Frame Of Reference
In the study of scattering theory, nuclear reactions, and collision processes, the distinction between the Laboratory (Lab) reference frame and the Centre-of-Mass (CM) reference frame plays a central role. These two frames provide different perspectives for describing the motion, momentum transfer, and angular distribution of interacting particles. Since observations in an experiment are made in the laboratory frame, but theoretical simplicity often arises in the centre-of-mass frame, understanding the transformation between these two coordinate systems becomes essential.
The Laboratory frame is the frame in which the experimental apparatus is at rest. In scattering experiments, a projectile particle (like an electron, neutron, or α-particle) is accelerated toward a stationary target. Measurements such as scattering angle, energy of outgoing particles, and cross-sections are typically recorded in this frame. The dynamics here are often complicated due to the asymmetry introduced by one particle being initially at rest.
The Centre-of-Mass frame, on the other hand, is the frame in which the total momentum of the entire system is zero. Here, both particles move toward one another with equal and opposite momenta before collision. This symmetry significantly simplifies calculations of scattering amplitude, angular momentum, and differential cross-sections. Quantum scattering theory, especially partial-wave analysis, is most conveniently formulated in the CM frame because conservation laws take their simplest form.
Transformation between the two frames involves relating the velocities, energies, and scattering angles. For a projectile of mass $ m_1 $ colliding with a target of mass $ m_2 $, the velocity of the centre of mass is
\[\mathbf{V}_{CM} = \frac{m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2}{m_1 + m_2}.\]In most practical cases, the target is initially at rest, leading to
\[\mathbf{V}_{CM} = \frac{m_1}{m_1 + m_2}\mathbf{v}_1.\]Energies also transform differently between the two frames. The CM kinetic energy,
\[E_{CM} = \frac{1}{2}\mu v_{\text{rel}}^2,\]depends on the reduced mass $ \mu = \frac{m_1 m_2}{m_1 + m_2} $ and the relative velocity $ v_{\text{rel}} $. The laboratory kinetic energy of the projectile is
\[E_{Lab} = \frac{1}{2}m_1 v_1^2.\]Thus,
\[E_{CM} = \frac{m_2}{m_1 + m_2} E_{Lab},\]which shows that only a fraction of the laboratory energy contributes to the actual interaction.
One of the most important aspects is the relation between scattering angles:
\[\tan\theta_{Lab} = \frac{\sin\theta_{CM}}{\cos\theta_{CM} + \frac{m_1}{m_2}}.\]For elastic scattering involving equal masses $ m_1 = m_2 $, this simplifies dramatically, and the maximum lab scattering angle becomes $ 90^\circ $. For high mass targets, the CM and lab angles become nearly identical.
Supplementary Readings
1. Centre-of-Mass Velocity
Consider two particles of masses $ m_1 $ and $ m_2 $, velocities $ \mathbf{v}_1 $ and $ \mathbf{v}_2 $. The CM velocity:
\[\mathbf{V}_{CM} = \frac{m_1\mathbf{v}_1 + m_2\mathbf{v}_2}{m_1 + m_2}.\]For a stationary target: $ \mathbf{v}_2 = 0 $,
\[\mathbf{V}_{CM} = \frac{m_1}{m_1 + m_2}\mathbf{v}_1.\]2. Relative Velocity and Reduced Mass
The relative velocity is:
\[\mathbf{v}_{rel} = \mathbf{v}_1 - \mathbf{v}_2 = \mathbf{v}_1.\]The reduced mass $ \mu $ is:
\[\mu = \frac{m_1 m_2}{m_1 + m_2}.\]3. Energy Transformation
Laboratory frame projectile energy:
\[E_{Lab} = \frac{1}{2}m_1 v_1^2.\]CM frame kinetic energy:
\[E_{CM} = \frac{1}{2}\mu v_1^2.\]Thus,
\[E_{CM} = \frac{m_2}{m_1 + m_2} E_{Lab}.\]4. Transformation of Scattering Angles
The relation between CM and Lab angles (elastic scattering):
\[\tan\theta_{Lab} = \frac{\sin\theta_{CM}}{\cos\theta_{CM} + \frac{m_1}{m_2}}.\]For $ m_1 = m_2 $:
\[\tan\theta_{Lab} = \frac{\sin\theta_{CM}}{\cos\theta_{CM} + 1}.\]For $ m_2 \gg m_1 $ (heavy target):
\[\theta_{Lab} \approx \theta_{CM}.\]5. Differential Cross-Section Transformation
Cross-section transforms as:
\[\left(\frac{d\sigma}{d\Omega}\right)_{Lab} = \left(\frac{d\sigma}{d\Omega}\right)_{CM} \left|\frac{\sin\theta_{CM}}{\sin\theta_{Lab}}\cdot \frac{d\theta_{CM}}{d\theta_{Lab}}\right|.\]This relation is essential for interpreting experimental scattering data using theoretical predictions derived in the CM frame.