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Proton–proton chain - Wikipedia

The proton–proton chain, also commonly referred to as the p-p chain, is one of two known sets ... The proton-proton chain is, like a decay chain, a series of reactions. ... will annihilate with an electron from the environment into two gamma rays. ... is the same as the PEP reaction, see below) has a Q value (released energy) of ...

For more information, see Proton–proton chain - Wikipedia

Energy of photon \Delta E = h \upsilon , h - Planck's constant, \upsilon - frequency

Step 1: Calculating the frequency of the gamma particle

Form the figure

\gamma _1 decays from energy level E_1 \text{ to } E_3   

Energy difference \Delta E = E_1 - E_2 = 1.088 MeV - 0 = 1.088 MeV = 1.088 * 10^{6} * 1.6 * 10^{-19}        \because 1 eV = 1.6 * 10^{-19} J

\Delta E_1 = 1.7408 * 10^{-13} joules

Frequency of \gamma_1, \upsilon_1 = \frac{\Delta E}{h} = \frac{ 1.7408 * 10^{-13}}{6.626 * 10^{-34}}

\upsilon_1 = 2.63 *10^{20} Hz

\gamma _2 decays from energy level E_2 \text{ to } E_3

Energy difference \Delta E = E_2 - E_3 = 0.412 MeV - 0 = 0.412 MeV = 0.412 * 10^{6} * 1.6 * 10^{-19}

E_2 = 6.592 * 10^{-14} MeV

Frequency of \gamma_2, \upsilon_2 = \frac{\Delta E}{h} = \frac{6.592 * 10^{-14}}{6.626 * 10^{-34}}

\upsilon_2 = 9.987 * 10^{19}   Hz

\gamma _3 decays from energy level E_1 \text{ to } E_2

Energy difference \Delta E = E_1 - E_2 =   1.088 MeV - 0.412 MeV = 0.676 MeV = 0.676 * 10^{6} * 1.6 * 10^{-19}

\Delta E = 1.0816 * 10^{-13}

\Delta E = 1.0816 * 10^{-13}

Frequency of \gamma_3, \upsilon_3 = \frac{\Delta E}{h} = \frac{1.0816 * 10^{-13}}{6.626 * 10^{-34}}

\upsilon_3 = 1.632 * 10^{20} Hz

Step 2: Determine the maximum kinetic energy of the \beta particle  

Given that

m(^{198}_{78} Au) = 197.968233 u

m(^{196}_{80} Hg) = 197.966760 u

Energy E = \Delta m * c^2 where, \Delta m - mass defect

Mass defect [math]\Delta m=\left[m(_{78}^{198}Au)-m(_{80}^{196}Hg)\right][/math]

[math] E_a = [197.968233 - 197.966760] c^2 * 931.5 MeV/c^2 [/math]

E_a = 0.001473 *931.5 MeV

E_a = 1.372 MeV

\beta_1^- decays from E_a \text{ to } E_1

Maximum kinetic energy of the \beta_1^-, K.E =E_a - E_1

K.E = 1.372 - 1.088 MeV = 0.284 MeV

\beta_2^- decays from E_a \text{ to } E_2

Maximum kinetic energy of the \beta_2^-, K.E =E_a - E_2

K.E = 1.372 - 0.412 MeV = 0.96 MeV.