In a nucleus, the probability of the process occur ring due to incident neutron per cm2 is defined as the nuclear cross-section (σ). A microscopic cross-section is what it’s called. Macroscopic cross-section refers to the overall cross-section of all the nuclei in a unit volume of 1 cm3 MEANING
Microscopic cross-section (σ) is the probability that a single nucleus will interact with an incident neutron. It has units of barns (1 barn = 10⁻²⁴ cm²) and represents an effective "target area" that a nucleus presents to an incoming neutron.
Macroscopic cross-section (Σ) is the total interaction probability per unit path length for a neutron traveling through a material. It accounts for ALL nuclei in a 1 cm³ volume.
The Relationship
$$\Sigma = N \cdot \sigma$$
Where:
- Σ = Macroscopic cross-section (units: cm⁻¹)
- N = number density of nuclei (nuclei/cm³)
- σ = microscopic cross-section (cm²)
Simple Analogy
Imagine firing bullets at a wall of targets:
| Concept | Analogy |
|---|---|
| Microscopic σ | The size of one individual target |
| N (number density) | How many targets are packed into the wall |
| Macroscopic Σ | The total target area per unit thickness of wall - the combined "blocking" effect |
Physical Meaning of Σ
Σ = probability per unit path length (cm) that a neutron will interact.
- If Σ = 2 cm⁻¹, there is a 2% chance of interaction per 0.01 cm of travel.
- Mean free path λ = 1/Σ - the average distance a neutron travels before interacting.
Example
For natural uranium (²³⁸U):
- σ (absorption) ≈ 2.7 barns = 2.7 × 10⁻²⁴ cm²
- N ≈ 4.8 × 10²² atoms/cm³
- Σ = N × σ = (4.8 × 10²²) × (2.7 × 10⁻²⁴) ≈ 0.13 cm⁻¹
In Short
| Term | Symbol | Units | Meaning |
|---|---|---|---|
| Microscopic cross-section | σ | cm² (barns) | Interaction probability for one nucleus |
| Macroscopic cross-section | Σ | cm⁻¹ | Interaction probability per cm of path through bulk material |
The macroscopic cross-section essentially "scales up" the single-nucleus probability to the bulk material level, making it practical for reactor physics and neutron transport calculations.
SPECIFIC ACTIVITY DEF
Specific Activity - Definition
Specific Activity is the activity (radioactive decay rate) per unit mass of a radioactive substance.
$$SA = \frac{A}{m}$$
Where:
- A = Activity (decays per second = Becquerels)
- m = Mass of the radioactive material (grams)
Units
| System | Unit |
|---|---|
| SI | Bq/g or Bq/kg (Becquerels per gram) |
| Traditional | Ci/g (Curies per gram) |
Formula from First Principles
$$SA = \frac{\lambda \cdot N_A}{M}$$
Where:
- λ = decay constant = ln2 / t½
- N_A = Avogadro's number (6.022 × 10²³ mol⁻¹)
- M = atomic mass (g/mol)
- t½ = half-life
So in full:
$$SA = \frac{0.693 \times N_A}{t_{1/2} \times M}$$
Key Insight
- Shorter half-life → Higher specific activity (decays faster per gram)
- Longer half-life → Lower specific activity (decays slowly per gram)
| Nuclide | Half-life | Specific Activity |
|---|---|---|
| Po-210 | 138 days | ~4,500 Ci/g (very HIGH) |
| Co-60 | 5.27 years | ~1,130 Ci/g (high) |
| Cs-137 | 30.2 years | ~87 Ci/g (moderate) |
| U-238 | 4.5 × 10⁹ years | ~3.4 × 10⁻⁷ Ci/g (very LOW) |
Physical Meaning
Specific activity tells you how radioactive a given mass of material is. It is important in:
- Radiation protection - determining hazard per gram ingested/inhaled
- Nuclear medicine - ensuring a radiopharmaceutical dose delivers enough activity in a small volume
- Reactor physics - characterizing irradiated fuel and waste
- Isotope production - measuring the purity/potency of a produced isotope
Reference Point
By definition, 1 gram of Ra-226 has an activity of exactly 1 Curie (Ci) = 3.7 × 10¹⁰ disintegrations/second. This is the historical basis of the Curie unit itself.
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