To understand why the correct answer is option 'D', let's analyze the concept of lattice vacancies and their relation to temperature.

Introduction to Lattice Vacancies:

In a crystal lattice, vacancies are the empty spaces where an atom is missing. These vacancies can occur due to various reasons such as thermal fluctuations or crystal defects. The creation of a lattice vacancy requires an input of energy, known as the vacancy formation energy.

Given Information:

According to the question, the energy required to create a lattice vacancy in the crystal is 1 eV. Additionally, we are provided with the temperatures at which the crystal is at equilibrium, 1200 K and 300 K.

Equilibrium at 300 K:

At equilibrium, the number of vacancies created is balanced by the number of vacancies annihilated. At 300 K, the crystal is at a lower temperature, which implies that the thermal energy available is less. Therefore, the number of vacancies created is comparatively smaller.

Calculating the Ratio of Number Densities:

To calculate the ratio of number densities of vacancies at 1200 K and 300 K, we need to consider the Boltzmann distribution. According to this distribution, the probability of finding an atom in a particular energy state is given by the exponential of the negative energy divided by the product of Boltzmann's constant and the temperature.

Let's denote the ratio of number densities as n(1200 K)/n(300 K). Using the Boltzmann distribution, we have:

n(1200 K)/n(300 K) = exp[(E_vacancy)/(k*T_1200)] / exp[(E_vacancy)/(k*T_300)]

where:
- E_vacancy is the vacancy formation energy (1 eV)
- k is Boltzmann's constant (8.617333262145 x 10^-5 eV/K)
- T_1200 is the temperature at 1200 K
- T_300 is the temperature at 300 K

Simplifying the Equation:

Substituting the given values into the equation, we get:

n(1200 K)/n(300 K) = exp[(1 eV)/(8.617333262145 x 10^-5 eV/K * 1200 K)] / exp[(1 eV)/(8.617333262145 x 10^-5 eV/K * 300 K)]

n(1200 K)/n(300 K) = exp[1/(8.617333262145 x 10^-5 * 1200)] / exp[1/(8.617333262145 x 10^-5 * 300)]

n(1200 K)/n(300 K) = exp(1/103.408) / exp(1/25.808)

n(1200 K)/n(300 K) ≈ exp(0.00967) / exp(0.03876)

n(1200 K)/n(300 K) ≈ 1.00972 / 1.0395

n(1200 K)/n(300 K) ≈ 0.9703

Approximating the Result:

The ratio of number densities is approximately 0.9703. Since the value is less than 1, it means that at 1200 K, there are fewer vacancies compared to 300 K.

Converting the Ratio to Exponential Form:

To convert the ratio