Add Obsidian knowledge base (docs/) and update README with docs structure

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---
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
```