# Presentation: 392116

##  decay and the 

***ORNL***

** ****	****	****                    ****Outline**
**Where we are today**
**Potential ways to improve the calculations**
**Increasing the Hilbert space**
**Effective operators for small spaces**
**Using Factorization schemes**
**Using Coupled Cluster theory**
**Conclusions**

****** ****decay and the **
**nuclear many-body problem**

**With input from Jon Engel (UNC)**

## Nuclear structure landscapes

**Main theory goals:**
** ****Identify/investigate many-body **
**  ****methods that will extend to RIA**
** ****Generate effective interactions**
** ****Make reliable predictions**
** ****Guide experimental efforts**
** ****Use NN and 3N forces to build nuclei**
**Various approaches to  **
**low-energy nuclear theory:**
** ****Coupled-Cluster theory**
** ****Shell Model Monte Carlo**
** ****DMRG/Factorization**
** ****shell model diagonalization**
** ****Continuum shell models**
** ****HFB**
** ****QRPA**
** ****TDHF**

## One approach to the problem: Green Function Monte Carlo

**        ****(ANL/LANL/UIUC)**

**Since 1982:**
** ****algorithms**
** ****Variational MC**
** ****AV18**
** ****Computing**

**For A=10, 1.5 Tflop-hours/state**
**For A=12, 50 Tflop-hours/state**

**Indicate the need for **
**3 (and 4?) body interactions**
**Future prospects:**
** ****A=12 by 2003/2004**
** ****triple alpha burning**
** ****Reaction aspects **
** ****NNN studies**

## Two basic approaches have been applied 

**to ********-decay problem**

## Nuclear physics of the problem

## Present 

**published **
**results **

**Kill outliers**
**Factor of **
**3 in C****mm**
**Assume **
**T1/2 =4E-27 years**

## What the calculations predict

**Caurier et al**

## Begin with a bare NN (+3N) Hamiltonian

**nuclear**
**structure**
**theory**

**Begin with a bare NN (+3N) Hamiltonian**

**Solve the quantum many-body problem:**
** ****Easier said than done due to combinatorial**
**   ****growth of the problem as a function of particles. **

| Nucleus | 4 shells  | 7 shells |
| --- | --- | --- |
| 4He | 4E4 | 9E6 |
| 8B | 4E8 | 5E13 |
| 12C | 6E11 | 4E19 |
| 16O | 3E14 | 9E24 |

**Oscillator**
**single-particle basis states**

**Many-body basis states**

## Choice of model space and the G-matrix

**Choice of model space and the G-matrix**

- Q-Space

- P-Space

**ph intermediate**
**states**

## Solving the quantum many-body problem in a basis

**Methods of solution**
**Diagonalize H******
**Determine the optimal (sometimes correlated) basis (Papenbrock)**
**Reformulate problem as a path-integral (AFMC: SMMC)**
**Resum of quantum many-body perturbation theory diagrams**

**Many-body basis states**

**Reference Slater determinant**

## Diagonalization: configuration-interaction, interacting shell model 

**Yields eigenfunctions which are linear combinations of**
**particle-hole amplitudes**

**1p-1h**

**2p-2h**

**Advantages**
** ****Detailed spectroscopic information available**
** ****Wave functions calculated and stored**
**Disadvantages**
** ****Dimension of problem increases dramatically with the **
**   ****number of active particles (combinatorial growth). **
** ****disconnected diagrams enter if truncated**

## Efficient basis set selection

**(Other slide show)**

**Papenbrock & Dean, PRC67,051303(R) (2003)**
**Papenbrock, Juodagalvis, Dean, PRC69, 024313 (2004)**

## Use Many-body perturbation theory to modify

**the operator**

**The transformation operator**

**Effective operators**

**For initial application see Engel and Vogel: nucl-th/0311072 **

## Another fascinating tool:

**Coupled Cluster Theory**

**Some interesting features of CCM:**
** ****Fully microscopic**
** ****Size extensive: **
**  ****only linked diagrams enter **
** ****Size consistent: **
**   ****the energy of two non-interacting **
**   ****fragments computed separately is the same as that **
**   ****computed for both fragments simultaneously**
** ****Capable of systematic improvement**
** ****Amenable to parallel computing**

**Computational chemistry: 100’s of publications in 2002**
**(Science Citation Index) for applications and developments. **

## Coupled Cluster Theory

**Correlated Ground-State**
**wave function**

**Correlation **
**operator**

**Reference Slater **
**determinant**

**Energy**

**Amplitude equations**

** ****With all T’s the spectrum of H is the **
**   ****same as the spectrum of the **
**   ****similarity transformed H; formally valid**
** ****In practice E closely approximates a**
**  ****variational theory when T is truncated**

**Dean & Hjorth-Jensen, PRC submitted 2003**
**Kowalski, Dean, Hjorth-Jensen,**
**Papenbrock, Piecuch, PRL, in press 2004**

## Ground states of oxygen

**Dean & Hjorth-Jensen (PRC, submitted)**

**Use realistic interactions, G-matrix renormalization CCSD results**

## Correcting the CCSD results by non-iterative methods

**Find a method that will yield the complete**
**diagonalization result in a given model space**
**How do we obtain the triples correction? **
**How do our results compare with ‘exact’ results in a given**
**model space, for a given Hamiltonian? **
**“****Completely Renormalized Coupled Cluster Theory”**
**P. Piecuch, K. Kowalski, P.-D. Fan, I.S.O. Pimienta, and**
**M.J. McGuire, Int. Rev. Phys. Chem. 21, 527 (2002)**

## 16O in four major oscillator shells

**	****a) T****1**** and T****2**** of similar order**
**	****b) T****1****T****2**** disconnected **
**	****  ****	****   ****>> T****3**** connected triples**
**	****c) diff between CISD and CISDT**
**	****	****  ****comes mainly from T****1****T****2**
**	****d) If T****3**** were large CCSD(T) **
**	****	****would be far below CCSD**
**2) Size extensive nature of CC **
**3) CCSD + CR-CCSD(T) bring**
**	****T****1****3****T****2****, T****1****T****2****2**** T****2****3 ****not in CISDTQ**
**4) Scaling**

**16****O in four major oscillator shells**

| Method | Energy (MeV) |
| --- | --- |
| CCSD | -139.310 |
| CR-CCSD(T),a | -139.465 |
| CR-CCSD(T),a/D=1 | -139.621 |
| CR-CCSD(T),b | -139.375 |
| CR-CCSD(T),b/D=1 | -139.440 |
| CR-CCSD(T),c | -139.391 |
| CR-CCSD(T),c/D=1 | -139.467 |
| CISD | -131.887 |
| CISDT | -135.489 |
| CISDTQ | -138.387 |

**Kowalski, Dean, Hjorth-Jensen, Piecuch, **
**PRL in press (2004)**

## Conclusions and Perspectives

-  **Solutions to nuclear many-body problems requires extensive use of **
**   ****computational and mathematical tools. Numerical analysis becomes**
**   ****extremely important; methods from other fields do help us.**
** ****Nuclear theory progress is not presently tied to bb-decay; it is tied to**
**  ****RIA and low-energy nuclear structure experiments.  **
** ****Big problems include correlations in Hilbert space, size of space,**
**   ****derivation of effective interactions and operators....**
** ****bb-decay, near term plans....**
** ****Do Ge76 by opening the f7/2 and including the d5/2 (others?)**
**   ****(with factorization method)**
**	****limits (from small space, and larger space). **
** ****Solve ‘left’ wave function in CCSD... Compute  **
**	****matrix element (Moore’s law). **
** ****Reproduce/predict as much data as possible on a given nucleus**
**(GT, 2nu-bb, M1,.......)**