Download Assessing uncertainties of theoretical atomic transition probabilities

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
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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:
Out:
 RCG
In:
Out:
 RCE
In:
Out:
Z, Nel, configurations
Slater and CI parameters P,
Transition matrix elements M
P, M
Eigenvalues E, Eigenvectors V,
Wavelengths λ, Line Strengths S, P
Derivatives ∂P/∂E
E, V, P, ∂P/∂E, experimental energies Eexp
Fitted parameters PLSF, eigenvalues ELSF,
eigenvectors VLSF
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?
Adapted from
S. Enzonga Yoca and
P. Quinet, JPB 47
035002 (2014)
E1:
gA = 2.03×1018 S / λvac3
M1:
gA = 2.70×1013 S / λvac3
E2:
gA = 1.12×1018 S / λvac5
Wrong!
Compare S and S*
Test case: M1 and E2
transitions in Fe V (Ti-like)
1S1
120
E, 1000 cm−1
100
1D1
34 levels
590
transitions
80
60
40
3P1, 3F1
1D2
3G
20
0
1G1
1F
1G2, 3D, 1I, 1S2
3P2, 3H
5D
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
Plan of Monte-Carlo
experiment with Cowan codes
 Vary E2 transition matrix elements (1%
around ab initio values)
 Vary P (ΔPLSF around PLSF)
Vary parameters randomly with normal
distribution
 Make trial calculations with varied
parameters
 recognize resulting levels by eigenvectors
 rescale A from S using Eexp instead of E
 Analyze statistics
First test: Vary only E2 matrix
elements
δA/A* (percent)
100
Each point
represents 400
random trials
10
1
1.E-10
1.E-08
1.E-06
1.E-04
1.E-02 1.E+00
S
Vary E2 matrix elements and Slater
parameters
Cancellation Factor
CF = (S+ + S−)/(S+ + |S−|)
−1 ≤ CF ≤ 1
|CF| ≪ 1 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
Statistical distributions of A values
What quantity has best statistical
properties (A, ln(A), Ap)? 1000 trials
590000 points
n=
δA/std(A)
Box-Cox transformation
𝐟=
𝐴/𝐴∗ 𝑝 − 1
,𝑝 ≠ 0
𝑝
ln 𝐴/𝐴∗ , 𝑝 = 0
Despite piecewise definition, f(p) is
a continuous function!
Statistical parameters
𝑛
(𝑥
𝑖=1 𝑖
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝜎 2 =
𝑠𝑘𝑒𝑤𝑛𝑒𝑠𝑠 =
𝑘𝑢𝑟𝑡𝑜𝑠𝑖𝑠 =
𝑛
(𝑥
𝑖=1 𝑖
(𝑛 − 1)
− 𝑥)3
(𝑛 − 1)𝜎 3
𝑛
(𝑥
𝑖=1 𝑖
(𝑛 −
− 𝑥)4
1)𝜎 4
− 𝑥)2
−3
Normal probability plots
Same 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 p
1000 trials, 590 000 data points
Distribution of optimal p
10 000 trials, 5 900 000 data points
Statistics of outliers compared
to normal distribution
10 000 trials, 5 900 000 data points
n=
δA/std(A)
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 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
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.