based on the

method of second quantization of the oscillator quanta.The WF of CM motion is presented as

† †

, , , , 000ˆ ˆ( ) ( ) ( ) ( ),i i i i

i i i i i i i

A N L A

N L M i N L N L iR N Y R

where †̂ – creation operator of the oscillator quantum,

and

,

4( 1)

( )!!( 1)!!i i

i i

N L

N L

i i i i

NN L N L

thus

† †

, , , 000ˆ ˆ| ( ) ( ) ( ) ( ) .i i i

i i i i i i i i i i i

N L Ashell shell

A N L A A N L A N L i AR N Y R

Formula000/ ( )

i

i i

Ashell

A A iR

is the definition of the translationally-invariant WF here.

8

The just presented components (1) are not

orthogonal one to another and to the shell-model

components.

The basis of cluster-channel terms (1) incorporating

all channels of a certain fragmentation A1 + A2 (a

complete set of internal states of each cluster) is

complete.

Moreover this basis is overload and even linear

dependent.

The basis may be exploited by itself or being added

to a certain number of shell-model WF (polarization

terms). In the latter case a hybrid basis appears.

9

The next step in shaping of a basis of general type is to

build orthonormalized WFs including the cluster terms of

several channels and the polarization terms. The WFsare obtained by diagonalization of the matrix

( )

, ,

1,2

( ) 2

, , ' , ' , ' , ,

1,2 1,2 1,2

ˆ ( )

ˆ ˆ( ) ' ( ) ' ( )

1 ii i i i

i i i

i i i i i i i i i i i i

Aj

pol N L M i A

i

A A Aj

pol N L M i A N L M i A N L M i A

i i i

A R

A R R A R

in which the terms of the products are expressed in the

form of superpositions of SDs by use of the formula (2).

Eigenvectors of the matrix normalized by its eigenvalues

shape the desirable basis taking the form of SD linear

combinations. This basis may be employed in computing

of spectra of halo, clustered, resonance states and otherobservables.

(3)

10

ADVANTAGES OF THE BASIS

1. Calculations of matrix elements in the basis are similar

in design to ordinary shell-model computations being

reduced to the work with SDs. So an arbitrary

microscopic (ab initio or effective, including two-, three-,

etc. nucleon forces) Hamiltonian may be explored.

2. The cluster part of the basis is rather short. Limitations

in use of the approach are imposed by the dimensionality

of a typical basis vector.

3. The approach as a whole is very flexible due to

possibilities to vary: number of cluster channels;, relative

motoon, A-, A1, A2-nucleon model spaces, (nmax, N(A)

max);

basis may be orthogonal, non-orthogonal and, if

necessary, overloaded. This presents a way to take into

account various halo, cluster and other properties of asystem.

11

CLUSTER FORM FACTORS AND

SPECTROSCOPIC FACTORS

Terms (1) represented in the proper form may be usedfor the purposes other than spectral, etc.computations. The formalism is convenient for thecalculations of the cluster form factors (CFF) and thespectroscopic factors (SF) of arbitrary solutions of A-nucleon problemΨA . These values are obtained by:1. Projection of ΨA onto the non-orthonormalizes terms (1)

1 2 1 2

ˆ| { ( ) } .nlAA A A A nl AC A

2. Diagonalization of sub-matrix contained in right-lower

quadrant of the matrix (3) which is reduced by additional

condition ΨAi =Ψ’Ai:

12

The object

1/ 2( ) ( );nl kl k MDC lk nk

C f

is called CFF. As a result SF takes the form:

1 1 1 2 1 2

2 2 1 '( ) '

'

( ) | .nl n l k kAA A l l k AA A AA A nl n lk nn

S d C C B B

Its eigenvalues and eigenfunctions take the forms:

ˆ ˆ ˆ{ ( ) } |1| { ( ) } ;k kk D l C D l CA f A f

( ) ( ).k kl nl nln

f B

This definition of CFF and SF plays an important role

in the theory of nuclear reactions. The authors of the

idea (T. Fliessbach, and H. J. Mang, Nucl. Phys. A263 (1976) 75) called the values “new” CFF and SF.

(5)

1 2 1 2

2' 000 000 '

ˆ|| || ( ) ( ) | | ( ) ( ) .nn A nl A A n l AN R A R

(4)

13

AGGREGATE AMOUNT OF CLUSTERING

The “new” SF is mathematically well-defined, its values are

limited by 1. A number of sum rules are the consequences

of its definition. That is why it was proposed in (R. G. Lovas

et all, Phys. Rep. 294 (1998) 265) to consider “new” SF as

a quantitative measure (statistical weight, probability,

“amount “) of clustering.

The WF (CFF) of different channels are nonorthogonal even

after the just presented procedure. That is why a treatment

of the multi-channel problem is a delicate task. For these

purposes it is necessary to introduce a more general

definition of aggregate amount of clustering (AOS) .

14

The generalization procedure is the following. Right-

lower sub-matrix of (3) contains WF of different states of

fragments determining certain channels in this case:

1 1 1 2

2' 000 000 '

ˆ|| || ( ) ( ) | | ( ) ' ( ) ' .nn A nl A A n l AN R A R

After the diagonalization of it orthnormalized set of

coupled-channel A-nucleon cluster WF appear. The sum

of squared overlaps of the wave function ψA with these

WF provides a proper definition of the aggregate amount

of clustering.

Contrary to this definition equality (5) may be called the

one-channel amount of clustering.

15

RESULTS AND DISCUSSION

Both realistic phenomenological potential JISP16 based

on complete set of 2N-data (A. M. Shirokov, J. P. Vary, A.

I. Mazur, and T. A. Weber, PLB 644 (2007) 33) and

potential Daejeon16 which is built starting from N3LO

forces (A.M. Shirokov, I.J. Shin, Y. Kim et al, PLB 761,

87 (2016) are explored. Both potential are well-tested in

the calculations of A-nucleon systems (A≤16).

Codes Antoine and Bigstick are used for shell-model

computing of the polarization terms and the internal WFs

of clusters.

Two strongly clustered systems 8Be and 7Li are

considered.

16

CHOICE OF BASES

1. Conventional basis of NCSM – mod1.

2. Pure cluster one-channel basis (both 4He clusters in

GS, truncation level – Nmax = 0) – mod2.

3. Three- or two-channel basis incorporating the realistic

WFs of the first and the second 0+ states of 4He with

truncation level Nmax =2 – mod3.

4, 5. Two hybrid bases containing complete set of NCSM

WF up to the truncation level of 8Be and 7Li Nmax-2:

one of them is doped with one cluster component Nmax

corresponding to mod2 conditions, another one – doped

with three (or two) components Nmax for 8Be (or 7Li)

corresponding to mod3 conditions. They are called mod4

and mod5.

17

JISP16, ħω =15 MeV. Red – mod1, green – mod2, blue– mod3.

GS BINDING ENERGIES OF 8Be NUCLEUS

18

Daejeon16, ħω =15 MeV. Red – mod1, green – mod2,blue – mod3.

19

AOS OF α-PARTICLES in 8Be

Daejeon16

JISP16

N=4 N=6 N=8 N=10 N=12

α+α 0.582 0.406 0.801 0.833 0.843

α+α , α*+α 0.871 0.829 0.807 0.846

α+α , α*+α , α*+α * 0.923 0.856 0.827 0.847

N=4 N=6 N=8 N=10 N=12

α+α 0.068 0.765 0.866 0.861 0.875

α+α , α*+α 0.442 0.793 0.868 0.868

α+α , α*+α , α*+α * 0.992 0.864 0.879 0.873

20

Daejeon16

JISP16

N=4 N=6 N=8 N=10 N=12

mod1 36.20 46.47 52.17 54.62 55.72

mod4 39.00 47.16 52.34

mod5 38.98 46.82 52.22

N=4 N=6 N=8 N=10 N=12

mod1 22.21 34.93 44.62 49.68 52.25

mod4 27.07 36.27 45.02

mod5 24.14 35.08 44.78

GS ENERGY of 8Be NUCLEUS

21

nNmax, (7Li) 11 13

E 38.53 38.90

S, Nmax (4He) =0 0.840 0.836

S, Nmax (4He) =0 0.848 0.857

GS BINDINDG ENERGIES and SF of α+t CHANNEL in 7Li

22

GS binding energy of 7Li nucleus. Solid line – mod1,

dotted – mod2, dashed-dotted – mod3, dashed – mod5.

23

SUMMARY

1. The approach adopted for microscopic and ab initio

description of the clustered systems, systems with halo

nucleon, narrow resonance states is developed. The

mathematics is based on the cluster coefficients technique.

2. The method provides a way to perform calculations using

various bases including the shell-model components

together with the cluster terms of several channels i. e. to

construct versions of the basis conforming to the properties

of systems under study.

3. Calculations of the ground states of 8Be and 7Li nuclei in

the framework of ab initio approach show that habituated

two-cluster view on strongly-clustered systems gives but a

rough idea of its structure.

24

Indeed:

A. “Non-clustered” components contribute significantly to

the binding energies. For large values N it is typical that

the relative contribution of these components (additional

energy per additional SF) is larger that the one of the

cluster components.

B. Including of several “excited” channels does not

change a pattern because of their strong non-

orthogonality.

C. A few cluster terms with smaller values of N

determines the most part of the cluster contribution to the

SF and binding energies.

In spite of that the form of CFF calculated by use of

NCSM with moderate Nmax reliably determines the

asymptotic behavior of the resonances and weakly bound

cluster states. That will be presented in the next talk.

25

THANK YOU FOR YOUR ATTENTION!