E. Hiyama (RIKEN) · 2015. 5. 24. · Kyushu Univ. RIKEN KEK Nara Women’s Univ. (strangeness...

Gaussian Expansion Method and its

application to atomic and nuclear physics

E. Hiyama (RIKEN)

Kyushu Univ.

RIKEN

KEK

Nara Women’s Univ. (strangeness nuclear physics laboratory)

3) 4)

5)

1) Born in Fukuoka 2) Ph.D at Kyushu Univ. 3) Research associate at KEK(2000～2004) 4) Associate Professor at Nara Women’s Univ.(2004～March 2008) 5) Associate Chief Scientist at RIKEN (April 2008 ～ )

1) , 2)

I shall explain about RIKEN.

Japan

Wako branch

2F

Main buliding

5 PDs Y. Funaki H. Suno P. Gubler N. Sakumichi M. Isaka +3 graduated students

April (2014) 2PDs+ 1 graduate students

Special Postdoctal Researchers Program at RIKEN http://www.riken.jp/en/careers/programs/spdr/ In 2014, the calling announcement will start in May. The deadline to submit the application is June. When you succeed in getting the position, you can start to work in April in 2015 for three years. If you are interested in this position, please tell me.

Sec.3

Sec.4

Outline

4-nucleon system

3N + Λ, 3N + Σ　

perticle conversion

Nuclear physics

Hypernuclear physics

Sec.1 Introduction Sec.2 Gaussian expansion method for few-body systems (GEM)

Sec.5 neutron-rich nucles+Λ hypernucleus

Section 1

Introduction

　One of the most important subjects in physics：　　to calculate (three- and four-body ) Schrödinger equation accurately 　By solving the equation, we can predict various

observable before measurement and can get new understandings.

　For this purpose, it is necessary to develop the method

to calculate three- and four-body problems precisely and to apply to various fields such as nuclear physics

as well as atomic physics.

　

Interaction V(r)

Two-body Three-body Four-body

Atomic physics

electron nucleus

H molecule etc.

He atom etc.

In atomic physics, constituent particles are electrons and nuclei.

melectron << mnucleus

Coulomb interaction is weak.

For example, using adiabatic approximation, we can perform three- and four-body calculations more easily than those in the case of nuclear physics.

Interaction V(r)

Two-body Three-body Four-body

Nuclear Physics

However, in nuclear physics, since interaction between nucleon and nucleon is strong, we have no approximation method to solve three- and four-body problems accurately. Then, We should develop the method for solving various few-body problem accurately.

interaction V(r) Two-body problem Three-body Four-body

Nuclear physics

In the two-body problem, since how to calculate two-body Schrödinger Equation is written in the textbook of quantum mechanics, everybody can calculate the two-body problem exactly. However, it used to be difficult to calculate three-body and four–body problem.

Simple in two-body system

In the three-body system, there are many rearrangement channel.

Strongly coupling

Loosely coupling

If one more particle is added to…. Three-body problem：complicated!

(2+1) coupling　　　　

Three-types

Strongly coupling

Loosely coupling

Four-body problem is more complicate.

Strongly binding　

　　 Loosely binding

(3+1) binding four tyes　

(2+2 )binding three types

We should take account of these rearrangement and solve accurately.　　

Few-body research group to solve four-body problem

Only 7 group in the world!

・RIKEN & Kyushu Univ.(Japan) ・Niigata Univ. (Japan) ・Ruhr Univ. (Germany) ・Pisa Univ. (Italy) ・Arizona Univ. (U.S.A.) ・Argonne National Laboratory (U.S.A.) ・Trento Univ. (Italy)

(in 2001)

・A variational method using Gaussian basis functions

・Take all the sets of Jacobi coordinates

High-precision calculations of various 3- and 4-body systems:

Our few-body caluclation method

Gaussian Expansion Method (GEM) , since 1987

Review article : E. Hiyama, M. Kamimura and Y. Kino, Prog. Part. Nucl. Phys. 51 (2003), 223.

Developed by Kyushu Univ. Group, Kamimura and his collaborators.

,

Light hypernuclei, 3-quark systems,

Exotic atoms / molecules , 3- and 4-nucleon systems,

multi-cluster structure of light nuclei,

My research purpose:

i) To apply our own method (Gaussian Expansion Method) to N-body problems. 10-body problems before my retirement

・To calculate any interactions such as central force, spin-orbit force, tensor force, momentum dependent force, quadratic spin-orbit force etc.

・To calculate particle conversion interactions such as ΛN－ΣN, ΛΛーΞN－ΣΣ etc.

・To calculate bound states, resonant states and to treat continuum states

ii) To establish the following framework

Present status

・any interactions ・particle-conversion ・bound state ・resonant state ・continuum state

３-body　　４-body　　　５-body

partly

partly

not yet

partly

partly partly partly

6-body problem 　　　(this year)

Few-nucleon systems has been encouraging in order to develop my method.

done done done

done

done done

done done

Furthermore, hypernuclear physics provide us many challenging subjects.

In hypernuclear physics, we have realistic interactions such as Nijmegen model (Nijmegen soft core 97, Extended soft core 08, etc)

・ To have high repulsive core

・particle conversion interaction such as ΛN-ΣN coupling.

n

p p

Λ n

p

Λ

n

4He Λ 4H Λ

Next, I shall explain our method, Gaussian Expansion Method.

Gaussian Expansion Method (GEM)

for Few-Body Systems

Section 2

0 0

1

100

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=

In order to solve the Schrödinger equation, we use Rayleigh-Ritz variational method and we obtain eigen value E and eigen function Ψ.

Here, we expand the total wavefunction in terms of a set of L2-integrable basis function {Φn:n=1,….,N)}

The Rayleigh-Ritz variational principle leads to a generalized matrix eigenvalue problem.

Next, by solving eigenstate problem, we get eigenenergy E and unknown coefficients　Cn .

( Hi n) - E ( Ni n ) Cn =0

Hin= <Φi | H | Φn > Nin = <Φi | 1 | Φｎ >

Where the energy and overlap matrix elements are given by

r R 1

1

r R

r

R 2 2

3

3

C=1 C=2 C=3

r R 1

1

r R

r

R 2 2

3

3

C=1 C=2 C=3

Basis functions of each Jacobi coordinate


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Determined by diagonalizing H


CNL,lm

r R 1

1

r R

r

R 2 2

3

3

C=1 C=2 C=3

r R

2

2

For Example

C=2 is suitable to describe this rearrangement.

Only using C=1 channel, we can get energy. But, in order to get accurate energy, we need a large number of basis functions.

An important issue of the variational method is how to select a good set of basis functions.

What is good set of basis functions?

(1) To describe short-ranged correlation and long-range tail behaviour, highly oscillatory character of few-body wave functions, etc.

(2) Easily to calculate the matrix elements of Hamiltonian

Hin= <Φi | H | Φn >, Nin = <Φi | 1 | Φｎ >

The key point of solving three-body problem: To propose basis function to calculate the matrix elements easily between basis function of different channels

　Using this three-body basis function, we calculate the matrix element of Hamiltonian.

The suited basis function is Gaussian basis function proposed by Kyushu Univ. Group (afterwards, we proposed ‘infinitesimally Gaussian Lobe basis function.

Channel c=a channel c=b

For this purpose, we use the following basis function:

νn=(1/rn )2

rn=r1an-1 (n=1-nmax)

The Gaussian basis function is suitable not only for the calculation of the matrix elements but also for describing short-ranged correlations, long-ranged tail behaviour.

Φlmn(r)

If the interaction between composed particle is central force, we can calculate the three-body matrix element very easily.

However, it is very difficult to apply to four-body problems and three-body problem with complicated interaction such as spin-orbit force and tensor force.

The method has been applied to nuclear physics successfully.

Gaussian basis function proposed by Kyushu Univ. Group

Why difficult?

I shall explain why.

r R 1

1

r R

r

R 2 2

3

3

C=1 C=2 C=3

Coordinate transformation

Gaussian shape after transformation

We have 10 Spherical harmonics’. This integration over all of angular coordinates seems hard work.

JM

r1

The overlap matrix element using

<Φ(1)JM(r1,R1)| 1 | Φ(2)

JM( r2,R2)> =<r1

lR1Le-νr e –λR Ylm(r1)YLM(R1) | 1 |

X r2

l R2L e –νr e-λR Ylm(r2)YLM(R2) >

^ ^

^ ^ dr1dR1

2 2

2 2

=∑ ∑ ∑ ∑ γλ δl-λγ’Λδ L’-λ x √(2l-1)!(2L’-1)!/(2λ!)(2(l’-λ)!)(2Λ)!(2(L’-Λ)!(2I+1)!(2J+1)!

I K L l χ ^ ̂ ^ ̂ Λ l’-λ l’

Λ L’-Λ L’ I K J

(-)l+I+J(λΛ00|I0)(l’-λ L’-λ|K0)

x∑W(IlKL;jJ)(KL00|j0)(Il00|j0)(2m+2j+1)!!/2m+j+n+4 m!(ξ-ξ2/η)j+n+j/2

x ∑ (2p+2j+2n+1)!!/(2P+2j+1)!! m p

(ξ2/(ζη-ξ2))p

It is laborious to integrate over the angular coordinate of spherical harmonics. a）In the case of calculating the matrix element of central Force, we have 10 spherical harmonics’. It is hard work for beginner to do it. 　

b）In the case of complicated interaction such as spin-orbit Force and tensor force, we have more than 10 spherical Harmonics’. 　

c）In the case of four-body problem, we have 24 spherical harmonics at least. 　　　　

Three-body matrix element Namely,

a）three-body problem using complicated interactions b）four-body problem 　It is difficult even for expert to apply the method to a) and b) cases. 　

Difficult points using

We should overcome this difficulty and make the method to be applicable for beginners such as master course students.

Infinitesimally shifted Gaussian Lobe basis function

Infinitesimally shifted Gaussian Lobe basis function

X

Gaussian function around the origin Describe angular dependence

by superposing the shifted Gaussian function without spherical harmonics


Shifted parameter C, D can be easily determined.

Infinitesimally shifted Gaussian Lobe Function (ISGL function)

-a 0 a

Construction of the ISGL function

Z

-a

0 a

For example l=1 m=0

re-νr Y10(r)=N1(ν)Ze-νr 2

N1(ν)=[2ν(ν/π)3/2]1/2

Φ10=N1(ε,ν)[e -ν(z-εa) – e -ν(z+εa) ] e -ν(x +y ) 2 2 2 2 ~ ~

=N1(ε,ν)e-νa –νr (e2νεaz - e-2νεaz) 2 2

Taylor expansion

2 N1(ν)

=N1(ε,ν)e-νa –νr (1+2(νεaz)+(2νεaz)2/2!+( )3/3!+・・・) 2 2

- N1(ε,ν)e-νa –νr (1-2(νεaz)+(-2νεaz)2/2!+(- )3/3!+・・・) 2 2

=N1(ε,ν)e-νa -νr (4νεaz+8/3(νεaz)3+・・・・) 2 2

Mixture of more than l=3

~

~

~

N1(ε,ν) ~ N1(ν)/4ενa ε→0

=N1(ν)ze-νr 2

=∑Clm,n e -ν(r-εD ) lm,n 2

n=1

2

D10,1=(0,0,a), D10,2=(0,0,-a) C10,1=1, C10,2=-1

For any L, we get ISGL function. rle-νrYlm(r)= =ΣAlm,nZl-m-k(x+iy)m(x2+y2+x2)2k e-νr

n

2 lim Nl(ε)ΣClm,ne-ν(r-εa) =

Since the ISGL function is Gaussian function, then it is easy to calculate the matrix element of any interaction such as

It is possible to apply the method to 4- 5-body problem as well as 3-body problem.


VT(r)

L・（σ1+σ2) VLS(r). 2

2

Two-body overlap matrix element Φlm(r)= ε->0 n

2 limNl(ε)ΣClm,ne-ν(r-εa)

<Φlm|1|Φlm>=lim Nl(ε)Nl(ε’) x < ΣClm,ne-ν(r-εa)

2

n |1|

ΣClm,ne-ν(r-εa’) 2

n’

* >

=lim Nl(ε)Nl(ε’) x ΣΣ Clm,n Clm,n’ ( π

ν+ν’ ) 3/2

e -ν+ν’(a-a’) εε’ νν’

2

≡

X

When we compute the matrix element using ‘FUNCTION EXP(x)’, we encounter the following problem. finite ε,ε’->mixture of angular momentum Very small ε,ε’-> serious round-off error

We use ‘FORTRAN’.


In fact, in quantum chemistry, the gaussian function has often been used since 1960’s when εis not infinitesimal but finite. The function is called ‘Gaussian Lobe basis function’.

However, at that time, this function was not used due to the weak points of angular momentum mixture and serious round-off error for small ε.

Two-body overlap matrix element

limNl(ε)ΣClm,ne-ν(r-εa) rle-νr2Ylm(r)= 2

ε->0 εl


2

n |1|

ΣClm,ne-ν(r-ε’a’) 2

n’

* >


ν+ν’ ) 3/2


2

≡

X

ε’l

εl

e xεε’ = 1+x(εε’)1+・・・・xl

1 l!

(εε’)l+・・・

Vanish inε->0 n n’

Lower power than (εε’)l is cancelled by Σ Clm,n

(εε’)l term only is left by ΣClm,n. Pick up only this term

Two-body overlap matrix element

limNl(ε)ΣClm,ne-ν(r-εa) rle-νr2Ylm(r)= 2

ε->0 εl


2

n |1|

ΣClm,ne-ν(r-ε’a’) 2

n’

* >


ν+ν’ ) 3/2


2

ε’l

εl

n n’

=lim Nl(ε)Nl(ε’) x ΣΣ Clm,n Clm,n’ π

ν+ν’ ) 3/2

( -ν+ν’(a-a’) εε’ νν’

( )l n n’

Namely, after matrix calculation by hand、

we pick up (εε’)l term .->Then we solved the problem about ‘round-off error’. Also,ε、ε’->0, the problem about mixture of higher angular momentum was solved.　　　　　　　　

As a results, the problem since 1960’s in quantum chemistry has been solved! 　　　　　　　　　　　　　　　　　　　　　　　　　

The method using ISGL function is applicable for especially nuclear physics with high accuracy.

The merit of this method: (1) To calculate the energy of bound state very accurately (2) To calculate the wavefunction very precisely

one successful examples

4He

4-nucleon bound state NN: AV8’

p p

n n

Benchmark-test calculation to solve the 4-nucleon bound state

1. Faddeev-Yakubovski (Kamada et al.) 2. Gaussian Expansion Method (Kamimura and Hiyama ) 3. Stochastic varitional (Varga et al.) 4. Hyperspherical variational (Viviani et al.) 5. Green Function Variational Monte Carlo (Carlson at al.) 6. Non-Core shell model (Navratil et al.) 7. Effective Interaction Hypershperical

HarmonicsEIHH (Barnea et al.)

PRC 64, 044001(2001)

7 different groups (18 co-authors)

　１）subject: Let’s solve the 4-body problem and obtain the binding energy and wavefunction of 4He by using their own calculation methods.

　２） We all should send our results to Prof. Gloeckle at Ruhr Univ. until 1st February, 2001 by e-mail. Therefore, we all know ONLY our own result and do not know other groups’ results (a blind-calculation test)

　3） As soon as Prof. Gloeckle got all results, he wrote the paper and submitted it to Physical Review C (PRC). We all knew each other’s results after this paper was submitted to PRC (What a terrible test !).

4He

Benchmark-test calculation of the 4-nucleon bound state

Within the 7 groups

n n p p

This competition is very severe and stressful !

Happiest

Happier

Happy

All of our results are in good agreement with each other.

All of our results are different from each other. We do not know which one is correct.

Divided into 2 groups. We do not know which group give

the correct result.

unhappy The groups except me give the correct result. My result only is wrong!

This benchmark test was so stressful! Because, if I give a different result from other groups,

then, general readers in the world will consider that

my calculational method is not reliable.

If so, I must decide to quit my research. And I would really regret to join this bench-mark test.


But, my superviser, Prof. M. Kamimura who is co-author, said to me, “You do not need to quit your research! You should continue surviving in this research field. Because you are young.

I said, “NO! Great professor, please do not say so! I will quit my research.”

Professor said, “I will quit my research”

Instead, I will quit my research, since I shall soon retire from Kyushu Univ. within 2 years. This would give much less damage to us.”

Realistic NN force: AV8’

4He

4 nucleon bound state

Benchmark-test 4-body calculation : Phys. Rev. C64 (2001), 044001

by 7 groups ①

②

③

④

⑤

⑥

⑦

n

p p

n

Good agreement among 7different methods In the binding energy, r.m.s. radius and wavefunction density


ours

GEM

N N

4He

Λ

Replace n

p p

Λ

n

p

Λ

n

4He Λ

4H

N N

ΛＮーΣＮ　coupling effect in single Λ　hypernuclei

Λ

neutron

u d

u d u d

proton: 3 quarks

spin:1/2 isospin:1/2 Mass: 938 MeV

No charge +charge

hyperon: including strangeness quark

u d s

Λ、Σ

u s s

Ξ

s s s

Ω

What is Λ particle?

I focus on Λ particle. The mass of Λ is similar with neutron And no charge. Life time ～ 10-10 sec

Section ４

ΛH and ΛHe 4 4

Non-strangeness nuclei N Δ

N N

N

Δ

250MeV

80MeV Λ

Σ

On the other hand, the mass difference between Λ and Σ is much smaller, then there is significant probability of Σ in Λ hypernuclei.

Nucleon can be converted into Δ. However, since mass difference between nucleon and Δ is large, then probability of Δ in nucleus is not large.

Interesting Issues for the ΛN－ΣN particle conversion in hypernuclei

(1)  How large is the mixing probability of the Σ particle in the hypernuclei?

(2) How important is the ΛＮーΣＮ coupling in the binding energy of the Λ hypernuclei?

(3) How large is the Σ-excitation as effective three-body ΛNN force?

-BΛ -BΛ

0 MeV 0 MeV 3He+Λ 3H+Λ

1+

0+

-2.39

-1.24

-2.04 0+

1+

-1.00

Exp. Exp.

N

N

N

Λ

4He Λ

4H Λ

study of and is the most useful because both of the spin-doublet states are observed.

4He Λ 4H Λ

NNNΛ + NNNΣ

1) E. Hiyama et al., Phys. Rev. C65, 011301 (R) (2001).

2) A. Nogga et al., Phys. Rev. Lett. 88, 172501 (2002).

3) H. Nemura et al., Phys. Rev. Lett. 89, 142502 (2002).

Full 4-body calculations :

For precise studies of and , it is highly desirable to perform full 4-body calculations taking both the NNN 　and NNN∑　　

channels explicitly.

Λ 4H 4He Λ

Λ

+ N N

N Λ

N N

N Σ

So far, the following authors succeeded in performing this type of

difficult 4-body calculation and pointed out that the ΛN－∑N particle

conversion is very important to make these A=4 hypernuclei bound.

N N

N Λ 4He, 4H Λ Λ

VNN : AV8 potential

VYN : Nijmegen soft-core ’97f potential

no ∑

3-body force

∑-mixing

2.1 %

1.0 %

(CAL)

Although the ∑-mixing probability is small, we find that the ∑-mixing plays an essential role to make critical stability in these A-4 hypernuclei.

+ N N N Λ

N N N Σ

Section ５

Neutron-rich Λ hypernucleus

α Λ

7He Λ

n n

In neutron-‐rich and proton-‐rich nuclei,

When some neutrons or protons are added to clustering nuclei, addi7onal neutrons are located outside the clustering nuclei due to the Pauli blocking effect.

As a result, we have neutron/proton halo structure in these nuclei. There　are many interes7ng phenomena in this field as you know.

Nuclear cluster

Nuclear cluster

Nuclear cluster

Nuclear cluster

nn

nn

nnn n

Nucleus

Ques7on:How is the structure modified when a hyperon, Λ par7cle, is injected into the nucleus?

Λ

Hypernucleus

Λ par7cle can reach deep inside, and aIract the surrounding nucleons towards the interior of the nucleus.

No Pauli principle Between N and Λ

Λ

Glue-like role of Λ　particle provides another interesting phenomena.

Λ

Λ

Nucleus Hypernucleus

There is no Pauli Pricliple between N and Λ.

Λ particle can reach deep inside, and attract the surrounding nucleons towards the interior of the nucleus.

Λ

γ nucleus

hypernucleus

Due to the attraction of Λ N interaction, the resultant hypernucleus will become more stable against the neutron decay. Neutron decay threshold

Nuclear chart with strangeness

Λ

Multi-strangeness system such as Neutron star

Extending drip-line!

Interesting phenomena concerning the neutron halo have　been observed near the neutron drip line of light nuclei.

How is structure change when a Λ　particle is injected into neutron rich nuclei?

α

n n

7He

Λ

Λ

E. Hiyama, Y. Yamamoto, T. Motoba and M. Kamimura,PRC80, 054321 (2009)

α Λ

7He Λ

n

Observed at JLAB, Phys. Rev. LeI. 110, 12502 (2013).

n

α 6He

n n 6He : One of the lightest n-‐rich nuclei

7He: One of the lightest n-‐rich hypernuclei

Λ

0+

2+

γ

α+n+n

-‐1.03 MeV

0 MeV

γ

1/2+

3/2+ 5/2+

5He+n+n Λ

α+Λ+n+n 0 MeV

-‐6.19

-‐4.57

Halo states

6He 7He Λ

Prompt par7cle decay

CAL: E. Hiyama et al., PRC53, 2075 (1996), PRC80, 054321 (2009)

Exp:-0.98

5/2+ →1/2+ 3/2+ →1/2+

・Useful for the study of the excita7on mechanism in neutron-‐rich nuclei ・Helpful for the study of the excita7on mechanism of the halo nucleus

In n-‐rich nuclei and n-‐rich hypernuclei, there will be many examples such as Combina7on of 6He and 7He. I hope that γ-‐ray spectroscopy of n-‐rich hypernuclei will be performed at J-‐PARC.

Λ

Tokyo

J-PRAC ～ 100km away from Tokyo

J-PARC: Japan proton Accelerator Research Complex

We will have many hypernuclei at J-PARC facility.

Nuclear chart with strangeness

Λ

Multi-strangeness system such as Neutron star

Λ J-PARC

Future prospects of my research

Section 6

My research purpose:

i) To apply our own method (Gaussian Expansion Method) to N-body problems. 10-body problems

・To calculate any interactions such as central force, spin-orbit force, tensor force, momentum dependent force, quadratic spin-orbit force etc.

・To calculate particle conversion interactions such as ΛN－ΣN, ΛΛーΞN－ΣΣ etc.

・To calculate bound states, resonant states and to treat continuum states

ii) To establish the following framework

4-nucleon system

3N + Λ, 3N + Σ　

perticle conversion

Nuclear physics

Hypernuclear physics

α

n n

7He

Λ

Λ

My own research method: “Gaussian Expansion Method” 　

Hypernuclear

Physics

Unstable nuclear physics

Physics on Few-nucleon Systems

Hardon Physics

Muon Catalyzed Fusion

Contribution Feedback to develop my method

My Research

Strategy

Something New physics

Ultra cold atom

For this research project,

I welcome you to join my laboratory (strangeness nuclear physics laboratory) and let’s explore new physics using our few-body calculation method together with me!

Thank you!

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E. Hiyama (RIKEN) · 2015. 5. 24. · Kyushu Univ. RIKEN KEK Nara Women’s Univ. (strangeness...

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