Resolution in electron microscopy

The resolution of an electron microscope is determined by the wavelength of the electrons being used. The shorter the wavelength, the higher the resolution and the better the microscope can distinguish between closely spaced objects. The wavelength of electrons can be calculated using the de Broglie equation:

λ = h / p

Where λ is the wavelength, h is Planck's constant (6.63 x 10^-34 J s), and p is the momentum of the electrons.

Calculating the electron's momentum

The momentum of an electron can be calculated using the equation:

p = mv

Where p is the momentum, m is the mass of an electron (9.11 x 10^-31 kg), and v is the velocity of the electrons.

Calculating the velocity of the electrons

The velocity of the electrons can be found using the equation for kinetic energy:

eV = 1/2 mv^2

Where eV is the energy of the electrons, m is the mass of an electron, and v is the velocity.

Calculating the energy of the electrons

The energy of the electrons can be calculated using the equation for potential energy:

eV = qV

Where eV is the energy of the electrons, q is the charge of an electron (1.6 x 10^-19 C), and V is the potential difference through which the electrons are accelerated.

Substituting the values and solving for velocity

Substituting the given values into the equation, we get:

eV = 1/2 mv^2

(1.6 x 10^-19 C)(200,000 V) = 1/2 (9.11 x 10^-31 kg)v^2

Solving for v, we find:

v^2 = (2)(1.6 x 10^-19 C)(200,000 V) / (9.11 x 10^-31 kg)

v^2 ≈ 6.63 x 10^15 m^2/s^2

v ≈ 2.58 x 10^8 m/s

Calculating the wavelength and resolution

Now that we have the velocity of the electrons, we can calculate the wavelength using the de Broglie equation:

λ = h / p

λ = (6.63 x 10^-34 J s) / (9.11 x 10^-31 kg)(2.58 x 10^8 m/s)

λ ≈ 2.52 x 10^-12 m

Finally, to find the resolution, we can use the equation:

Resolution = 0.61λ / NA

Where NA is the numerical aperture of the microscope. The numerical aperture depends on the design of the microscope and typically ranges from 0.1 to 1.0.

Assuming a numerical aperture of 1.0, the resolution can be calculated as:

Resolution ≈ 0.61(2.52 x 10^-12 m) / 1.0

Resolution ≈ 1.54 x 10^-12 m

Converting the resolution to nanometers (1 m = 10^9 nm), we get:

Resolution ≈ 1.54 x 10^-3 nm

Rounding to two decimal places, the best possible resolution of the microscope