Speaker
Description
To discuss the determination of model errors we use for simplicity mass models ("mass tables"). Usually their errors are characterized by the root-mean-square deviation $ \sigma_{\mathrm{rms}} = \left( \left[ \sum_{i = 1}^n \left( M_{\mathrm{exp}}^i - M_{\mathrm{th}}^i \right)^2 \right] / n \right)^{1/2} $. However, we immediately (should) see
that the experimental error contributes to the rms value so it is an overestimate of the model error. What to do? We asked AI and the first suggestion that came up was: Multiply each squared residual by a weight $ w_i = 1/ \sigma_{\mathrm{exp}}^2 $ where $ \sigma_{\mathrm{exp}}^2 $ is the standard deviation for that point. But when applied to masses and especially in fitting model parameters then data points with tiny experimental errors would completely determine the outcome, which obviously is not what we want. Actually the mass of $ {}^{12} \mathrm{C} $ has zero error because it defines the atomic mass unit!! The failure of AI shows it is not optimum to hunt for some expression to use, instead we must apply real intelligence. A first step is to figure out what a model error really is. If you have a lot of model data points they usually follow a Gaussian distribution around the correct value. So, what we like to determine is the σ of this distribution. This is accomplished by applying the maximum likelihood method, and was presented already 40 years ago in [1], but few use it. I will apply it to some older and more current mass models and discuss how they perform for data that was not known when they were published. Another misconception that persists in the community is that constrained HFB calculations "automatically" converge towards fission barrier saddle points and that it is not necessary to calculate the energy versus many shape degrees of freedom. For example [2] states They require only one constraint to force a stretching of the system along the various stages while all other details of the rearrangements and shapes result automatically of the calculation. It is true that a curve is obtained but it is often not related to the actual fission barrier [3].
References
[1] P. Möller and J. R. Nix, Atomic Data Nucl. Data Tables 39 (1988) 213, and LosAlamos preprint LA-UR-3983 (with more complete tables).
[2] J. Erler, K. Langanke, H.P. Loens, G. Martinez-Pinedo, and P.-G. Reinhard, Phys. Rev. C 85 024802 (2012).
[3] P. Möller, A. J. Sierk, R. Bengtsson, H. Sagawa, and T. Ichikawa, Phys. Rev. Lett. 103 (2009) 212501.
| Category | Theory |
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