beta_shape_pnpi/docs/02-phase-space.md

---

title: "02 – Phase Space" date: 2026-04-26 tags:

• beta-spectrum/correction/phase-space status: active aliases: [Phase Space, Baseline Spectrum] cssclasses: [] related_components: corrections/01-fermi-function, corrections/06-recoil

---

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 (corrections/06-recoil)
• New physics (Fierz interference term; see Hayen et al., Rev. Mod. Phys. 90, 015008 (2018))

Calculator Implementation

[!tip] Code mapping Module: beta_spectrum/components/phase_space.py → PhaseSpace(W0, transition_type) class

Evaluates p \times W \times (W_0 - W)^2 multiplied by a transition-type-dependent forbidden factor. Supports optional neutrino mass.

Transition Type and Forbidden Factors

The PhaseSpace calculator accepts a transition_type parameter that determines the multiplicative forbidden factor applied to the baseline phase space:

transition_type	Forbidden Order	Forbidden Factor
A, F1	0, 1	1
F1U, F2	1, 2	p_\nu^2 + p_e^2
F2U, F3	2, 3	p_\nu^4 + \frac{3}{10}p_\nu^2 p_e^2 + p_e^4
F3U, F4	3, 4	p_\nu^6 + 7p_\nu^4 p_e^2 + 7p_\nu^2 p_e^4 + p_e^6

The forbidden factor modifies the energy dependence of the spectrum, becoming significant at higher electron energies. The full nuclear shape factor C(Z,W) is external nuclear-structure theory, out of scope for this repository: the calculator treats the forbiddenness classes above as input configuration.

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

from beta_spectrum import PhaseSpace, T_to_W

ps = PhaseSpace(W0=T_to_W(5.0), transition_type="A")  # allowed transition, 5 MeV endpoint
values = ps(W_grid)                                    # returns p·W·(W₀-W)² × forbidden_factor