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Improved Local and Global Mass Relations W J M F Collis

Improved Local and Global Mass Relations W J M F Collis. Take any 2 x 2 “square” from the Chart of the Nuclides. The sums of the 2 atomic masses in each diagonal are approximately equal! 24690 + 20750 -22287 – 23143=10 kev. The approach.

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Improved Local and Global Mass Relations W J M F Collis

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  1. Improved Local and Global Mass RelationsW J M F Collis Take any 2 x 2 “square” from the Chart of the Nuclides. The sums of the 2 atomic masses in each diagonal are approximately equal! 24690 + 20750 -22287 – 23143=10 kev Collis: Siena 2012

  2. Collis: Siena 2012 The approach • A simple empirical approach which takes NO ACCOUNT of any underlying physics!! • No nuclear “theory” is involved, (but some scholars have attempted rationalize why it works).

  3. Garvey KelsonTransverse Mass relation (1966) • Local Model Error 215 keV(1501 isotopes) • M(N,Z) = fZ(Z)+fN(N)+ fA(N+Z) Global Model Error 328 keV(2120 isotopes) Collis: Siena 2012

  4. Collis: Siena 2012 Judgement of theScientific Establishment • The Garvey Kelson local relations have a minor role to play in verifying atomic mass measurements. • Corresponding global equations are less accurate and are prone to fail for isotopes far from stability. • Too many parameters! Unreliable. • Few improvements made after 40 years.

  5. Collis: Siena 2012 Model Error • The Model is the same as the standard error but not only takes account of the number of arbitrary parameters (increasing the error above the RMS value) but also removes the effect of measurement errors. • In this presentation all the Model Errors are close to the standard errors (just a few keV less)..

  6. Data & Methods Latest atomic masses: Georges Audi and Wang Meng, "The Ame2011 atomic mass adjustment", Private Communication April 2011 Experimental errors are 25 keV RMS Software Proprietary least squares regression programmed in “C” with optimizations for sparse matrices. Collis: Siena 2012

  7. Collis: Siena 2012 Strategy • Search for optimal local relations (partial difference mass equations) similar to Garvey-Kelson relations with least squares • Refine local relation with integer coefficients • “Guess” corresponding partial differential equation • Find general solution(s) to equation • Find optimal coefficients for general solution by least squares regression

  8. Collis: Siena 2012 “Guessing” the corresponding differential equation. This looks like ∂4 M = 0 ∂Z2∂N2 Model Error is 314 keV

  9. Collis: Siena 2012 Model Error is 193 keV over 2120 isotopes Better than any known global equation! Corresponding global solution • M(N,Z) = fZ(Z)+ fN(N)+ NfNZ(Z)+ ZfZN(N)

  10. Collis: Siena 2012 The arbitrary functions f(x) • These are point functions with integer arguments (N, Z etc.) • The points can be determined optimally by least squares regression. • There are as many parameters as there are different values of N and Z. • But there may be fewer parameters than other models (e.g. Janecke Masson) and predictions are more accurate and robust.

  11. Collis: Siena 2012 An dramatic improvement(But what has changed?) Model Error is 82 keV! ∂4 M = k(-1)N+Z ∂Z2∂N2

  12. Collis: Siena 2012 A superior local relation. Model Error is 71 keV This is the best local mass relation on the 5 x 5 plane.

  13. Collis: Siena 2012 Model Error is 174 keV Better still ! Corresponding global solution M(N,Z) = fZ(Z)+ fN(N)+ NfNZ(Z)+ ZfZN(N) + odd

  14. Collis: Siena 2012 A step too far? ∂6 M = k(-1)N+Z ∂Z3∂N3 • M(N,Z) = fZ(Z)+fN(N)+ NfNZ(Z)+ ZfZN(N)+N2fN2Z(Z)+Z2fZ2N(N)+ odd Model Error 102 keV with 748 parameters

  15. Collis: Siena 2012 Reducing the number of parameters. • There’s an arbitrary parameter for ever “point” in the point function. • We can reduce the number of points by linear interpolation. • We can eliminate entire functions if we want to calculate nucleon separation energies. • We could try to add “semi-empirical” terms (but Coulomb Enegy etc. do NOT help!)

  16. Collis: Siena 2012 Nucleon separation Energies M(N,Z)-M(N-1,Z) = fN(N)+ NfNZ(Z)+ ZfZN(N) + odd fZ(Z) term eliminated

  17. Collis: Siena 2012 The “Odd” function odd(N,Z) = k . (-1)N+Z A simple periodic function based on Atomic Number (A=N+Z). Perhaps this “explains” the mass number / isospin dependence in Janecke Masson formulas?

  18. Collis: Siena 2012 Conclusions • Can replace Garvey Kelson and Janecke Masson formulas • Sufficient accuracy for super-nova modelling. • Particularly suitable for calculating nucleon separation energies. • Good for verifying experimental measurements. Thank you. mr.collis@physics.org

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