One form of nuclear radiation, beta decay, occurs when a neutron changes into a proton,

an electron, and a neutral particle called an antineutrino: n $$ p$++$ e$+$e$,$ where $$e

is the symbol for an antineutrino. When this change happens to a neutron within the

nucleus of an atom, the proton remains behind in the nucleus while the electron and

neutrino are ejected from the nucleus. The ejected electron is called a beta particle.

One nucleus that exhibits beta decay is the isotope of hydrogen $3$H$,$ called tritium,

whose nucleus consists of one proton $($making it hydrogen$)$ and two neutrons $($giving

tritium an atomic mass m $=3$u$).$ Tritium is radioactive, and it decays to helium:

$3$H $3$He $+$ e$+$e$.$

$($a$)$ Is charge conserved in the beta decay process? Explain.

$($b$)$ Why is the final product a helium atom? Explain.

$($c$)$ The nuclei of both $3$H and $3$He have radii of $1\mathrm{.}51015$ m$.$ With what minimum

speed must the electron be ejected if it is to escape from the nucleus and not fall

back?

$2$

$7.$ The sun is powered by fusion, with four protons fusing together to form a helium

nucleus $($two of the protons turn into neutrons$)$ and, in the process, releasing a large

amount of thermal energy. The process happens in several steps, not all at once. In one

step, two protons fuse together, with one proton then becoming a neutron, to form the

$$heavy hydrogen$$ isotope deuterium $(2$H$).$ A proton is essentially a $2\mathrm{.}4$ fm$-$diameter

sphere of charge, and fusion occurs only if two protons come into contact with each

other. This requires extraordinarily high temperatures due to the strong repulsion

between protons. Recall that the average kinetic energy of a gas particle is $3$

$2$ kB T $.$

$($a$)$ Suppose two protons, each with exactly the average kinetic energy, have a head$-$on

collision. What is the minimum temperature for fusion to occur?

$($b$)$ Your answer to part a is much hotter than the $15$ million K in the core of the sun.

If the temperature were as high as you calculated, every proton in the sun would

fuse almost instantly, and the sun would explode. For the sun to last for billions of

years, fusion can only occur in collisions between two protons with kinetic energies

much higher than average. Only a very tiny fraction of the protons have enough

kinetic energy to fuse when the collide, but that fraction is enough to keep the sun

going. Suppose two protons with the same energy collide head$-$on and just barely

manage to fuse. By what factor does each proton$$s energy exceed the average

kinetic energy at $15$ million K$?$