Add Obsidian knowledge base (docs/) and update README with docs structure
This commit is contained in:
@@ -0,0 +1,68 @@
|
||||
---
|
||||
title: "02 – Phase Space"
|
||||
date: 2026-04-26
|
||||
tags:
|
||||
- beta-spectrum/correction/phase-space
|
||||
status: active
|
||||
aliases: [Phase Space, Baseline Spectrum]
|
||||
cssclasses: []
|
||||
related_components: [[01-fermi-function]], [[07-recoil-effects]]
|
||||
---
|
||||
|
||||
# Phase Space — Baseline Spectral Shape $p W (W_0 - W)^2$
|
||||
|
||||
## The Unmodified Spectrum
|
||||
|
||||
The baseline β spectrum shape without any corrections is pure phase space:
|
||||
|
||||
$$\frac{dN}{dW}\bigg|_{\text{PS}} = p W (W_0 - W)^2 \tag{4, first factor}$$
|
||||
|
||||
where:
|
||||
- $p = \sqrt{W^2 - 1}$ — electron momentum ($m_e c$ units)
|
||||
- $W$ — total electron energy ($m_e c^2$ units)
|
||||
- $W_0$ — endpoint (Q-value + $m_ec^2$)
|
||||
|
||||
## Neutrino Mass Effects
|
||||
|
||||
If the neutrino has non-zero mass $m_\nu$, the spectrum is modified near the endpoint:
|
||||
|
||||
$$p \to p_e = \sqrt{W^2 - 1}, \quad p_\nu = \sqrt{(W_0 - W)^2 - m_\nu^2}$$
|
||||
|
||||
The phase space factor becomes:
|
||||
$$(W_0 - W)\sqrt{(W_0 - W)^2 - m_\nu^2}$$
|
||||
|
||||
This produces a characteristic distortion near $W = W_0 - m_\nu$, used in neutrino mass experiments (e.g., KATRIN with tritium β decay).
|
||||
|
||||
## Kurie Plot
|
||||
|
||||
The phase space subtracted spectrum gives the **Kurie function**:
|
||||
|
||||
$$K(W) = \sqrt{\frac{N(W)}{p W F(Z,W)}} \approx W_0 - W \quad (\text{linear for allowed transitions})$$
|
||||
|
||||
Deviations from linearity near the endpoint indicate:
|
||||
- Non-zero neutrino mass
|
||||
- Weak magnetism / recoil effects ([[07-recoil-effects]])
|
||||
- New physics (Fierz term [[Hayen2017_summary#The Fierz Interference Term]])
|
||||
|
||||
## Calculator Implementation
|
||||
|
||||
> [!tip] Code mapping
|
||||
> Module: `beta_spectrum/components/phase_space.py` → `PhaseSpace(W0)` class
|
||||
>
|
||||
> Simple evaluation of $p \times W \times (W_0 - W)^2$ with masking near endpoint for numerical safety. Optionally includes neutrino mass support.
|
||||
|
||||
## Energy Range
|
||||
|
||||
The phase space factor:
|
||||
- Vanishes at $W = 1$ ($T = 0$, the threshold) — goes as $(W-1)^{1/2}$ via momentum
|
||||
- Vanishes at $W = W_0$ (the endpoint) — quadratic behavior from neutrino energy
|
||||
- Peaks roughly at $W \approx W_0 / 3$ to $W_0 / 2$ depending on $Z$
|
||||
|
||||
## Code API
|
||||
|
||||
```python
|
||||
from beta_spectrum import PhaseSpace, T_to_W
|
||||
|
||||
ps = PhaseSpace(W0=T_to_W(5.0)) # endpoint = 5 MeV
|
||||
values = ps(W_grid) # returns p·W·(W₀-W)²
|
||||
```
|
||||
Reference in New Issue
Block a user