Problem 12 - Classical Mechanics - An Undergraduate Text

Question

Symmetric V-shaped molecule Figure \(15.7\) shows the symmetric V-shaped triatomic molecule \(\mathrm{X} \mathrm{Y}_{2}\); the \(\mathrm{X}-\mathrm{Y}\) bonds are represented by springs of strength \(k\), while the \(\mathrm{Y}-\mathrm{Y}\) bond is represented by a spring of strength \(\in k\). Common examples of such molecules include water, hydrogen sulphide, sulphur dioxide and nitrogen dioxide; the apex angle \(2 \alpha\) is typically between \(90^{\circ}\) and \(120^{\circ}\). In planar motion, the molecule has six degrees of freedom of which three are rigid body motions; there are therefore three vibrational modes. It is best to exploit the reflective symmetry of the molecule and solve separately for the symmetric and antisymmetric modes. Figure \(15.7\) (left) shows a symmetric motion while (right) shows an antisymmetric motion; the displacements \(X, Y, x, y\) are measured from the equilibrium position. Show that there is one antisymmetric mode whose frequency \(\omega_{3}\) is given by $$ \omega_{3}^{2}=\frac{k}{m M}\left(M+2 m \sin ^{2} \alpha\right) $$ and show that the frequencies of the symmetric modes satisfy the equation $$ \mu^{2}-\left(1+2 \gamma \cos ^{2} \alpha+2 \epsilon\right) \mu+2 \epsilon \cos ^{2} \alpha(1+2 \gamma)=0 $$ where \(\mu=m \omega^{2} / k\) and \(\gamma=m / M\). Find the three vibrational frequencies for the special case in which \(M=2 m, \alpha=60^{\circ}\) and \(\epsilon=1 / 2\).

Step-by-Step Solution

Answer

The solution involves firstly understanding both equations given in the problem, especially the quadratic equation concerning the symmetric modes frequencies. The three vibrational frequencies can be found by substituting given values \(M=2 m\), \(\alpha=60^{\circ}\), \(\epsilon=1 / 2\) into the derived equations in previous steps.

Step 1: Find \(\omega_{3}^{2}\)

The exercise provides the equation for the antisymmetric mode frequency as \(\omega_{3}^{2}=\frac{k}{m M}\left(M+2 m \sin ^{2} \alpha\right)\). This equation does not require any deformation, as it was directly given in the problem statement.

Step 2: Analyze \(\mu^2\) equation

The equation \(\mu^{2}-\left(1+2 \gamma \cos ^{2} \alpha+2 \epsilon\right) \mu+2 \epsilon \cos ^{2} \alpha(1+2 \gamma)=0\) is quadratic in nature with \(\mu\) as the target variable, where \(\mu=m \omega^{2} / k\) and \(\gamma=m / M\). Solving this type of quadratic equation, we usually apply solutions for \(\mu = \frac{-b ± \sqrt{b^{2} - 4ac}}{2a}\). In this case, \(a = 1\), \(b = -\left(1+2 \gamma \cos ^{2} \alpha+2 \epsilon\right)\), and \(c = 2 \epsilon \cos ^{2} \alpha(1+2 \gamma)\). Solving the quadratic equation will provide us the solutions for \(\mu\) which corresponds to the two symmetric mode frequencies.

Step 3: Find vibrational frequencies for special case

We are given that \(M=2 m\), \(\alpha=60^{\circ}\), \(\epsilon=1 / 2\). We apply these values into the equations obtained from the previous steps. This will simplify the expressions and provide concrete numerical values for the vibrational frequencies.

Key Concepts

Symmetric V-shaped Molecule

In the study of vibrational modes, the symmetric V-shaped molecule, commonly represented as \( \text{XY}_2 \), plays an interesting role due to its unique structure. These molecules, like water \( (\text{H}_2\text{O}) \) and sulfur dioxide \( (\text{SO}_2) \), are often viewed as having a central atom bonded to two other atoms in a V shape. This configuration is crucial because it determines the molecule's vibrational modes.
The V shape can be visualized with two X-Y bonds, represented as springs, and a Y-Y bond. The bond stiffness is crucial for determining vibrational frequencies. Understanding this shape helps us know how the molecule vibrates when energy is applied, with movements categorized into symmetric and antisymmetric modes.

• Symmetric modes: Both branches of the 'V' move together.
• Antisymmetric modes: Branches move in opposite directions.

This symmetry has big implications for solving related mathematical models of these molecules.

Degrees of Freedom

Degrees of freedom in a symmetric V-shaped molecule refer to the different independent movements possible within the molecule. A triatomic molecule has a total of six degrees of freedom:

• Three translational: Movement in three-dimensional space.
• Three rotational/vibrational: Define how the molecule vibrates and rotates.

However, since three are rigid body motions that don't lead to vibration, we're left with three vibrational modes in this context.
Understanding degrees of freedom is crucial as it allows scientists to predict how a molecule will behave in different physical environments, influencing its chemical properties and reactions.

Quadratic Equation Solutions

Solving quadratic equations is a fundamental skill in finding vibrational frequencies for symmetric V-shaped molecules. The equation given in the exercise \[ \mu^{2} - \left(1 + 2\gamma \cos^{2} \alpha + 2 \epsilon\right) \mu + 2\epsilon \cos^{2} \alpha(1 + 2\gamma) = 0 \] can be broken down using the quadratic formula:
\[ \mu = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]
This formula helps find solutions for \( \mu \), which then are used to calculate the vibrational frequencies. Here, different constants like \( a \), \( b \), and \( c \) are determined by the properties of the molecule such as angles and atomic masses.
Solving these provides insight into the symmetric vibrational modes of the molecule.

Reflective Symmetry

Reflective symmetry in the context of symmetric V-shaped molecules refers to the symmetry about a central axis. This means that if you imagine folding the molecule along this axis, both sides would match perfectly.
This symmetry makes solving for vibrational modes easier, as we can separately calculate symmetric and antisymmetric vibrations.
In practical terms, using reflective symmetry allows us to break down complex problems into smaller parts, analyzing how each component behaves relative to the whole. It’s this simplification that aids in solving the mathematical models related to vibrational studies.