# Presentation: 119269

## Terascale computational atomic physics for the plasma edge

*PSACI PAC *
*PPPL*
*June 5-6, 2003*

**Mitch Pindzola**
*Department of Physics *
*Auburn University*
- and
**David R. Schultz**
*Physics Division*
*Oak Ridge National Laboratory*

## Mitch Pindzola

- Francis Robicheaux
- Eugene Oks
- James Colgan
- Stuart Loch
- Michael Witthoeft
- Turker Topcu
*Auburn University*

- Don Griffin
- Dario Mitnik
- Connor Ballance
*Rollins College*

- Hugh Summers
- Nigel Badnell
- Alan Whiteford
*Strathclyde University *
*JET, UK*

- Phil Burke
- Brendan McLaughlin
*Queen’s University, Belfast*

- Cliff Noble
*Daresbury Laboratory*

- David Schultz
- Tatsuya Minami
- Jack Wells
*ORNL*

- Keith Berrington
*Sheffield Hallam University*

- Klaus Bartschat
*Drake University*

**Notes:**
This is title of our SciDAC project – emphasize again that SciDAC funded by Dept of Energy. 
This slide shows our (many) collaborators both in the U.S. and the U.K.
Also that we use some of the largest non-defense supercomputers in the country, and use substantial proportions of computing time available.
(millions of CPU hours)

## Direct Dissemination of Atomic Data 

**Controlled Fusion Atomic Data Center, ORNL**
**www-cfadc.phy.ornl.gov**

**A****tomic ****D****ata and ****A****nalysis ****S****tructure**
**International consortium lead by Strathclyde University and JET**
**adas.phys.strath.ac.uk/adas/docs/**
**   ****manual**
**H. P. Summers, ADAS Users Manual (2****nd**** Edition, 2003)**

**Notes:**
So how do we go about providing atomic data for these (fusion) plasma processes?
One way is through setting up of atomic databases: eg ORNL and ADAS.
Provide data on all collisional radiative processes: e-impact excitation/ionization, 
Charge exchange, proton/heavy particle impact excitation, recombination (dielectronic and radiative) etc. 
 More available information on websites shown.

Emphasize that most of the atomic calculations made by our group result in (possibly large quantities) of atomic data which is then posted on these websites in formats suitable for use by plasma modelers. We aim to provide high-quality atomic data and so we use some leading theoretical methods

## Scientific Advances, Accomplishments

**Importance of continuum coupling in electron-impact excitation of light elements**
**H (JPB 2000), He, He****+****, Li (PRA 2001), ...**
**Importance of three-body dynamics in the electron-impact ionization of light elements**
**Li, Li********* (PRL 2001), He****+**** (PRA 2003), Li****2+**** (PRA 2002)**
**Importance of correlation and coupling in the dielectronic recombination of atomic ions**
**DR project (2003), Cl****13+**** (PRL 2003)**
**Importance of continuum coupling in proton scattering from light elements**
**H (PRL 1999), H (PRA 2002), Li********* (PRA 2003)**
**Importance of three-body dynamics in the double photoionization of atoms and molecules**
**He (JPB 2001), Be (PRA 2002), (2********,2e) He (PRL 2002), H****2**
**Importance of four-body dynamics in the triple photoionization of atoms**
**Li photoionization**
**Behavior of solitons in Bose-Einstein Condensates (JPB 2003)**
**Simulations of the behavior of ultracold plasmas (PRL 2002)**

**Notes:**
These are some fundamental atomic scattering problems to which we apply large numerical calculations. Most have relevance to processes which can take place in fusion devices.

 Specific projects which we are/have been working on recently are listed. Will talk about some of them in detail in the coming slides.

 Other atomic scattering problems which we’ve also looked at – ranging from cold plasma studies, to BEC’s, to others touching on astrophysics

## Electron-impact excitation of Li and Li+

**	**

***Electron-impact excitation of Li and Li******+***

- R-matrix with Pseudo States (RMPS) method has been used for many electron-impact excitation cross section calculations
- In these calculations use of pseudo states reflected the importance of accurately representing the coupling to the continuum
- Systems of importance in fusion research, dramatic revision of accepted rate coefficients
**Li**: Griffin et al, Phys. Rev. A (2001)
**Li****+****:** Ballance et al, J. Phys. B (2003)
**C****2+****:** Mitnik et al, J. Phys. B (2003)
**C****3+****:** Griffin et al, J. Phys. B (2000)

**Notes:**
So, studied electron-impact excitation of many light atoms and atomic ions; some examples given here. Chiefly used RMPS method although for some calculations these have been confirmed by selected TDCC calculations.

However, including these states can give an order of magnitude increase in the size of the matrices which would lead to three orders of magnitude increase in the time required for the calculations 

Importance of coupling of the bound states to the continuum in R-matrix calculations

Y-axis is collision strength which is closely related to the cross section (find out exactly how?)

## Electron-impact excitation of heavier ions

**	**

***Electron-impact excitation of heavier ions***

- Large R-matrix calculations (with no pseudo states) contain many levels to represent the low-lying bound states accurately
- Ne+: Griffin et al, J. Phys. B (2001)
- Ne4+: Griffin et al, J. Phys. B (2000)
- Ne5+: Mitnik et al, J. Phys. B (2001)
- Fe20+: Badnell & Griffin, J. Phys. B (2001)
- Fe21+: Badnell et al, J. Phys. B (2001)
- Fe23+: Whiteford et al, J. Phys. B (2002)
- Fe24+: Whiteford et al, J. Phys. B (2002)
- Fe25+: Ballance et al, J. Phys. B (2002)

**Notes:**
Massively parallel computer codes needed to run these large calculations
Neon calculations made in ICFT representation

Point to make here is that previous calculations (on Li, C, etc) had relatively small structure components (few electrons) and large pseudo state expansions to get accurate continuum coupling effects incorporated.
These calculations on neon and iron type systems need large structure (also with intermediate coupling) to get low-lying bound states accurately. These massive calculations cannot, yet, have pseudo state incorporated into them, so we only can examine transitions to low lying states.

Also, big effort of course to look at Fe systems due to its importance astrophysically. (Ask Mitch; should I mention Iron Project of Berrington/Badnell UK group?) 

## Electron-impact ionization of lithium

- Demonstrated importance of treating the three-body Coulomb problem using      non-perturbative methods
- Large differences with perturbative calculations on the order of 50 – 100 % in ionization cross section
***J. Colgan, M. S. Pindzola, D. M. Mitnik, D. C. Griffin, and I. Bray,  Phys. Rev. Lett. (2001)***

***Li (2s)***

***Li (2p)***

**Notes:**
 Here say experimental measurements (of Zapesochnyi and Aleksakin) are from 1969(!). It seemed distorted-wave theory (perturbative) agreed well with them. When this was revisited, recently, it was found that new, state-of-the-art theoretical calculations were significantly below the old experiment and perturbative theory. 3 completely independent theories now were in excellent agreement, highlighting need for new experimental measurements, and also showing that previous plasma models using older theoretical/experimental data would be using data in error by ~50% or more.

 Make sure that I explain what ‘new’ theoretical calculations are; TDCC time-dependent (wave-packet) close-coupling calculations. RMPS are based on well-known R-matrix theory (of Burke), and CCC are convergent close-coupling calculations.

 Ionization of excited state lithium (from 2p). NO experimental measurements. Again perturbative theories way too high – almost by a factor of 2 at the peak of the cross section. Although population of excited states will probably be lower than that of ground state, magnitude of the cross sections shows that it will contribute significantly to the ionization rate (coefficients).

 good agreement with a more recent experiment in a higher energy range (away from the peak of the cross section).

## Electron-impact ionization of Be

**Red line: TDCC**** **
**Green line: DWIS(N)**
**Black line: DWIS(N-1)**
**Blue circles: CCC**
- Large differences near ionization threshold and cross section peak between the perturbative and nonperturbative calculations
- No experimental measurements with which to compare for either configuration
***	******J. Colgan, S. D. Loch, M. S. Pindzola, C. P. Ballance, and D. C. Griffin, Phys. Rev. A (2003)***

***Be(1s******2******2s******2)***

***Be(1s******2******2s2p)***

**Notes:**
Needs to be obvious what each line is – bigger legend or put it in right hand box.
Explain 2 different types of DW; DWIS(N) is both incident and scattered electrons calculated in VN potential;
DWIS(N-1) is both incident and scattered electrons calculated in VN-1 potential

Also mention (if pushed) that lower graph has TDCC calculations from the 2p sub-shell of Be only! Lower line shows the DW contribution from the 2s sub-shell.

Also talk about difference in rate coefficient with distorted-wave – around a factor of 10 at low energy. Got a figure if required?

## Dielectronic Recombination

***Dielectronic Recombination***

- Project well underway to calculate dielectronic recombination rate coefficients for complete isoelectronic sequences
- Currently completed H-like through Be-like sequence
- Moving onto second row of periodic table
- Theoretical approach most accurate at high temperatures where recombination project is focused towards fusion plasma applications
_***Theory & Methodology:***_*** Badnell et al, A&A (2003)***
_***H-like ions:***_*** Badnell et al, A&A (2003)***
_***He-like ions:***_*** Badnell et al, A&A (2003)***
_***Li-like ions:***_*** Colgan, Pindzola, and Badnell, A&A (2003)***
_***Be-like ions:***_*** Colgan, Pindzola, Whiteford, and Badnell, A&A (2003)***

**Notes:**
Many experiments for Li-like systems; good agreement found
Again, this project has astrophysical relevance (as well as fusion) as DR is probably dominant electron-ion recombination mechanism in astrophysical plasmas. 
Plays important role in determining the level populations and ionization balance of both high and low-temperature non-LTE plasmas for a wide range of electron densities
Important therefore in modeling of transient astrophysical plasmas such as solar flares
Cannot no longer assume coronal approximation; that all the ion population is in the ground state – there may well be significant population in metastable states

Could mention (if they don’t know) that:
DR can be viewed as a two-step process:
1) free electron excites an electron in an ion and is captured into an autoionizing state of the ion (reversible)
2) an electron can make a spontaneous radiative transition, leaving the ion in a non-autoionizing ie. a bound state – or – 
3) an electron autoionizes leaving the ion in a bound state
e- + X+z  X+z-1  (X+z)* + e-  					
       	 (X+z)* + h
DR can proceed via many intermediate autoionizing states (possible entire Rydberg series) – underlining its importance  

## Dielectronic Recombination of Cl13+ 

- Dielectronic recombination rate coefficients for Be-like Cl13+
- Comparisons of our theoretical calculations with experiment of Heidelberg group at the TSR
- Experimental confirmation of DR project results
- Trielectronic recombination (core excitation of two electrons) observed for the first time
***	******Schnell et al. Phys. Rev. Lett.  (2003)***

- Rate coefficient

- Energy (eV)

**Notes:**
Emphasize these experimental measurements confirm our DR project results for Be-like system investigated here.

For Li-like systems many more experimental measurements; our DR project in good agreement with these, especially at higher electron temperatures.

Will probably need a little explanation of TR; core excitation of TWO electrons into a triply intermediate state (DR is core excitation of ONE electron). This process is automatically included in our DR calculations, but this is first time it has been isolated experimentally (as DR will in most situations, dominate it in the cross section, TR is significantly weaker).
TR resonances represent more complex features than DR resonances because radiative stabilization competes with additional autoionizing decay channels not present in DR.
Chlorine was picked as the TR resonances were most striking for this case. Was observed (though perhaps not as clearly) in other beryllium-like ions.

## Ion-Atom Collisions

*Large scale calculations have established benchmarks by which to judge and evaluate other calculations and experiments, have lead to significant revisions of accepted results*
*p + H:* Kolakowska, et al, Phys. Rev. A (1998)
*p + H:* Schultz, et al., Phys. Rev. Lett. (1999)
*p + Li:* Pindzola, Phys. Rev. A (1999)
*p + Li*:* Pindzola, Phys. Rev. A (2002)
*** **+ H:* Pindzola, Minami, and Schultz, Phys. Rev. A (2003)
- Currently working on:
- 	*Be**4+, **C**6+** + H total and state-selective charge transfer*
*	**p + H* in ExB fields**** ****with the goal of reaction and population control*

*Time-dependent evolution of electronic wave function during p + Li collision*

*p*

*Li*

**Notes:**
Time-dependent evolution of electronic wave function during collision – ask Mitch or look at papers; this is proton on Li(2p) I think? Yes. 
Proton on Li (2py) at 15 keV with b=5. This was proton-impact excitation also.
TD methods – following evolution of time-dependent wave function on a grid allows us to extract scattering information after collision is over. That’s what we are showing here.
Looked at many ion-impact situations as listed. Currently looking at some directly relevant to fusion plasmas, eg Be4+ on hydrogen atoms and effect of crossed fields on excitation and charge exchange rates.
One-electron two-center problem solved exactly on a numerical 3D lattice – large calculations even on a parallel computer.

## Double photoionization of helium

***Triple differential cross section at equal energy sharing between the ejected electrons***

***(************,2e) *****He:** Colgan, Pindzola, and Robicheaux, J. Phys. B (2001)
***(************,2e) *****He:** Colgan and Pindzola, Phys. Rev. A (2002)
***(************,2e) *****Be:** Colgan and Pindzola, Phys. Rev. A (2002)
***(2************,2e) *****He:** Colgan and Pindzola, Phys. Rev. Lett. (2002)
***(************,2e) *****He*:** Colgan and Pindzola, Phys. Rev. A (2003)

**Notes:**
This is a more fundamental atomic collision process – three-body Coulomb break-up problem – much experimental interest here on electron-electron correlation.
A single photon knocks out both electrons from the helium atom – so electron-electron correlation is crucial to correctly calculating the cross section for this process. The figure shows the differential cross section for helium calculated using the time-dependent method.

Calculated many related quantities of two-electron systems such as in He, Be, etc (Could also mention now looking a 4 body break up problems such as triple photoionization of lithium).

## Mode excitation of a BEC soliton state

***Density plot of a sodium condensate in a ***
***soliton state undergoing mode excitation****** ***

- Feder, Pindzola, Collins, Schneider, Clark, Phys. Rev. A (2000)
- Denschlag et al, Science (2000)
- Pindzola and Texier, J. Phys. B (2003)
- Currently working on BEC’s in wave-guides in collaboration with the MIT group of Pritchard and Ketterle

**Notes:**
Time evolution of Bose-Einstein condensates on a lattice. Direction numerical solution of Gross Pitaevskii equation in all 3 dimensions.

## Simulations of ultra-cold plasmas

***Inclusion of electron - Rydberg atom scattering and three-body recombination is necessary to properly simulate the expansion of an ultra-cold plasma***

- Robicheaux and Hanson, Phys. Rev. Lett. (2002)
- Robicheaux and Hanson, Phys. Plasmas (2003)

- Asymptotic expansion velocity

- Initial average ion energy

- Experiment

- With new effects

- Without new effects

**Notes:**
Looking at atomic physics phenomena in different plasma regime – ultra-cold plasmas. By ultra cold we mean in  micro-Kelvin range

Models of experiments which looked at plasma expansions and thermalizations required atomic physics calculations (electron Rydberg scattering and 3-body recombination) to accurately model the experiment.

## Science Applications

**Li beam diagnostics at General Atomic’s DIIID**
**Rate data: Physica Scripta (2003), modeling comparisons: Phys. Rev. E (2003)**
**Be wall studies at PISCES-B experiment at UCSD **
**Rate data to be published in Physica Scripta**
**He beam diagnostic at University of Wisconsin**
**PPPL, GA Beam penetration modeling**
**GA Charge Exchange Recombination Spectroscopy**
**Kr diagnostics at Tore-Supra **
**Kr****q+**** (PRA 2002)**
**W, Hf wall erosion studies at JET**
**X-ray spectra from Chandra and XMM-Newton**
**Fe****20+**** (JPB 2001), Fe****21+**** (JPB 2001), Fe****23+**** (JPB 2002), ...**
**Early universe isotope abundances**
**He (PRA 2002), Li (PRA 2003)**
**Hypernovae spectra**
**Fe****3+**** (JPB 2003)**

**Notes:**
These are some fundamental atomic scattering problems to which we apply large numerical calculations. Most have relevance to processes which can take place in fusion devices.

 Specific projects which we are/have been working on recently are listed. Will talk about some of them in detail in the coming slides.

 Other atomic scattering problems which we’ve also looked at – ranging from cold plasma studies, to BEC’s, to others touching on astrophysics

## Electron-impact ionization data for fusion plasmas

- Program initiated to completely revise atomic collisional data for lithium
- Calculations have been used to perform modeling in support of plasma edge diagnostics made by Todd Evan’s group at ***DIIID ***
- Accurate atomic data essential for collisional radiative modeling to be meaningful
- Plasma modeling calculations also improve edge diagnostics at ***JET***

**Notes:**
It was realized, in recent years, that the current atomic data for lithium and its ions was pretty much inadequate. There has been little recent experimental measurements of fundamental processes (such as ionization of Li etc) and, before 5/10 years ago, most theoretical calculations
were perturbative in nature; also it was not clear how accurate they were.

Also there was a demand for high quality data since lithium has been used (or proposed) as a plasma diagnostic within fusion devices. Indeed experiments have already been made using such diagnostics. Also liquid lithium was also considered as a plasma wall component, although it is unclear how viable this is!

So our group decided to carry out a series of calculations, using several different theoretical techniques, for scattering processes from lithium and all its ions.

## Effective ionization rates for Li

- New data for excitation, ionization, and recombination has been incorporated into an ADAS plasma modeling calculation
- Also available at CFADC database
- Very large differences compared to model using older atomic data (log scale!)
- Significant increase in the accuracy of the plasma modeling for lithium
***Loch et al, Physica Scripta, (2003)***

***Effective ionization rate for Li at an electron density of 10******14****** cm******–3 ******as a function of electron temperature***

**Notes:**
Emphasize log scale on both axis. LARGE difference in the effective ionization rates from old data and rates from new calculations.
Increase in the accuracy of plasma modeling for lithium now.

Ask Stuart – what exactly is ‘effective’ ionization rate?
Effective is density dependent; also it gives the total loss out of a given ion stage; i.e. includes ionization from all excited states of that ion.
So this would include ionization from all excites states (2p,3l,4l) etc. 

## Li emissivity coefficients

- Emissivity coefficients as a function of electron temperature and density
- Obtained from ADAS modeling calculation
- Data such as this is used in a wide range of fusion plasma edge diagnostics
- Also comparing these and similar plasma characteristics with modeling calculations made using Los Alamos set of computer codes
***Loch, Fontes, et al, Phys. Rev. E,  (2003)***

**Notes:**
Mention that we are currently working on comparing our ADAS plasma modeling calculations (incorporating our new data) with plasma modeling calculations based on the Los Alamos suite of computer codes. LA data typically uses plane-wave Born/distorted wave type of data but they usually look at heavier elements – are their models less accurate than ours for light systems?

Number of photons lost:
This is 2p-2s transition.
A-value times the upper level population.... No of photons emitted per second per unit volume from a given transition
As you increase the temperature the 2p level can be reached – so the emissivity turns on. Also as you increase the electron density the emissivity initially increases. At high densities, it drops as the lithium is preferentially ionized instead of excited.

## Collisional Data for modeling plasmas  

- ITER may use Be as a plasma wall component – our calculations made in support of the ITER fusion development program of Alberto Loarte (MPI – Garching)
- Collisional Radiative modeling also made for the PISCES-B fusion device which will use Be as a potential plasma-facing wall component – in collaboration with Russ Doerner’s group at UCSD

***ITER ***

**Notes:**
In current calendar year we started new program to examine similar scattering quantities for beryllium. Important to fusion as it has been proposed as a first wall component in plasma devices. Thus interaction with plasma (in the form of excitation, ionization, and charge transfer) is important.

Just recently submitted to PRA 

Large differences found between TDCC calculations and perturbative distorted-wave calculations for neutral Be
Program currently underway to revise electron-impact excitation and ionization data for beryllium

## Be data applications

- 	Ratio of ionization rate coefficients for Be
- 	Large difference between typically employed DW calculations and new TDCC results show as much as a factor of 10 difference in the rate coefficient

## Revision of TRANSP and ONETWO collision data  

- Data needed to update the inelastic, heavy-particle collision database used by TRANSP (PPPL, McCune) and ONETWO (GA, Murakami) for neutral beam deposition, transport modeling
- Priorities include charge transfer, ionization/excitation of light impurity ions (C6+, O8+, Be4+) between 10 eV and 150 keV colliding with hydrogen
- Next phase, excited hydrogen targets, other light impurities, helium
- Synergy with charge exchange recombination spectroscopy data production

**Notes:**
In current calendar year we started new program to examine similar scattering quantities for beryllium. Important to fusion as it has been proposed as a first wall component in plasma devices. Thus interaction with plasma (in the form of excitation, ionization, and charge transfer) is important.

Just recently submitted to PRA 

Large differences found between TDCC calculations and perturbative distorted-wave calculations for neutral Be
Program currently underway to revise electron-impact excitation and ionization data for beryllium

## Status of recommended charge transfer cross section for C6+ + H

***Status of recommended charge transfer cross section for C******6+****** + H***

## LTDSE calculations to address charge transfer data needs

- Asymmetric nuclear charges necessitate larger grids with high-order methods, 2453 and 5123 Fourier collocation – ORNL Cheetah IBM SP4
- First case Be4+ + H, H(2s)
- Impurity charge exchange database project,  ions + H, H(2s)

***LTDSE calculations to address charge transfer data needs***

## LTDSE results for Be4+ + H charge transfer

***LTDSE results for Be******4+****** + H charge transfer***

- State-selective charge transfer results are providing stringent test of other theories
- First results show significant deviations from best existing calculations
- State-selective results will benefit spectroscopy diagnostics

## New results relevant to CHERS

***New results relevant to CHERS***

- For highly charged ions of great importance to diagnostics (e.g. C6+), optical transitions are detected between relatively high-lying n-levels
- MPP computer have enabled half-billion trajectory simulations to produce new results pertaining to GA diagnostics
- C, O, N, Ne, ..., data will be generated for H and H*

## Heavy species plasma studies 

- Plasma transport models study heavy species impurity transport studies
- Our calculations generated collisional rate coefficients for all ion stages of krypton
- Work has impacted plasma modeling of Mario Mattioli and coworkers at the CEA, Cadarache, France

***Reverse field pinch plasma device (RFX) at Padova, Italy***

## Computational Aspects

**NERSC support by Ng and Lamoureux on memory access by lattice codes – Winter 2001/02**
**Fourier transform method by Robicheaux and Colgan for lattice codes – Spring 2002**
**Comparison study of TDSE1 and TDSE2 codes by Minami – Summer 2002**
**Variable mesh studies by Pindzola and Witthoeft – Fall 2002**
**Propagator studies by Robicheaux and Pindzola for lattice codes – Winter 2002/03**
**Comparisons of ADAS and Los Alamos plasma modeling codes by Loch and Fontes – Spring 2003**
**ORNL PERC evalution of R-matrix and LTDSE codes – Summer 2003**
**NERSC support by Schwartz on 3-d visualization for four-body codes – Summer 2003**
**Comparison study of PRMAT1 and PRMAT2 codes by Ballance and McLaughlin – Summer 2003**

**Notes:**
These are some fundamental atomic scattering problems to which we apply large numerical calculations. Most have relevance to processes which can take place in fusion devices.

 Specific projects which we are/have been working on recently are listed. Will talk about some of them in detail in the coming slides.

 Other atomic scattering problems which we’ve also looked at – ranging from cold plasma studies, to BEC’s, to others touching on astrophysics

## LTDSE calculations of proton-lithium excitation

- Parallel time-dependent evolution of electronic wavefunction during collision

- Parallel computation of time evolutions for required impact parameters  cross sections

- 2p  3s

- 2p  3p

- 2p  3d

- p + Li(2p)

## Scaling of the time-dependent codes on Seaborg

- Calculation of proton Li charge exchange cross sections
- LTDSE finite differences code

## Science Education

**PhD students: **
**M. C. Witthoeft (BS Kansas State University)**
**T. Topcu (BS Mamara University)**
**Post-doctoral Fellows: **
**J. Colgan (PhD Queen’s University, Belfast)**
**D. M. Mitnik (PhD Hebrew University)**
**S. D. Loch (PhD Strathclyde University)**
**C. P. Ballance (PhD Queen’s University, Belfast) **
**T. Minami (PhD Tokyo University)**
**Long term visitor: C. Fontes (LANL)**
**Short term visitors: **
**H. P. Summers (Strathclyde University)**
**N. R. Badnell (Strathclyde University)**
**M.  O’Mullane (JET)**
**K. Berrington (Sheffield-Hallam University)**
**B. M. McLaughlin (Queen’s University)**
**T. W. Gorczcycz (Western Michigan University)**

**Notes:**
These are some fundamental atomic scattering problems to which we apply large numerical calculations. Most have relevance to processes which can take place in fusion devices.

 Specific projects which we are/have been working on recently are listed. Will talk about some of them in detail in the coming slides.

 Other atomic scattering problems which we’ve also looked at – ranging from cold plasma studies, to BEC’s, to others touching on astrophysics