with Monte Carlo random trials Alexander Kramida National Institute of Standards and Technology Gaithersburg Maryland USA Parameters in atomic codes Transition matrix elements Slater parameters ID: 815057
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Slide1
Assessing uncertainties of theoretical atomic transition probabilities with Monte Carlo random trials
Alexander Kramida
National Institute of Standards and Technology,Gaithersburg, Maryland, USA
Slide2Parameters in atomic codes
Transition matrix elementsSlater parametersCI parametersParameters of effective potentialsDiagonal matrix elements of the HamiltonianFundamental “constants”Cut-off radii…
Slide3Cowan’s atomic codes
RCN+RCN2 In: Z, Nel, configurations Out: Slater and CI parameters P, Transition matrix elements MRCG In: Out: Eigenvalues E, Eigenvectors V, Wavelengths λ, Line Strengths S, Derivatives
∂P/∂ERCE In: E, V, P, ∂P/∂E, experimental energies Eexp Out: Fitted parameters eigenvalues ELSF, eigenvectors VLSF
P
,
M
P
P
LSF
,
Slide4Uncertainties of fitted parametersΔP
LSF = ∂P/∂E (Eexp − E)
Slide5How to estimate uncertainties of S (or
A, f)?Compare results of different codesCompare results of the same code Length vs Velocity forms With different sets of configurations With varied parameters
Slide6What to compare?
E1:gA = 2.03×1018 S / λvac3M1:gA = 2.70×1013 S / λvac3E2:gA = 1.12×1018
S / λvac5Adapted fromS. Enzonga Yoca and P. Quinet, JPB 47 035002 (2014)
Wrong!
Slide7Compare S and S*
Slide8Test case: M1 and E2 transitions in Fe V (Ti-like)
Slide9Test case: Fe VMore complexity
Interacting configurations: 3d4 3d3(4s+5s+4d+5d) 3d2(4s2+4d2+4s4d)38 E2 transition matrix elements86 Slater parameters Eav ϛ3d, ϛ4d
F2,4(nd,nʹd) G0,2,4(nl,nʹlʹ) α3d, β3d, and T3d61 CI parameters
Slide10Plan of Monte-Carlo experiment with Cowan codes
Vary E2 transition matrix elements (1% around ab initio values)Vary P (ΔPLSF around PLSF) Make trial calculations with varied parameters recognize resulting levels by eigenvectors rescale A from S
using Eexp instead of EAnalyze statisticsVary parameters randomly with normal distribution
Slide11First test: Vary only E2 matrix elements
Slide12Vary E2 matrix elements and Slater parameters
Slide13Cancellation Factor
CF = (S+ + S−)/(S+ + |S−|)−1 ≤ CF ≤ 1|CF| means strong cancellationDegree of cancellationDc = δCF
/|CF|where δCF is standard deviation of CFDc ≥ 0Dc ≥ 0.5 means really strong cancellation
Slide14Statistical distributions of A
values
Slide15What quantity has best statistical properties (A
, ln(A), Ap)?n = δA/std(A)
1000 trials 590000 points
Slide16Box-Cox transformation
Despite piecewise definition, f(p) is a continuous function!
Slide17Statistical parameters
Slide18Normal probability plots
Same transition, same trial data (A-values) Different parameter p of Box-Cox transformation
Slide19Two methods of optimizing p
(a) Maximizing the correlation coefficient C of the normal probability plot(b) Finding p yielding zero skewness of distribution of f(A, p)
Slide20Distribution of optimal p
1000 trials, 590 000 data points
Slide21Distribution of optimal p
10 000 trials, 5 900 000 data points
Slide22Statistics of outliers compared to normal distribution
n = δA/std(A)10 000 trials, 5 900 000 data points
Slide23Abnormal transitions
Normal probability plots with optimal parameter p of Box-Cox transformation
Slide24Main conclusions (so far)
Standard deviations σ are not sufficient to describe statistics of A-valuesKnowledge 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 normalIn addition to σ, parameter p of optimal Box-Cox transformation is sufficient to characterize statistics of most transitions
Slide25Required statistics size10 compared datasets:
σA differs from true value by >20% for 99% of transitions100 datasets: “wrong” σA for 3% of transitions1000 datasets: “wrong” σA for 1% of transitions10000 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 transitions100 datasets: “wrong” σA
for a few of 590 transitions
Slide26Strategy for estimating uncertainties
Investigate internal uncertainties of the model by varying its parameters and comparing resultsInvestigate 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)
Slide27Further 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.
Slide28Final conclusionThe “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.