An R-matrix approach for plasma modelling and the interpretation of

astrophysical plasmas

July 27th , Queen's University, ICPEAC 2011

Connor Ballance

Auburn University

Collaborators : T G Lee, S D Loch, M S Pindzola (AU) : N R Badnell ( Strathclyde ) : B M McLaughlin (QUB) : M A Bautista (WMU)

Supported by : US DoE Fusion Energy Sciences

Overview

● Introduction : Comprehensive approach to plasma modelling, our current capabilities and future directions.

● Electron-impact ionisation : high n shell ionisation

● Electron-impact excitation : scripted R-matrix calculations : parallel DARC code : Fe-Peak elements and beyond

● Photo-ionisation of Mid-Z elements : parallel dipole (DARC) codes

Introduction

● In recent years, EIE R-matrix calculations, have moved beyond the isolated, one-off serial calculations to parallel calculations along entire iso-nuclear/iso-electronic sequences

Witthoeft et al 2007 (J. Phys. B Vol 40) Liang and Badnell 2010 (Astron. Astrophys. Vol 518 A64)

Perl-scripted calculations, automatically calculate tabulated every effective collision strength for all transitions from user given structure.

● This data is stored in a well-prescribed format that includes the atomic configurations, the energy levels, the A-Values for all E1,E2,M1,M2 transitions, Maxwellian averaged collision strengths for a range of temperatures and the Born/Bethe infinite energy limit points. (why?).

http://www-cfadc.phy.ornl.gov/home.html

Introduction

● Modern computing archictectures have over 100,000 cpu cores

Kraken, NICS Oak Ridge, TN (Cray ) Hopper, NERSC, Berkley, CA (Cray)

and if utilised correctly, can support PetaFlop/sec calculations.

● The serial codes (1973-1999) could take a week(s) to calculate an ion stage. The first generation of parallelcodes (2-3 hundred levels, 1-2 thousand channels) a day

● Now we need adapt to (1-3 thousand level calculationswith 5-20 thousand channels) if we are to address open p-and-d shell systems.

R-matrix/R-matrix with Pseudostates (RMPS) review

                 

                

The importance of excited state ionisation

●Effective ionisation rates include the contribution of excited state ionisation,which becomes computationally demanding for non-perturbative methods such the RMPS

Allain et al 2004Nucl. Fusion 44, 655 (2004)

How do we employ the RMPS method effectively, for high n shell ionisation ?

The calculation of ionisation from every term of a high n can be an order of magnitudemore computationally demandingthan the groundstate alone.

 

                 

                

There are many non-perturbative ionisation cross-sections from the

groundstate For example, consider the closed shell groundstate of neutral neon

Groundstate Excited terms

However systems with several valence electrons soon become problematic

Consider the all LS coupling electron-impact ionisation (both ground and excited) state from a boron-like system such as B I / C II

This will require ionisation from : 1s^2 2s^2 nl (where n=2-4, l=0-3) : 1s^2 2s 2p^2 : 1s^2 2p^3 (for C II )

In addition to the above spectroscopic terms, we shall require minimum pseudostate expansions of the form:

1. 1s^2 2s^2 nl (where n=5,14,l=0-6)2. 1s^2 2s 2p nl (where n=5,14,l=0-6)3. 1s^2 2p^2 nl (where n=5,14,l=0-6)

If you want to calculate

a) Direct ionisation of the outer shell electronb) Direct ionisation of the 2s electron c) All the excitation-autoionisations from every term ie. e + 1s^2 2s^2 3s --> 1s^2 2s2p 3s

Well, 1444 terms , approximately 5000 close-coupled channels and 5 Tb of Hamiltonian matrices requiring diagonalisation poses an interesting challenge ...

This represents the current work, that extends beyond naively splitting the serial problem over more processors, to one in which the parallel code adapts to a particular problem

Hamiltonian formation

1. Serial : Each partial wave is calculated consecutively ( 50-100 ) .... a month

2. Naive parallelism : each partial wave is carried out concurrently .... 3 days (remember a single partial wave > 200 Gbs) .... 100 procs

3. Adaptive parallelism : As well as each partial wave being carried out concurrently the target terms are grouped into their L S Pi groups (perhaps 20-40 unique groups) .... 2000-4000 procs … 4hrs

RESULT : Hamiltonian formation is reduced scattering from a set of target terms with the same L S Pi values.

Hamiltonian diagonalisation

1.Serial : Impossible ! Every eigenvalue of a 200 K by 200 K Hamiltonian 2. Naive parallelism : sequential parallel diagonalisation using Scalapack, possible, but regardless of diagonalisation time, you must read 5 Tb .... 4 days

3.Adaptive parallelism : Each Hamiltonian is concurrently diagonalised in parallel , with an n^3 scaling law controlling the distribution of processors … 5 hrs

RESULT : Adaptive diagonalisation ---> 1 Hamiltonian read, 1 diagonalisation

Better load balancing as processing power is distributed to where it is needed

Electron-impact ionisation of neutral Boron , n=3 shell

Notice: large excitation-autoionisation (EA) i.e. e + 1s^2 3l --> 1s^2 2s 2p 3l + e

Large EA destroys any n^4 scaling law (or rescaling of an empirical formula) needed to extrapolate to higher n shell ..... solution ?

3s

3p

3d

If we can explicitly caculate to high enough n shell, that the direct ionisation completely dominates over EA , then we can scale simple ionisation expressions from the last explicitly calculated n shell shell of the RMPS

B

B

B

+

2+

n=4

n=4

n=5

Below , we have statistically averaged the term resolved RMPS results used to rescale an expression, such as the Burgess-Vriens for the higher n shells.

Excited state ionisation from higher charged ions

Higher partial waves begin to dominate the cross section, for high n , multiply charged systems

Convergence is very slow

Comparisons with distorted wave are favourable, again

RMPS becomes computational Demanding, requiring 110-130 pseudo-orbitalsranging from n=6-16,18 and l=0-10to achieve convergence

Time : distorted wave (15 mins) RMPS (4 hrs)

Distorted wave (Younger Potential)Distorted Wave (Macek Potential) RMPS

Electron-impact excitationand how we adapt to future challenges.

Scripted Semi-relativistic ICFT (Intermediate Coupling Frame Transformation) R-matrix calculations

● A perl-script has been developed that automatically takes structure calculations for atomic ions Z < 36 , through to effective collision strengths for ALL ions along an iso-electronic sequence

● Intermediate Coupling Frame Transformation, developed by D C Griffin is an R-matrix approach primarily carried out in LS coupling but including mass-velocity and Darwin terms, that transforms term-resolved K-matrices into level-level K-matrices, and ultimately level resolved excitation rates. Provides comparable cross sections to the Breit-Pauli R-matrix codes.

Example Papers:

G. Y. Liang et al Astron. Astrophys. 528 A69(15) (2011). Li-like sequenceG. Y. Liang et al Astron. Astrophys. 518 A64(20) (2010). Neon sequenceG. Y. Liang et al J. Phys. B 42 225002(12) (2009). Na-like sequenceM. C. Witthoeft et al J. Phys. B 40 2969-93 (2007). Fl-like sequence

Typically 100-300 levels calculations, with 1000-1500 channels

However, eventually as Z increases we must adopt the a parallel version of relativistic R-matrix codes.

DARC (Dirac Atomic R-matrix Code)● These codes have been modified in an analguous way to the non-relativistic codes - distribution of integral generation - multi-layered parallelism of the Hamiltonian formation - Concurrent adaptive diagonalisation of the Hamiltonian

0.6 billion Racah coefficients 0.6 billion Racah coefficients per symmetry per symmetry

Excitation/ionisation of Mo II

8000-10000 channelsstill a challenge

4d^5,4d^4 4f-5f

This is preparationcalculation for futureW ion stages.

e.g. DARC calculations for Fe III

Target : 3d^6, 3d^54s, 3d^54p, 3p^43d^8, 3p^53d^7 (2-3 hundred level)ideally now we would add (3p^54d, 3p^43d^74s ,3p^43d^74p)

which if we keep to 0.0-0.5 Ryds, provides a satisfactory description of Fe III

e.g. DARC calculations for Fe II

Before After

(Ith -Iobs)/Iobs

Collision strengths : Zhang 1992A-values : Nahar and Pradhan

Collision stengths : parallel DARC A-values : HFR (cowan)

A typical collision strength within the groundstate complex of Fe III illustrates the need for a very fine energy mesh !

ICFT DARC

However, if we want continued scaling to 5-12 thousand close-coupledchannels, reconsider electron-impact ionisation of B I * 1444 terms * 5000 channels ,* Hamiltionian Matrices close to 200,000 by 200, 000 in size* over 1 million possible transitions

But the formation of the R-matrix is crippling in both memory and the time required! .... cannot be left to a single processor

RRij =

k

W * Wik kj

E - Ek

=

w ik / E - E k

wkj

200 000 * 5000 * 8 * 2

= 16 Gb ! (takes mins )

5000*5000*8

0.2 Gb N slices

928/61440 = 0.015 sec per R-matrix formation

Test run : 4 partial wavesSe III (225 levels): 437,737,748,1017 channels

12,000 pts. CPU TIME= 1.380 MIN -- processors=: 250 sub world 0 CPU TIME= 3.851 MIN -- processors=: 250 sub world 1 CPU TIME= 3.990 MIN -- processors=: 250 sub world 2 CPU TIME= 6.496 MIN -- processors=: 250 sub world 3 Suggested Proc distribution: 89 245 253 413

CPU TIME= 3.563 MIN -- processors=: 245 sub world 1 CPU TIME= 3.868 MIN -- processors=: 413 sub world 3 CPU TIME= 3.869 MIN -- processors=: 89 sub world 0 CPU TIME= 3.977 MIN -- processors=: 253 sub world 2 -------------------------------------------------------------

Another 2.5 mins saved .... better balance of resources

RECAP

We can also adapt the distribution of processors to energy points in the outer region

Photoionisation of Mid-Z atomic ions

Posters : Thu 140,Thu 143Posters : Thu 140,Thu 143

An overview of the parallel DARC dipole photoionisation codes

An overview of the parallel DARC dipole photoionisation codes● The parallel dipole suite of codes, benefit from the changes made from excitation

All the eigenvectors from a pair of dipole allowed symmetries are required for bound-free dipole matrix formation

●The current approach ensures that every dipole matrix pair is carried out concurrently with groups of processors assigned to an individual dipole.

RESULT : ALL photoionisation, dielectronic-recombination or radiative-damped excitation takes the time for a single dipole formation

● The capacity to perform photoionisations calculations with over 500 levels, improves the residual ion structure, the ionisation potential and resonance structure associated with over 3500 channels.

● As the code scales to 100,000 processors , we can have a resolution of 10^(-8) Rydbergs ie (6-30 million pt) in the incident photon energy, which is vital when comparing with ALS measurements at 4,6,9 meV.

● Of course, theoretical calaculations can guide experimental measurement and determine metastable fractions in experimental measurement.

Example : Groundstate Photoionisation of Ca II (work in progress)

Target : 3p^6, 3p^5[4s-5p] 3p^4 3d 4s 3p^4 3d 4p 3P^4 3d 4d which gives rise to 513 levels

Theoretical : DARC R-matrix

Experiment : Lyon (1987) ALS(2010)

Currently, the theoretical modelis only from the groundstate,with metastable calculations ongoing.

hv + Ca II (^2 S) 3p^64s

Example : Photoionisation of Kr II and Xe II

● Used a 326 level model for the residual ion in both cases

● Target configurations for K III include : 4s^2 4p^44s 4p^5 4p^64s^2 4p^2 4d^24s^2 4p^3 4d4s 4p^4 4d 4p^4 4d^2

● Target configurations for Xe III (exactly the same : switching n=4 to 5)

● In the following graphs the R-matrix results have been statistically averaged over the initial p^5 levels

The agreement with Xe II is equally good

Come by the poster for more details … TH 143

Thank-you for your attention