### Half life

N(t) = N₀(1/2)^(t/th)
or N(t) = N₀2^(–t/th)
N₀ is initial amount
N(t) is the amount remaining after time t
th is the half life time, ie, time for half the amount to decay
solving for t or th, t/th = –log₂(N/N₀)

Half life can also be defined as the time for the activity to fall to half it's original rate. In that case:
A(t) = A₀(1/2)^(t/th)
or A(t) = A₀2^(–t/th)
t/th = –log₂(A/A₀)

Equivalent formulas
N(t) = N₀(e)^(–t/τ)
τ is a positive number called the mean lifetime of the decaying quantity,
th = τ ln 2 = (1/λ) ln 2
τ = th / ln2

N(t) = N₀(e)^(–λτ)
λ is a positive number called the decay constant of the decaying quantity.
λ = 1/τ = ln2/th

Some quantities decay by two exponential-decay processes simultaneously. In this case, the actual half-life T1/2 can be related to the half-lives t1 and t2 that the quantity would have if each of the decay processes acted in isolation:
1/T₁₂ = (1/t₁) + (1/t₂)

1 rem = 0.01 Sv = 10 mSv
1 Gy = 1 Sv = 1 J / kg

Relating half life to activity
Total activity— A, is number of decays per unit time of a radioactive sample.
N(t) = N₀(1/2)^(t/th)
dN/dt = N₀(1/th)(1/2)^t(ln(1/2))
dN/dt = N₀(1/th)(1/2)^t(–0.6931472)
put the HL in seconds, the activity, number of decays per second is
A = dN = (ln(1/2))(1/th)(N₀)

## Planck's law

I(v,T) = 2 hv³ / (c²}(e^(hv/kt)–1)

where
I(ν,T) is the energy per unit time (or the power) radiated per unit area of emitting surface in the normal direction per unit solid angle per unit frequency by a black body at temperature T;
h is the Planck constant
c is the speed of light in a vacuum
k is the Boltzmann constant
ν is the frequency of the electromagnetic radiation
T is the absolute temperature of the body

1. alpha particles, which are basically helium nuclei. The are slow and can be stopped by a piece of aluminum foil.

2. beta particles, which are high speed electrons. They have some penetrating particle.

3. gamma rays, which are basically high energy X-rays. These are the most dangerous.

## Photon energy

Energy in one photon
E is photon energy in J
h is the Planck constant, 6.62607e−34 Js
c is the speed of light in vacuum in m/s
λ is the photon's wavelength in m
f is the photon's frequency

E = hc/λ
or since λf = c
E = hc/(c/f) = hf
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