Difference between revisions of "Stretched Exponentials"
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In between exponential <math>\exp(-\lambda t)</math> and gaussian <math>\exp[-(\sigma t)^2]</math> relaxation (and indeed extending beyond either) is the much-abused empirical "'''''stretched exponential'''''" function, <center><math>\exp[-(\Lambda t)^\beta]</math></center> |
In between exponential <math>\exp(-\lambda t)</math> and gaussian <math>\exp[-(\sigma t)^2]</math> relaxation (and indeed extending beyond either) is the much-abused empirical "'''''stretched exponential'''''" function, <center><math>\exp[-(\Lambda t)^\beta]</math></center> |
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I don't much care for it. It will ''fit'' a wide variety of ZF-µSR relaxation functions, but what do the results ''mean''? What do they tell us about the physics? There are cases where "root exponential" relaxation functions (<math>\beta = 1/2</math>) suggest a rapidly fluctuating spin glass, but fit results with <math>\beta</math> all the way from 0.25 to 3 have been published without any hint of irony. Please don't resort to this. |
I don't much care for it. It will ''fit'' a wide variety of ZF-µSR relaxation functions, but what do the results ''mean''? What do they tell us about the physics? There are cases where "root exponential" relaxation functions (<math>\beta = 1/2</math>) suggest a rapidly fluctuating spin glass, but fit results with <math>\beta</math> all the way from 0.25 to 3 have been published without any hint of irony. Please don't resort to this. |
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<math>G(t) = \exp[-(\Lambda t)^\beta]</math> for <math>\Lambda = 1</math> and <math>\beta = 0.25, 0.5, 1, 2, 3</math>. Only <math>\beta = 1, 2</math> and sometimes <math>0.5</math> have familiar physical meanings. |
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Revision as of 12:04, 29 August 2022
Relaxonomy --> here
In between exponential <math>\exp(-\lambda t)</math> and gaussian <math>\exp[-(\sigma t)^2]</math> relaxation (and indeed extending beyond either) is the much-abused empirical "stretched exponential" function,
I don't much care for it. It will fit a wide variety of ZF-µSR relaxation functions, but what do the results mean? What do they tell us about the physics? There are cases where "root exponential" relaxation functions (<math>\beta = 1/2</math>) suggest a rapidly fluctuating spin glass, but fit results with <math>\beta</math> all the way from 0.25 to 3 have been published without any hint of irony. Please don't resort to this.
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<math>G(t) = \exp[-(\Lambda t)^\beta]</math> for <math>\Lambda = 1</math> and <math>\beta = 0.25, 0.5, 1, 2, 3</math>. Only <math>\beta = 1, 2</math> and sometimes <math>0.5</math> have familiar physical meanings. |
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