82 lines
2.9 KiB
Markdown
82 lines
2.9 KiB
Markdown
---
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title: "02 – Phase Space"
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date: 2026-04-26
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tags:
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- beta-spectrum/correction/phase-space
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status: active
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aliases: [Phase Space, Baseline Spectrum]
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cssclasses: []
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related_components: [[01-fermi-function]], [[07-recoil-effects]]
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---
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# Phase Space — Baseline Spectral Shape $p W (W_0 - W)^2$
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## The Unmodified Spectrum
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The baseline β spectrum shape without any corrections is pure phase space:
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$$\frac{dN}{dW}\bigg|_{\text{PS}} = p W (W_0 - W)^2 \tag{4, first factor}$$
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where:
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- $p = \sqrt{W^2 - 1}$ — electron momentum ($m_e c$ units)
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- $W$ — total electron energy ($m_e c^2$ units)
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- $W_0$ — endpoint (Q-value + $m_ec^2$)
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## Neutrino Mass Effects
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If the neutrino has non-zero mass $m_\nu$, the spectrum is modified near the endpoint:
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$$p \to p_e = \sqrt{W^2 - 1}, \quad p_\nu = \sqrt{(W_0 - W)^2 - m_\nu^2}$$
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The phase space factor becomes:
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$$(W_0 - W)\sqrt{(W_0 - W)^2 - m_\nu^2}$$
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This produces a characteristic distortion near $W = W_0 - m_\nu$, used in neutrino mass experiments (e.g., KATRIN with tritium β decay).
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## Kurie Plot
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The phase space subtracted spectrum gives the **Kurie function**:
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$$K(W) = \sqrt{\frac{N(W)}{p W F(Z,W)}} \approx W_0 - W \quad (\text{linear for allowed transitions})$$
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Deviations from linearity near the endpoint indicate:
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- Non-zero neutrino mass
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- Weak magnetism / recoil effects ([[07-recoil-effects]])
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- New physics (Fierz term [[Hayen2017_summary#The Fierz Interference Term]])
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## Calculator Implementation
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> [!tip] Code mapping
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> Module: `beta_spectrum/components/phase_space.py` → `PhaseSpace(W0, transition_type)` class
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>
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> Evaluates $p \times W \times (W_0 - W)^2$ multiplied by a **transition-type-dependent forbidden factor**. Supports optional neutrino mass.
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## Transition Type and Forbidden Factors
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The `PhaseSpace` calculator accepts a `transition_type` parameter that determines the multiplicative forbidden factor applied to the baseline phase space:
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| `transition_type` | Forbidden Order | Forbidden Factor |
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|---|---|---|
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| `A`, `F1` | 0, 1 | $1$ |
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| `F1U`, `F2` | 1, 2 | $p_\nu^2 + p_e^2$ |
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| `F2U`, `F3` | 2, 3 | $p_\nu^4 + \frac{3}{10}p_\nu^2 p_e^2 + p_e^4$ |
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| `F3U`, `F4` | 3, 4 | $p_\nu^6 + 7p_\nu^4 p_e^2 + 7p_\nu^2 p_e^4 + p_e^6$ |
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The forbidden factor modifies the energy dependence of the spectrum, becoming significant at higher electron energies. See `[[10-nuclear-structure]]` for the full shape factor treatment.
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## Energy Range
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The phase space factor:
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- Vanishes at $W = 1$ ($T = 0$, the threshold) — goes as $(W-1)^{1/2}$ via momentum
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- Vanishes at $W = W_0$ (the endpoint) — quadratic behavior from neutrino energy
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- Peaks roughly at $W \approx W_0 / 3$ to $W_0 / 2$ depending on $Z$
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## Code API
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```python
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from beta_spectrum import PhaseSpace, T_to_W
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ps = PhaseSpace(W0=T_to_W(5.0), transition_type="A") # allowed transition, 5 MeV endpoint
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values = ps(W_grid) # returns p·W·(W₀-W)² × forbidden_factor
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```
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