β-decay spectrum of Iodine-128

β decay spectrum and half-life of

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$d\mathrm{\lambda }\left(\mathrm{p}\right)=\frac{g^2{\left|M_{\mathit{fi}}\right|}^2}{2{\pi }^3{\mathrm{\hslash }}^7{c}^3}{(Q–T_e)}^2\times F(Z^{‘},p)p^2\mathit{dp}$

                                      (6)

Where F(Z’, p) is the Fermi function that accounts for Coulomb force on the electron as it travels away from the nucleus, (Q-T_e)² p² is a statistical factor coming from the density of states,

$M_{\mathit{fi}}$

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is the nuclear matrix element and is related to

$I_{\mathit{fi}}$

by a factor of g/V. This equation is then simplified and is the basis of a Fermi-Kurie plot.

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7.1 Fermi-Kurie Analysis

A Fermi-Kurie plot, developed by Franz N. D. Kurie, is a common method used for studying beta decay. They are made by plotting kinetic energy, T_e, against

$\sqrt{N\left(p\right)/F\left(Z^{‘},p\right)p^2}$

( from (6)). This simplification assumes the nuclear matrix element doesn’t influence the spectrum’s shape, which is generally true for allowed decays, and so it is left out. The other terms are constant and so included in the constant of proportionality.

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In this experiment it is the kinetic energy of the electron, not the momentum, that is measured and so it’s the number of counts in the kinetic energy bins that are received. Therefore, a relation between kinetic energy and momentum was found by equating E_e² = p²c² + m_e²c⁴ and E_e = T_e  +m_ec² to give

$p=\frac{\sqrt{T_e^2+2T_e{m}_e{c}^2}}{c}$

                                                                 (7)

It is found that

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$\frac{\mathrm{d}{\mathrm{T}}_{\mathrm{e}}}{\mathrm{dp}}=\frac{\sqrt{T_e^2+2T_e{m}_e{c}^2}\mathit{ c}}{(T_e+m_e{c}^2)}$

,                                                             (8)

to give a plot of

x axis = T_e

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y axis =

$\sqrt{\frac{N\left(T_e\right)}{F(Z^{‘},p)p^2}\times \frac{\sqrt{T_e^2+2T_e{m}_e{c}^2}\mathit{ c}}{(T_e+m_e{c}^2)}}$

.    (9) 

The Fermi values are found from a published table [10], but are in fact of the form

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$f$

(Z’, ) and for Iodine-128 this is

$f$

(54, ). Here =

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$\frac{p}{511\mathit{keV}/c}$

for the electron. 

$f\left(Z^{‘},\mathrm{\eta }\right)=\frac{{\mathrm{\eta }}^{2}F(Z^{‘},\mathrm{\eta })}{\phi \left(Z^{‘}\right)}$

The scaling factor,

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$\phi \left(Z^{‘}\right)$

, however, is not dependent on momentum and so does not change the shape of the Fermi function. It, therefore, can be absorbed in the constant of proportionality giving ( ′, ) ≈ ² ( ′, ). [9] ( ′, ) can then replace

$F(Z^{‘},p)p^2$

in equation (9)

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8 Experimental method

The Iodine-127 in the detector undergoes neutron bombardment by submersion in a neutron bath. Supply voltage of the detector is raised slowly to 550V with course gain 10, fine 45. Job files are utilised in MAESTRO to repeat the scan so many spectra are made during a fixed time of the decay. The method for making a calibration plot is repeated after these scans, using strong calibration sources e.g. ⁶⁰Co and ⁴⁴Ti. The table of data from these scans are then moved to Excel where the subsequent analysis is completed. Electron count for each momentum bin is the product of the count number and the value of the momentum bin. Fermi data from ’Tables for the Analysis of Beta Spectra’[10, C Sawyer] is plotted against the corresponding momentum bins. A Fermi-Kurie plot is made as described in the theory section of this report. Multiple plots are made for the different data sets and values for half-life and end-point energies are found. It is known there should be 3 endpoint energies for this decay and said endpoints can be found by fitting linear functions to ranges where the slope varies. These regions must not include areas where the line is non-linear, i.e. too close to the predicted endpoint energies or within the low energy range. The endpoints then calculated from where the fit lines meet.

Half-life is found by

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9 Results

Fermi data is plotted against the corresponding momentum bins.