Phys191 Lecture Notes: Nuclear Decay and X-Rays

Nuclear decay is probabilistic.
Cannot predict when an individual nucleus will decay.
More nuclei lead to more decay.
Decay rate is proportional to the number of nuclei.
- $\frac{\Delta N}{\Delta t} = -\lambda N$
- $\lambda$ is the decay constant (units of per second).
The negative sign indicates the number of nuclei is decreasing.

If $y = 10^x$ , then $\log_{10}(y) = x$ .
Euler's number (e) is an irrational number.
- The value of the exponential curve is equal to the gradient of that curve.
If $y = e^x$ , then $\ln(y) = x$ (natural log).

The equation $\frac{dN}{dt} = -\lambda N$ defines exponential decay.
- Analogous to exponential growth, but in reverse.
Solution to the equation:
- $N = N_0 e^{-\lambda t}$
- $N_0$ is the original number of nuclei.
- $\lambda$ is the decay constant.
Larger $\lambda$ means faster decay.

Activity (A) is the decay constant times the number of nuclei.
- $A = \lambda N$
Activity also decays exponentially.
- $A = A_0 e^{-\lambda t}$
- $A_0$ is the original activity.

Attenuation through a material is also a probabilistic process.
Example: 90% of radiation is stopped by 1 cm of lead.
- After 2 cm, 99% is stopped.
- After 3 cm, 99.9% is stopped, and so on.
Attenuation also exhibits exponential decay.

SI unit of activity is the Becquerel (Bq).
- 1 Bq = 1 decay per second.
A more practical unit is the Curie (Ci).
- 1 Ci = $3.7 \times 10^{10}$ decays per second.
- Based on the decay rate of 1 gram of radium.
- The modern value is $3.61 \times 10^{10}$ decays per second for radium.

First woman to win a Nobel Prize in Physics.
First person to win two Nobel Prizes in different sciences (Physics and Chemistry).
Her husband, Pierre Curie, died in an accident.

Half-life ( $t_{1/2}$ ) is the time required for the number of radioactive nuclei to decay to half the starting value.
After each half-life, the remaining amount halves.
Relationship between half-life and decay constant:
- $N = N_0 e^{-\lambda t}$
- At $t = t<em>{1/2}$ , $N = \frac{N</em>0}{2}$
- $\frac{N}{N_0} = e^{-\lambda t} = \frac{1}{2}$
- $-\lambda t_{1/2} = \ln(\frac{1}{2}) = -\ln(2)$
- $t_{1/2} = \frac{\ln(2)}{\lambda} \approx \frac{0.693}{\lambda}$
- $\lambda = \frac{0.693}{t_{1/2}}$
Can express exponentials as exp(something).

Calculate the number of atoms in 1 gram of radium-226:
- Avogadro's number ( $N_A$ ) divided by the atomic weight.
- $N = \frac{N_A}{226} = \frac{6.022 \times 10^{23}}{226} = 2.66 \times 10^{21}$ atoms.
Calculate the activity of 1 gram of radium-226:
- First, calculate the decay constant $\lambda$ .
- $t_{1/2} = 1622$ years
- Conversion to seconds: 1 year $\approx \pi \times 10^7$ s.
- $\lambda = \frac{0.693}{t_{1/2}} = \frac{0.693}{1622 \times \pi \times 10^7} \approx 1.36 \times 10^{-11} s^{-1}$
- Activity: $A = \lambda N = (1.36 \times 10^{-11})(2.66 \times 10^{21}) \approx 3.61 \times 10^{10}$ Bq.
Cobalt-60 has a half-life of 5.26 years.
- What is the decay constant in units of per month?
- $\lambda = \frac{0.693}{5.26 \times 12} \approx 0.011 \text{ per month}$
- What will be the activity of a 5000 Curie sample after 4 years?
- Using the decay constant in per month: $A = 5000 e^{-0.011 \times 48} \approx 2950 \text{ Ci}$
- Another method: Calculate the number of half-lives: $4 / 5.26 \approx 0.76$
- $A = \frac{5000}{2^{0.76}} \approx 2950 \text{ Ci}$

Uranium dating: Comparing uranium isotopes and lead isotopes to determine the age of rocks.
- Oldest rocks on Earth and Moon are about 4.5 billion years old.
Carbon dating: Cosmic rays produce carbon-14 in the upper atmosphere.
- Carbon-14 has a half-life of about 6,000 years.
- Plants take up carbon dioxide, maintaining a ratio of one part in $10^{12}$ carbon-14 to carbon-12.
- After death, carbon-14 decays, allowing estimation of the age of organic material.
- The ratio decreases by a factor of two in 5,730 years.

Bremsstrahlung radiation (breaking radiation):
- Charged particle (e.g., electron) slows down and emits electromagnetic radiation.
- Maximum photon energy equals the kinetic energy of the electron.
Characteristic X-rays:
- Incoming electron knocks out an electron from an atom.
- Another electron drops to fill the space, emitting a photon.
- K-alpha, K-beta, etc., are characteristic X-rays from transitions to the n=1 level.
- L-alpha, L-beta, etc., are from transitions to the n=2 level.
- Approximation: Use $Z - 1$ as the effective charge in the Bohr model to calculate energy levels for heavy atoms.

Crookes tube: Anode and cathode in a gas-filled jar.
- Positive ions hitting the cathode generate X-rays.
Thermionic tube: Separate heating coil for the cathode to control the current independently of the accelerating potential.
- This allows independent control of the amount and energy of the X-rays.
- Typical setup involves a tungsten anode that is water cooled.
- Only $\approx$ 1% of the energy comes out as X-rays; the rest is heat.

Health risks associated with radiation.
Alpha radiation is absorbed quickly and does significant localized damage.
- Radon gas is an alpha emitter that can damage lungs.
- Smoke alarms use alpha emitters but are safe externally.
Beta radiation travels further and causes less localized damage.
Relative effectiveness of radiation depends on its ability to ionize.
Alpha and beta with the same energy do the same total amount of ionization, but over different distances.
Alpha particles (e.g., from radon in air) have a range of about 3.5 cm and ionize about 4000 pairs per mm.
Beta particles travel about a linear distance of 20m, with a random walk of about 40m and ionize about 3 to 4 pairs per mm.