# Presentation: 947706

## Nuclear Structure

**Erich Ormand**
**N-Division, Physics and Adv. Technologies Directorate**
**Lawrence Livermore National Laboratory**

**This work was carried out under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. **

- This document was prepared as an account of work sponsored by an agency of  the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes.

- 15thNational Nuclear Physics
- Summer School
- June 15-27, 2003

**Lecture #2**

- 26-Jun-01

## Low-lying structure – The interacting Shell Model

**The interacting shell model is one of the most powerful tools available too us to describe the low-lying structure of light nuclei**
**We start at the usual place:**
**Construct many-body states |***********i********* so that**
**Calculate Hamiltonian matrix *****H******ij******=****************j*****|*****H*****|***********i*******
**Diagonalize to obtain eigenvalues**

**We want an accurate description of low-lying states**

- 26-Jun-01

## Nuclear structure with NN-interaction

***NN*****-interaction determined from scattering and the deuteron**
- Argonne, Bonn, Paris, Reid, etc.

**Problem:**** ****Repulsion in strong interaction ******** ****Infinite space!**

- Phase shift and potential in 1S0 channel

- Strong repulsion at ~ 0.5 fm
- j|H|i large

- 26-Jun-01

## Can we get around this problem?Effective interactions

**Choose subspace of      for a calculation (*****P*****-space)**
**Include most of the relevant physics **
***Q *****-space (excluded - infinite)**
**Effective interaction:**
**Two approaches:**
**Bloch-Horowitz **
**Lee-Suzuki:**

**Q**

**P**

*H**eff*=**P**X*H*X-1**P**

**Q**X*H*X-1**P****=0**

- 26-Jun-01

## Effective interactions permit first-principles shell-model applications

**Impossible problem ******** Difficult problem**
**Two, three, four****, ... A-body operators **
**Compromise between size of P-space and number of clusters**
**Three-body clusters**

- 

- 26-Jun-01

## The general idea behind effective interactions

**The general idea behind effective interactions**

- H

- 26-Jun-01

## Interactions in real-world applications

**Interactions in real-world applications**

**Ideally, we would like to use these fundamental interactions in our theory calculations**
**In most cases this is not really practical as the the NN-interaction has a very strong repulsive core at short distances**
**This means that in many-body applications an infinite number of states are needed as states can be scattered to high-energy intermediate states**
**We need to use effective interactions**
**Derived from some formal theory**
**This is in principle possible and is difficult. But it is becoming practical now for light nuclei**
**Assume they exist as the formal theory stipulates and determine it empirically to reproduce data**
**This has permitted many studies in nuclear structure to go forward**

- 26-Jun-01

## Shell model applications 

**The practical Shell Model**
**Choose a model space to be used for a range of nuclei**
**E.g., the 0d and 1s orbits (sd-shell) for ****16****O to ****40****Ca or the 0f and 1p orbits for ****40****Ca to ****120****Nd**
**We start from the premise that the effective interaction exists**
**We use effective interaction theory to make a first approximation (G-matrix)**
**Then tune specific matrix elements to reproduce known experimental levels**
**With this empirical interaction, then extrapolate to all nuclei within the chosen model space**
**Note that radial wave functions are explicitly not included, so we add them in later**

**The empirical shell model works well!**
**But be careful to know the limitations!**
**More on exact treatments later.**

- 26-Jun-01

## The Shell Model

**We write the Hamiltonian as**
**Start with closed inner core, e.g., for ****24****Mg, close the p-shell**
**Active valence particles in a computationally viable model space, e.g., the 0d****5/2****, 0d****3/2****, 1s****1/2**** orbits for ****24****Mg**

- No excitations up here

**Single-particle energy ***********i***

**Two-body residual interactions**

**Energy is relative to ****16****O core**

- 26-Jun-01

## Building the shell-model basis states

**Need to construct the many-body basis states to calculate matrix elements of H**
**Choose states with definite parity, *****J******z***** and *****T******z****** *****and let the Hamiltonian do the rest**
**A very useful approach is a bit-representation known as the M-scheme**

**A single integer represents a complicated Slater Determinant**

- 26-Jun-01

## Basis states - How many are there?

**To do a calculation we need states with fixed *****J******z***
**All J-values are contained in *****J******z*****=0 or *****J******z*****=1/2**
**Counting the number of basis states**
**Order-of-magnitude estimate **
***n***** particles, and *****N******sps***** single-particle states**
***N******sps***** in the sd-shell = 12 (0d****5/2****=6, 0d****3/2****=4, 1s****1/2****=2)**
***N******sps***** in the fp-shell = 20 (0f****7/2****=8, 0f****5/2****=6, 1p****3/2****=4, 1p****1/2****=2)**
**Includes states of all *****J***** and *****J******z***
**Number of *****J******z*****=0 divide by a factor of ten**
**Number of states with a given J**

- 26-Jun-01

## Tools of the trade - Lanczos

**Setup Hamiltonian matrix *********j*****|*****H*****|*****i********* and diagonalize**
**Lanczos algorithm**
**Bring matrix to tri-diagonal form**
** ****n****th**** iteration computes 2n****th**** moment**
**But you can’t find eigenvalues with calculated moments**
**Eigenvalues converge to extreme (largest and smallest) values**
**~ 100-200 iterations needed for 10 eigenvalues (even for 10****8**** states)**

**Note that after each iteration, we need to re-orthogonalize!**

- 26-Jun-01

## Shell-model codes

**Oak Ridge (1969)**
**Coefficients of Fractional parentage (CFP)**
**Glasgow (1977)**
**Good J****z**** (M-scheme)**
**J restored in diagonalization**
**OXBASH (1985)**
**J-projected M-scheme**
**Smaller matrices**
**RITSSCHIL (1985)**
**CFP**

**DUSM (1989)**
**Permutation groups**
**ANTOINE (1991 & 1999)**
**M-scheme**
**Apply matrix on-the-fly**
**Large dimensions**
**NATHAN (1998)**
**J-projected similar to ANTOINE**
**“****Hybrid” M-scheme-CFP code**
**REDSTICK (now)**
**Similar to ANTOINE**
**M-scheme**
**Three-body interactions**
**Parallel architecture**

- 26-Jun-01

## REDSTICK – basis state ordering

**Ordering of basis states speeds up the calculation**
**Need a *****FAST***** lookup scheme**
**Construct proton and neutron many-body Slater determinants**
**Order by Parity, J****z**** and **

- For each proton SD store start in Lanczos vector: pos(*i*)
- For each neutron SD store relative position in Jz, parity list: pos(*j*)
- Position in Lanczos vector determined by summing two integers *k= *pos(*i*)+ pos(*j*)
- Same algorithm can be used to sort and find two-body matrix elements

**Limits truncations**
**Partition truncations not possible**
**Much faster!**
**Lanczos vector points to p & n SD’s (less memory)**

- Lanczos vector

- 26-Jun-01

## REDSTICK – Applying the Hamiltonian – pn partOn-the-fly

**Find states connected by one-body proton or neutron operator **
**pn-part is the hardest**
**Product of two one-body operators**
**Pre-sort connections by one-body operators; store operator and phase**

**REDSTICK ****– Applying the Hamiltonian – pn part****On-the-fly**

**Loop over all initial proton-neutron SD’s *********i*****=********p*****k****** *********n*****l***
**For each ********p*****k***********loop over all ********p*****m*****  ****connected by one-body operator**
**For each ********n*****l***** loop over all ********n*****n***** ****connected by one-body operator but limited in J****z****, parity, and        by final proton state ********p*****m***
**Update Lanczos vector: *****Position=***pos(*m*)+pos(*n*)

- 26-Jun-01

## REDSTICK – Applying the Hamiltonian Parallel execution

**Divide loop over N****CPU**** independent processors **
**For load balance use ****do i=1,N****dim****,N****CPU**
**Creates updated Lanczos vector on each processor**
**Total Lanczos vector with Global sum**

- REDSTICK – Applying the Hamiltonian Parallel execution

**Global Sum ~ N****CPU**** speed up**

- 26-Jun-01

## REDSTICK – ReorthogonalizationParallel execution

**Not that trivial (but also not the main bottleneck in the calculation)**
**Let each processor be responsible for a sub set of Lanczos vectors written to disk**
**Each processor has a copy of the current vector**
**Accumulate sum overlaps                         on each node, *****n***
**Global sum                   then                      and normalize**
**Not quite the same as sequential reorthogonalization, but seems to work fine (also used in MFD).**

- REDSTICK – ReorthogonalizationParallel execution

- 26-Jun-01

## Shell-model codes - Performance

**REDSTICK or ANTOINE**

- 26-Jun-01

## Simple application of the shell model

**A=18, two-particle problem with ****16****O core**
**Two protons: ****18****Ne (T=1)**
**One Proton and one neutron: ****18****F (T=0 and T=1)**
**Two neutrons: ****18****O (T=1)**

**Example: ****18****O**

**How many states for each *****J******z*****? How many states of each *****J*****?**
**There are 14 states with J****z****=0**
**N(J=0)=3**
**N(J=1)=2**
**N(J=2)=5**
**N(J=3)=2**
**N(J=4)=2**

**Question # 1?**

- 26-Jun-01

## Simple application of the shell model, cont.

**Example:**

**Question #2**

**What are the energies of the three 0****+**** states in ****18****O?**
**Use the Universal SD-shell interaction (Wildenthal)**

- 26-Jun-01

## Simple application of the shell model, cont.

**Example:**

**Set up the Hamiltonian matrix**
**We can use all 14 *****J******z*****=0 states, and we’ll recover all 14 J-states**
**But for this example, we’ll use the two-particle J=0 states**

**Finding the eigenvalues**

- |(0d5/2)2J=0

- |(1s1/2)2J=0

- |(0d3/2)2J=0

- 26-Jun-01

## Effective interactions with the Lee-Suzuki method

**Effective interactions with the Lee-Suzuki method**

- 26-Jun-01

## For an n-body cluster in Heff, we must first solve the n-body problem!

**Find *****H******n-eff*****  ****iteratively**
***n*****-particles bound in oscillator potential**
**Steps:**

***A=2***** **
***H******2-eff***** **

- 26-Jun-01

## Effective interactions really work

**4****He with the effective-field theory Idaho-A potential**

**Effective interactions improve convergence!**
**Are EFT potentials useful for nuclear-structure studies?**

- 26-Jun-01

## But, we do need to use big computers

**Complex computational problem!**
**Example: ****10****B, *****N******max*****=4**
***H******3-eff***
***39,523,066 *****3-particle***** *****matrix elements**
******j****|*****H*****|********i******** matrix dimension: *****581,740***** ******** *****581,740***** =1.7 ******** 10****11**
**Easy for *****H******2-eff***** ~  1-2 CPU-hr for lowest ten states **
***H******3-eff***** :**
******j****|*****H*****|********i******** has 2.2 ******** 10****9**** non-zero elements!**
**  ******** ****100 CPU-hr**
**Three-body effective interaction takes 24-48 hours**
**True three-body interaction ~ 1 week**
**The future:**
***H******3-eff***** with *****N******max*****=6**
***H******4-eff***** with *****N******max*****=4**

- 26-Jun-01

## Results with three-body effective interactions

- Binding energies with Av8’

**Results with three-body effective interactions**

**Three-body effective interactions represent a significant improvement and give results within 400 keV of the GFMC**

- 26-Jun-01

## Excitation spectra with NN-interactions

**So far, things are looking pretty good!**

- 26-Jun-01

## Excitation spectrum of 8Be

**Experiments hint at a new excited state in ****8****Be**
**Excitation energy ~10-30 MeV**
**Previous theory studies were unable to predict the existence of such a state**
**In large model spaces we find an intruder band**
**                ****model space and 2********10****8**** basis states!**
**No-core ANTIONE**

- 26-Jun-01

## Excitation spectrum of 8Be

**What is the nature of this state?**
**Is it real?**
**Stable 0****+****, 2****+****, 4****+**** rotational band**
** **
**E****x**** appears to be stabilizing**
**~10-15 MeV**
**Beta-vibration of the ground-state ******** cluster?**
**Excitation energy seems too high**
**Super-deformed prolate shape?**
** **

- 26-Jun-01

## The NN-interaction clearly has problems

**Note inversion of spins!**

**The NN-interaction by itself does not describe nuclear structure**
**Also true for A=11**

- 26-Jun-01

## How about the three-body interaction?

**Tucson-Melbourne **

- 26-Jun-01

## Three-body interaction in a nucleus

**The three-nucleon interaction plays a *****critical***** role in determining the structure of nuclei**

- 26-Jun-01

## More on three-body interactions in a nucleus

**Gamow-Teller  and M1 transitions**
**Sensitive to spin-orbit force**
**Because ********is a generator of SU(4) and transitions in different representations are forbidden, i.e., B(GT)=0!! **
**NN-interaction tends to preserve SU(4)**
**Spin-orbit breaks SU(4)**

**The three-nucleon interaction has a strong spin-orbit component**

- 26-Jun-01

## Summary of ab initio studies

**Significant progress towards an exact understanding of nuclear structure is being made!**
**These are exciting times!!**

- 26-Jun-01