This problem develops perturbation theory to the helium atom,)we should evaluate the integral a basic idea on evaluating two-electron integrals. in applying first-order cazez if drdr, yi,(ri,((r,) 4πε e) i2 x. where 1/2 z3 exp(-zr, / a,) = (1)"a and z 2 for the helium atom. let us do this integral. (a) let ri and r2 be the radius vectors of electron 1 and 2, respectively, and let 0be the angle between these two vectors. now this is generally not the e of spherical coordinates, but if we choose one of the radius vectors, say ri to be the z axis, then the two es are the same. using the law of cosines, -2cos e show that e) becomes z6 872 dre α,π2 -2 zr, la 7 dre la2 e0) 6 jo 4 te desin 1/2 (+2 cos , cos over d.) (hint: there are also integrals (b) letting x = cos 0, show that the integral over 0is cos dx d0sin 1/2 °(r+-2nr, cos0)2 (c) substituting this result into e), show that e2 1626 drze -22r lao dr, e 2 + -2z dre21/ e) j0 6 1 -2z5/a0 e2 4z3 dre 4πε, αο +- -22/a2 2 5 -z 4πε,α 2/a214

answer: the correct answer is 1.

explanation: there are various process in which a radioactive nuclei decays:

1) alpha decay: in this process, a heavier nuclei decays into lighter nuclei by releasing alpha particle. the mass number is reduced by 4 units.

2) electron capture: in this decay process, a parent nuclei absorbs an electron and gets converted into a neutron. simply, a proton and an electron combines together to form a neutron. mass number does not change in this process.

3) gamma ray emission: it is a decay process in which an unstable nuclei gives excess energy by a spontaneous electromagnetic process. this decay releases . this process does not change the mass number.

4) positron emission: it is a type of decay process, in which a proton gets converted to neutron and an electron neutrino. this is also known as -decay. in this the mass number remains same.

from the above information, we see that only alpha-decay is producing a change in the mass number of the nucleus.

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