Chemical analysis of a silicon crystal reveals gallium at a level of 3.091×10−4 atomic percent. Assuming that the concentration of thermally excited charge carriers from the Si matrix is negligible, what is the density of free charge carriers (free carriers/cm3) in this Si crystal?

To calculate the density of free charge carriers in the silicon crystal, we first need to determine the number of gallium atoms present in the crystal.

1. Calculate the total number of silicon atoms:
- Let's assume the crystal contains 1 mole of silicon atoms.
- The Avogadro's number, N_A, is approximately 6.022 × 10^23 mol^−1.
- Therefore, the number of silicon atoms per mole is N_A.

2. Convert the atomic percent of gallium to the concentration of gallium atoms:
- The atomic percent of gallium is 3.091 × 10^−4.
- Multiply this by the number of silicon atoms per mole (N_A) to get the number of gallium atoms per mole.

3. Calculate the number of gallium atoms per cubic centimeter (cm^3):
- Divide the number of gallium atoms per mole by the density of the silicon crystal.
- The density of silicon is approximately 2.33 g/cm^3.

4. Convert the number of gallium atoms per cm^3 to the density of free charge carriers:
- Each gallium atom contributes one free charge carrier in the crystal.
- Therefore, the number of gallium atoms per cm^3 is equal to the density of free charge carriers in the crystal.

Let's calculate the density of free charge carriers using these steps:

Step 1:
Number of silicon atoms per mole = Avogadro's number (N_A)

Step 2:
Number of gallium atoms per mole = (3.091 × 10^−4) × (N_A)

Step 3:
Number of gallium atoms per cm^3 = (Number of gallium atoms per mole) / (density of silicon crystal)

Step 4:
Density of free charge carriers = Number of gallium atoms per cm^3

Now, let's plug in the values and calculate the density of free charge carriers.