Assessing uncertainties of theoretical atomic transition probabilities

1. Assessing uncertainties of theoretical atomic transition probabilities with Monte Carlo random trials Alexander Kramida National Institute of Standards and Technology, Gaithersburg, Maryland, USA

2. Parameters in atomic codes • Transition matrix elements • Slater parameters • CI parameters • Parameters of effective potentials • Diagonal matrix elements of the Hamiltonian • Fundamental “constants” • Cut-off radii • …

3. Cowan’s atomic codes • RCN+RCN2 • In: Z, Nel, configurations • Out: Slater and CI parameters P, • Transition matrix elements M • RCG • In: • Out: Eigenvalues E, Eigenvectors V, • Wavelengths λ, Line Strengths S, • Derivatives ∂P/∂E • RCE • In: E, V, P, ∂P/∂E, experimental energies Eexp • Out: Fitted parameters eigenvalues ELSF, • eigenvectors VLSF P, M P PLSF,

4. Uncertainties of fitted parameters ΔPLSF = ∂P/∂E (Eexp − E)

5. How to estimate uncertainties of S (or A, f)? • Compare results of different codes • Compare results of the same code • Length vs Velocity forms • With different sets of configurations • With varied parameters

6. What to compare? Adapted from S. EnzongaYoca and P. Quinet, JPB 47 035002 (2014) E1: gA = 2.03×1018S / λvac3 M1: gA = 2.70×1013S / λvac3 E2: gA = 1.12×1018S / λvac5 Wrong!

7. CompareS and S*

8. Test case: M1 and E2 transitions in Fe V (Ti-like)

9. Test case: Fe V More complexity Interacting configurations: 3d4 3d3(4s+5s+4d+5d) 3d2(4s2+4d2+4s4d) 38 E2 transition matrix elements 86 Slater parameters Eav ϛ3d, ϛ4d F2,4(nd,nʹd) G0,2,4(nl,nʹlʹ) α3d, β3d, and T3d 61 CI parameters

10. Plan of Monte-Carlo experiment with Cowan codes • Vary E2 transition matrix elements (1% around ab initiovalues) • Vary P (ΔPLSF around PLSF) • Make trial calculations with varied parameters • recognize resulting levels by eigenvectors • rescale A from S using Eexp instead of E • Analyze statistics Vary parameters randomly with normal distribution

11. First test: Vary only E2 matrix elements

12. Vary E2 matrix elements and Slater parameters

13. Cancellation Factor CF = (S+ + S−)/(S+ + |S−|) −1 ≤ CF ≤ 1 |CF| means strong cancellation Degree of cancellation Dc = δCF/|CF| where δCF is standard deviation of CF Dc≥ 0 Dc≥ 0.5 means really strong cancellation

14. Statistical distributions of A values

15. What quantity has best statistical properties (A, ln(A), Ap)? 1000 trials 590000 points n = δA/std(A)

16. Box-Cox transformation Despite piecewise definition, f(p) is a continuous function!

17. Statistical parameters

18. Normal probability plots Same transition, same trial data (A-values) Different parameter p of Box-Cox transformation

19. Two methods of optimizing p (a)Maximizing the correlation coefficient C of the normal probability plot (b)Finding pyielding zero skewness of distribution of f(A, p)

20. Distribution of optimal p 1000 trials, 590 000 data points

21. Distribution of optimal p 10 000 trials, 5 900 000 data points

22. Statistics of outliers compared to normal distribution 10 000 trials, 5 900 000 data points n = δA/std(A)

23. Abnormal transitions Normal probability plots with optimal parameter p of Box-Cox transformation

24. Main conclusions (so far) • Standard deviations σare not sufficient to describe statistics of A-values • Knowledge of distribution shapes is required • Each transition has a different shape of statistical distribution. Most are skewed. • For most transitions, a suitable Box-Cox transformation exists, which transforms statistics to normal • In addition to σ, parameter p of optimal Box-Cox transformation is sufficient to characterize statistics of most transitions

25. Required statistics size 10 compared datasets: σA differs from true value by >20% for 99% of transitions 100 datasets: “wrong” σA for 3% of transitions 1000 datasets: “wrong” σA for 1% of transitions 10000 datasets: “wrong” σA for a few of 590 transitions (all negligibly weak) If requirement on accuracy of σA is relaxed to 50%, 10 datasets: “wrong” σA for 10% of transitions 100 datasets: “wrong” σA for a few of 590 transitions

26. Strategy for estimating uncertainties • Investigate internal uncertainties of the model by varying its parameters and comparing results • Investigate internal uncertainties of the method by extending the model and looking at convergence trends (not done here) • Investigate possible contributions of neglected effects (not done here) • Investigate external uncertainties of the method by comparing with results of other methods (not done here)

27. Further notes • Distributions of parameters were arbitrarily assumed normal. True shapes are unknown. • Unknown distribution width of E2 matrix elements was arbitrarily assumed 1%. • Parameters were assumed statistically independent (not true). • When results of two different models are compared, shapes of statistical distributions of A-values should be similar (unconfirmed guess). • Implication for Monte-Carlo modeling of plasma kinetics: A-values given as randomized input parameters should be skewed, each in its own way described by Box-Cox parameter p, and correlated.

28. Final conclusion • The “new” field of Statistical Atomic Physics should be developed. • Main topics: • statistical properties of atomic parameters; • propagation of errors through atomic and plasma-kinetic models.