Assessing uncertainties of theoretical atomic transition probabilities 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 CI parameters Parameters of effective potentials Diagonal matrix elements of the Hamiltonian Fundamental “constants” Cut-off radii …

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,

Uncertainties of fitted parameters

ΔPLSF = ∂P/∂E (Eexp − E)

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

What to compare?

E1:gA = 2.03×1018 S / λvac

3

M1:gA = 2.70×1013 S / λvac

3

E2:gA = 1.12×1018 S / λvac

5

Adapted fromS. Enzonga Yoca and P. Quinet, JPB 47 035002 (2014)

Wrong!

Compare S and S*

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

0

20

40

60

80

100

120E,

1000 c

m−

1

5D

3P2, 3H3G

1G2, 3D, 1I, 1S2

1D21F

3P1, 3F11G1

1D1

1S134 levels

590 transitions

Test 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 T3d

61 CI parameters

Plan 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 E

Analyze statistics

Vary parameters randomly with normal distribution

First test: Vary only E2 matrix elements

1E-10 1E-08 1E-06 1E-04 1E-02 1E+00 1E+021

10

100

S

δA/A

* (p

erce

nt)

Each point repre-sents 400 random trials

Vary E2 matrix elements and Slater parameters

Cancellation Factor

CF = (S+ + S−)/(S+ + |S−|)−1 ≤ CF ≤ 1

|CF| means strong cancellation

Degree of cancellationDc = δCF/|CF|

where δCF is standard deviation of CF

Dc ≥ 0

Dc ≥ 0.5 means really strong cancellation

Statistical distributions of A values

What quantity has best statistical properties (A, ln(A), Ap)?

n = δA/std(A)

1000 trials 590000 points

𝐟={[ ( 𝐴 / 𝐴∗ )𝑝−1𝑝 ] ,𝑝 ≠0

ln (𝐴 / 𝐴∗ ) ,𝑝=0}

Box-Cox transformation

Despite piecewise definition, f(p) is a continuous function!

Statistical parameters

𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠=∑𝑖=1

𝑛

(𝑥 𝑖− 𝑥)3

(𝑛−1)𝜎 3

𝑘𝑢𝑟𝑡𝑜𝑠𝑖𝑠=∑𝑖=1

𝑛

(𝑥𝑖−𝑥)4

(𝑛−1)𝜎4 −3

𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛𝜎 2=∑𝑖=1

𝑛

(𝑥 𝑖−𝑥)2

(𝑛−1)

Normal probability plotsSame transition, same trial data (A-values) Different parameter p of Box-Cox transformation

Two 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)

Distribution of optimal p1000 trials, 590 000 data points

Distribution of optimal p10 000 trials, 5 900 000 data points

Statistics of outliers compared to normal distribution

n = δA/std(A)

10 000 trials, 5 900 000 data points

Abnormal transitions

Normal probability plots with optimal parameter p of Box-Cox transformation

Main conclusions (so far)

• Standard deviations σ are 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

Required statistics size10 compared datasets: σA differs from true value

by >20% for 99% of transitions100 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

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)

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.

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.