I've seen the two forms of Geiger-Nuttall law. which is the $$log\;\omega = aE^{1/2} + b\hspace{5em}:(1)$$ and $$log\;\omega=a'logR_{\alpha}+b'\hspace{4em}:(2)$$ From the range-energy relation, we know that $$R_\alpha \propto E^{3/2}\hspace{8em}:(3)$$ But these relation are incompatible with each other, as using (3) on (1) results in $$log\;\omega=a''R^{1/3}+b$$ instead of (2), So which one is correct? or is there a different definition of energy for (1) and (3)?, and how can we derive (2) from (1)? what formula is used to convert (1) to (2)?
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1$\begingroup$ Please give information as to the location of these two formulae. $\endgroup$– FarcherCommented May 28 at 10:27
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$\begingroup$ @Farcher equation (1) is obtained through Gamow's theory of alpha decay, (2) is obtained from experiment and (3) is geigers law, it's derivation can be found here google.com/… $\endgroup$– CuSO4 NaOHCommented Jun 13 at 9:06
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$\begingroup$ (2) is obtained from experiment - Is this a result you obtained or is it in a published paper? $\endgroup$– FarcherCommented Jun 13 at 9:46
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$\begingroup$ It was obtained by Geiger and Nuttall themselves and is found in various papers as well. $\endgroup$– CuSO4 NaOHCommented Jun 14 at 6:34
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1 Answer
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The old form of the G-N law, $\log\,\lambda=a'\log R_{\alpha}+b'$ with $a'$ and $b'$ as constants, is an empirical law and detailed in these two papers: The ranges of the α particles from various radioactive substances and a relation between range and period of transformation and The ranges of α particles from uranium.
Here is the graphical representation of the law from one of the papers.
Is there a different definition of energy for (1) and (3)?
There is a small difference because the range, $R$, is related to the energy of the alpha whereas the energy, $E$, is $Q$ value (energy of both alpha and daughter) of the decay.
The Gamow theory of alpha decay predicted the law in terms of the $Q$ value of the decay, ie including the small amount of kinetic energy of the daughter nucleus.
The last two lines of the theoretical derivation are as follows,
What you will note is that $a'$ and $b'$ are not constant but depend on $Z_{\rm D}$, the atomic number of the daughter nucleus.
A later paper, On the Validity of the Geiger-Nuttall Alpha-Decay Law and its Microscopic Basis explores the "constants" in the slightly different form of the G-N law, $\log_{10}T_{1/2}=A(Z)Q_{\alpha}^{-1/2}+ B(Z)$, further and concludes that $A(Z)= kZ-\ell$ and $B(Z)= mZ-n$ where $k,\ell,m,n$ are constants.
So which one is correct? - All the relationships are slightly different from one another.
$\dots$ how can we derive (2) from (1)? what formula is used to convert (1) to (2)? are questions I cannot answer because of the subtle but significant different forms of the relationships.
