A silicon ingot, which should contain 10^16 boron atoms/cm3, is to be grown by the Czochralski technique. What

concentration of boron atoms should be in the melt to give the required concentration in the ingot ? If the initial load of Si in the crucible is 60 kg, how many grams of boron (atomic weight 10.8) should be added ? The density of molten silicon is 2.53 gm/cm3 and segregation co-eff ko for boron is 0.8.

To determine the concentration of boron atoms in the melt, we first need to calculate the concentration of boron atoms in the ingot.

Given:
Concentration of boron atoms in the ingot (C_ingot) = 10^16 boron atoms/cm^3
Initial load of silicon in the crucible (Si_load) = 60 kg
Atomic weight of boron (B) = 10.8 g/mol
Density of molten silicon (Si_density) = 2.53 g/cm^3
Segregation coefficient for boron (ko) = 0.8

Step 1: Calculate the volume of silicon in the crucible.
Volume of silicon in the crucible (V_crucible) = Si_load / Si_density
V_crucible = 60,000 g / 2.53 g/cm^3
V_crucible = 23,715.45 cm^3

Step 2: Calculate the number of boron atoms in the ingot.
Number of boron atoms in the ingot (N_ingot) = C_ingot * V_crucible
N_ingot = (10^16 atoms/cm^3) * (23,715.45 cm^3)
N_ingot = 2.371545 × 10^20

Step 3: Calculate the number of moles of boron needed.
Number of moles of boron (n_B) = N_ingot / ko
n_B = (2.371545 × 10^20) / 0.8
n_B = 2.9644325 × 10^20

Step 4: Calculate the mass of boron needed.
Mass of boron (m_B) = n_B * atomic weight of boron
m_B = (2.9644325 × 10^20) * 10.8 g/mol
m_B = 3.2040759 × 10^21 g

Therefore, approximately 3.204 × 10^21 grams of boron should be added.

To determine the concentration of boron atoms in the melt, we can use the segregation coefficient (ko) formula:

Concentration in ingot = ko × Concentration in melt
10^16 atoms/cm3 = 0.8 × Concentration in melt
Concentration in melt = (10^16 atoms/cm3) / 0.8
Concentration in melt = 1.25 × 10^16 atoms/cm3

Now we need to find the number of moles of silicon and boron. First, let's find the volume of molten silicon:

60 kg = 60000 g
Volume = mass / density
Volume = 60000 g / 2.53 g/cm3
Volume = 23715.42 cm3

Now, let's find the number of silicon atoms and their molar quantity:

Silicon atoms volume = Volume × density
Silicon atoms = 23715.42 cm3 × 5.0 × 10^22 atoms/cm3(Avogadro constant)
Silicon atoms = 1.1857 × 10^28 atoms

Number of Moles (silicon) = Number of atoms / (Avogadro constant)
Number of Moles (silicon) = 1.1857 × 10^28 atoms / (6.022 × 10^23 atoms/mol)
Number of Moles (silicon) = 196,938.19 moles

Now we find the amount of boron atoms in the molten silicon:
Boron atoms = Volume × Concentration in melt
Boron atoms = 23715.42 cm3 × 1.25 × 10^16 atoms/cm3
Boron atoms = 2.9644 × 10^21 atoms

Now, let's find the molar quantity of boron:
Number of Moles (boron) = Number of atoms / (Avogadro constant)
Number of Moles (boron) = 2.9644 × 10^21 atoms / (6.022 × 10^23 atoms/mol)
Number of Moles (boron) = 4,920.96 moles

Now we have the molar quantities of silicon and boron, and we can find the amount of boron in grams that should be added:

Grams (boron) = Number of moles × atomic weight
Grams (boron) = 4,920.96 moles × 10.8 g/mol
Grams (boron) = 53,146.36 g

Therefore, to achieve the desired concentration of boron atoms in the silicon ingot, 53,146.36 grams of boron should be added to the initial load of 60 kg silicon in the crucible.

To determine the concentration of boron atoms required in the melt, we need to consider the segregation coefficient and the desired boron concentration in the silicon ingot.

The segregation coefficient (ko) accounts for the difference in concentration between the solid ingot and the molten silicon. In this scenario, the segregation coefficient for boron (ko) is given as 0.8.

The desired boron concentration in the silicon ingot is 10^16 boron atoms/cm^3.

To find the concentration of boron atoms in the melt, we divide the desired concentration by the segregation coefficient:

Concentration in Melt = Desired concentration / Segregation coefficient
Concentration in Melt = 10^16 boron atoms/cm^3 / 0.8

Now, let's calculate the concentration in the melt:

Concentration in Melt = 1.25 x 10^16 boron atoms/cm^3

To determine the mass of boron to be added, we need to consider the density of molten silicon and the initial load of silicon in the crucible.

The density of molten silicon is given as 2.53 g/cm^3.

The initial load of silicon in the crucible is 60 kg, which is equivalent to 60,000 g of silicon.

To calculate the mass of boron to be added, we multiply the concentration in the melt by the volume of silicon in the crucible:

Mass of Boron = Concentration in Melt * Volume of Silicon
Volume of Silicon = Mass of Silicon / Density of Molten Silicon

Volume of Silicon = 60,000 g / 2.53 g/cm^3
Volume of Silicon = 23,715.41 cm^3

Now we can calculate the mass of boron to be added:

Mass of Boron = (1.25 x 10^16 boron atoms/cm^3) * (23,715.41 cm^3)
Mass of Boron = 2.967 x 10^17 boron atoms

The atomic weight of boron is given as 10.8 g/mol. Since the molar mass of boron is equal to 1 mole of boron atoms, the mass of boron to be added can be calculated by dividing the number of boron atoms by Avogadro's number (6.022 x 10^23 atoms/mol):

Mass of Boron = (2.967 x 10^17 boron atoms) / (6.022 x 10^23 atoms/mol) * (10.8 g/mol)

The final calculation will give us the mass of boron to be added in grams.