Derivations

Question 1

Derive the width of the depletion zone

Answer 1

Depletion region is created on either side of the junction

NDxn = NAxp

The electric potential at any point x along an axis perpendicular to the junction is given by the Poisson equation

d^2V/dx^2 - N(x)q/ε = 0

Integrating to find V(xn) and V(xp)

Combine them as V(xn) + V(xp) = Vb

Substitute for NDxn = NAxp and rearrange for xn and for xp

W = xn + xp

Question 2

Reflection optics: x rays

Answer 2

snells law n1cosθ1 = n2cosθ2

x-rays with n < 1, θc when θ2 = 0 and n1 = 1

=> cos θc = n2

Taylor expanding θc

1 - θc^2/2 = n2

n2 = 1- δ

=> θc^2/2 = 1- δ

=> θc = sqrt(2δ)

Question 3

Zone plate focal length

Answer 3

(pm + rm) - (p0 - r0) = mλ/2

pm = sqrt(p0^2 + Rm^2) ~ p0 + Rm^2/2p0

rm = sqrt(r0^2 + Rm^2) ~ r0 + Rm^2/2r0

Substitute pm and rm into line 1

=> 1/p0 + 1/r0 ~ mλ/Rm^2 = 1/f

Question 4

Zone plate radius

Answer 4

rm = r0 + mλ/2

rm = sqrt(r0^2 + Rm^2)

Plug in and rearrange

Rm^2 = (r0 + mλ/2) - r0^2

Rm^2 = mr0λ + m^2 λ^2/4

Question 5

Resolution of an electron microscope

Answer 5

Diffraction

d1 = 1.22 λ/α

Spherical aberration

d2 = 2Cs α^3

d^2(tot) = d1^2 + d2^2

then find d(d^2(tot))/dα = 0

rearrange for α

dmin = 1.83 λ^3/4 Cs^1/4

resolution = dmin/2