If the vibration frequency of atoms in a crystaal depends on the atomic mass m, the atomic spacing r and compressibility C, then it is proportional to

Options

A. `m^(1/2)`
B. `r^(1/2)`
C. `C`
D. all the above

Answer

To solve the problem of determining how the vibrational frequency of atoms in a crystal depends on atomic mass (m), atomic spacing (r), and compressibility (C), we will use dimensional analysis. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Variables
We know that the vibrational frequency \( f \) depends on:
- Atomic mass \( m \)
- Atomic spacing \( r \)
- Compressibility \( C \)

### Step 2: Write the Relationship
We can express the frequency as:
\[ f \propto m^{\alpha} r^{\beta} C^{\gamma} \]
where \( \alpha \), \( \beta \), and \( \gamma \) are the powers to which each variable is raised.

### Step 3: Write the Dimensional Formulas
Next, we need the dimensional formulas for each variable:
- Frequency \( f \) has the dimension \( [T^{-1}] \).
- Atomic mass \( m \) has the dimension \( [M] \).
- Atomic spacing \( r \) has the dimension \( [L] \).
- Compressibility \( C \) has the dimension \( [M^{-1} L T^{-2}] \).

### Step 4: Substitute Dimensional Formulas
Substituting the dimensional formulas into our relationship gives:
\[ [T^{-1}] \propto [M]^{\alpha} [L]^{\beta} [M^{-1} L T^{-2}]^{\gamma} \]

### Step 5: Expand the Right Side
Expanding the right side:
\[ [T^{-1}] \propto [M]^{\alpha - \gamma} [L]^{\beta + \gamma} [T]^{-2\gamma} \]

### Step 6: Equate Dimensions
Now we equate the dimensions on both sides:
1. For mass \( M \):
\[ \alpha - \gamma = 0 \quad \text{(1)} \]
2. For length \( L \):
\[ \beta + \gamma = 0 \quad \text{(2)} \]
3. For time \( T \):
\[ -2\gamma = -1 \quad \text{(3)} \]

### Step 7: Solve the Equations
From equation (3):
\[ \gamma = \frac{1}{2} \]
Substituting \( \gamma \) into equation (2):
\[ \beta + \frac{1}{2} = 0 \implies \beta = -\frac{1}{2} \]
Substituting \( \gamma \) into equation (1):
\[ \alpha - \frac{1}{2} = 0 \implies \alpha = \frac{1}{2} \]

### Step 8: Write the Final Relationship
Now we can write the final relationship for the frequency:
\[ f \propto m^{\frac{1}{2}} r^{-\frac{1}{2}} C^{\frac{1}{2}} \]

### Step 9: Conclusion
Thus, the vibrational frequency \( f \) is proportional to:
\[ f \propto \frac{\sqrt{C}}{\sqrt{m} \sqrt{r}} \]

### Final Answer
The correct proportionality is:
\[ f \propto \frac{1}{\sqrt{m} \sqrt{r}} \sqrt{C} \]
This indicates that the frequency is inversely proportional to the square root of atomic mass and atomic spacing, while being directly proportional to the square root of compressibility.

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