Thermonuclear Fusion: The Basic Process — Question 3

Question 3

A stationary target wastes part of the beam energy in bulk motion

A deuteron of mass $2\,\mathrm u$ and lab kinetic energy 100 keV strikes stationary tritium of mass $3\,\mathrm u$. Treat both nonrelativistically.

1. Find the center-of-mass collision energy and kinetic energy of center-of-mass motion.

2. Compare with two oppositely directed beams having equal momentum magnitudes and total lab kinetic energy 100 keV.

Question 3 - Solution

A stationary target wastes part of the beam energy in bulk motion

(a) Method and calculation

$$V_{\mathrm{cm}}=\frac{m_Dv}{m_D+m_T},\quad K_{\mathrm{cm\,motion}}=\frac{m_D}{m_D+m_T}K_{\mathrm{lab}}.$$$$\boxed{E_{\mathrm{rel}}=\frac35(100)=60\,\mathrm{keV},\quad K_{\mathrm{cm\,motion}}=40\,\mathrm{keV}}.$$

(b) Interpretation and further calculation

$$\mathbf p_D+\mathbf p_T=0\Rightarrow V_{\mathrm{cm}}=0.$$$$\boxed{E_{\mathrm{rel,opposed}}=100\,\mathrm{keV}}\quad\text{for equal and opposite momenta}.$$

Physical check

Equal momentum does not mean equal beam energies for unequal masses. Reaction rates depend on relative collision energy rather than total laboratory energy alone.