Question

Specific heat ($C_v$) of a material was found to depend on temperature as shown below. Which of the following statement(s) is/are true?

• A. The material is metallic
• B. The material is insulating
• C. The material is three dimensional
• D. The material is one dimensional

Hint:

- A straight line in $\frac{C_v}{T}$ vs $T^2$ with finite intercept indicates a metallic system.
- The $T^3$ phonon law confirms the system is three-dimensional.

Answer 1

The Correct Option is A, C

Step 1: Recall specific heat dependence in metals.
At low temperatures, the heat capacity of a metal has two contributions: \[ C_v = \gamma T + \beta T^3 \] where - $\gamma T$ = electronic contribution (linear in $T$), - $\beta T^3$ = lattice contribution (phonons, Debye $T^3$ law).
Step 2: Rearrange to given form.
Dividing by $T$: \[ \frac{C_v}{T} = \gamma + \beta T^2 \] This is a straight line in $\frac{C_v}{T}$ vs $T^2$ plot, with - Intercept = $\gamma$ (electronic term), - Slope = $\beta$ (phonon contribution).
Step 3: Interpret the graph.
- The graph matches exactly the expected linear form. - This confirms the presence of both electronic and lattice contributions $\Rightarrow$ material is a **metal**.
Step 4: Dimensionality check.
- Debye’s $T^3$ law is valid for **three-dimensional solids**. - For 1D or 2D materials, phonon contributions follow different power laws ($T$ for 1D, $T^2$ for 2D). - Since the data shows $T^3$ dependence $\Rightarrow$ it must be **3D**.

Step 5: Evaluate options.
(A) Metallic → Correct.
(B) Insulating → Wrong, because intercept $\gamma \neq 0$ shows metallic behavior.
(C) Three dimensional → Correct (phonon $T^3$ law).
(D) One dimensional → Wrong. Final Answer: \[ \boxed{\text{(A) and (C)}} \]