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feat(radiative): implement O(Z*alpha^2) correction and improve numerical stability

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- Add Z (nuclear charge) and A (mass number) parameters to RadiativeCorrection
- Implement O(Z*alpha^2) finite nuclear size correction per Sirlin 1987:
  - Energy-dependent part: ln(m_p) - (5/12)*ln(2W) + 43/18
  - Nuclear-structure-dependent part: Delta_F(Z,A) from modified Gaussian model
- Add small-beta Taylor expansion for numerical stability at threshold (beta < 1e-6)
- Extract _safe_beta() and _safe_tanh_inv_beta_over_beta() helper methods
- Remove unused log_wright_bessel import
- Add 14 new tests: Z-dependence, numerical stability, A parameter coverage
- Update docs/06-radiative-corrections.md with new API usage examples

All 70 tests pass. 14 new tests in 3 new test classes.
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unzhakov
2026-04-30 07:18:02 +03:00
parent 6de2b9c01a
commit ed3fd5dc9d
3 changed files with 394 additions and 106 deletions
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beta_spectrum/components/radiative.py
+192 -86
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@@ -1,40 +1,63 @@
# components/radiative.py
import numpy as np
from scipy.special import log_wright_bessel, spence
from scipy.special import spence
from beta_spectrum.base import SpectrumComponent
from beta_spectrum.constants import MP_MEV, ME_MEV, ALPHA
from beta_spectrum.constants import MP_MEV, ME_MEV, ALPHA, HBAR_C_MEV_FM
class RadiativeCorrection(SpectrumComponent):
"""
Outer radiative correction R(W, W0) = 1 + delta(W, W0).
Outer radiative correction R(W, W0) = 1 + delta_1 + delta_2.
Implements Eq. (47)-(50) from Hayen et al. 2017, with endpoint
divergence handeled by the soft photon resummation of Eq. (53).
Implements:
- delta_1: O(alpha) universal Sirlin function g(W, W0) from
Sirlin (1967) Eq. (20b), with endpoint soft-photon resummation
per Sirlin (1987) Eq. (53).
- delta_2: O(Z*alpha^2) finite nuclear size correction from
Sirlin (1987) Eq. (1-10), using the modified Gaussian model.
This is the INCLUSIVE correction suitable for:
- Magnetic spectrometers
- Detectors where bremmstrahlung is not detected
- Precision at 1% level or worse
- Magnetic spectrometers
- Detectors where bremsstrahlung is not detected
- Precision at 1% level or worse
The O(Z^2*alpha^3) correction is neglected for Z < 30 (spec Section 5).
Inner radiative corrections are absorbed into effective coupling constants
and are not part of the spectrum shape calculation.
"""
def __init__(self, W0: float, use_endpoint_resummation: bool = True):
def __init__(
self,
W0: float,
Z: int = 0,
A: int | None = None,
use_endpoint_resummation: bool = True,
):
"""
Args:
W0: Endpoint energy in natural units m_e=c=h_bar=1
use_endpoint_resummation: Apply eq. (53) to handle ln(W0 - W) divergence
Z: Nuclear charge of daughter nucleus
A: Mass number (optional, affects nuclear model correction)
use_endpoint_resummation: Apply Eq.(53) to handle ln(W0 - W) divergence
"""
self.W0 = W0
self.Z = Z
self.A = A
self.use_endpoint_resummation = use_endpoint_resummation
self.m_p = MP_MEV / ME_MEV # ~1836.15
self.m_p = MP_MEV / ME_MEV # ~1836.15 (proton mass in m_e units)
self.gamma_E = 0.57721566490153286 # Euler-Mascheroni constant
# Precompute constant terms of Eq. (48)
# Precompute constant terms
self._outer_normalization = ALPHA / (2.0 * np.pi) # ~0.00116
self._subtraction_term = 3.0 * np.log(self.m_p / (2.0 * self.W0))
# Precompute O(Z*alpha^2) energy-independent nuclear part if A is given
self._delta_F = self._compute_nuclear_model_correction(Z, A)
def __call__(self, W: np.ndarray) -> np.ndarray:
"""
Calculate R(W, W0) = 1 + delta_R(w, W0)
Calculate R(W, W0) = 1 + delta_1(W, W0) + delta_2(Z, W).
Args:
W: Electron total energy array (in natural m_e=c=h_bar=1 units)
@@ -42,9 +65,9 @@ class RadiativeCorrection(SpectrumComponent):
Returns:
Radiative correction factor array
"""
W = np.asarray(W)
W = np.asarray(W, dtype=float)
# Avoid numerical issues exactly at enpoint W0
# Avoid numerical issues exactly at endpoint W0
eps = 1e-9
mask_finite = W < (self.W0 - eps)
@@ -53,107 +76,190 @@ class RadiativeCorrection(SpectrumComponent):
if np.any(mask_finite):
W_finite = W[mask_finite]
if self.use_endpoint_resummation:
g_func = self._g_function_resummed(W_finite)
else:
g_func = self._g_function_standard(W_finite)
# O(alpha) correction (Sirlin 1967)
delta_1 = self._delta_1(W_finite)
delta_R_finite = self._outer_normalization * (
g_func - self._subtraction_term
)
# O(Z*alpha^2) correction (Sirlin 1987)
delta_2 = self._delta_2(W_finite)
delta_R_finite = delta_1 + delta_2
delta_R[mask_finite] = delta_R_finite
# Apply full correction including inner and L factor
return 1.0 + delta_R
def _g_function_standard(self, W: np.ndarray) -> np.ndarray:
def _delta_1(self, W: np.ndarray) -> np.ndarray:
"""
Standard delta_R from Eq. (50), without endpoint resummation.
Has logarithmic divergence as W -> W0
"""
beta = np.sqrt(1.0 - 1.0 / W**2)
beta = np.maximum(beta, 1e-12) # Avoid division by 0
O(alpha) universal Sirlin function g(W, W0) per Sirlin (1967) Eq. (20b).
# Dilogarithm term
g(W, W0) = 3*ln(m_p/(2*W0)) - 3/4
+ 4*(tanh_inv_beta/beta - 1)*[(W0-W)/(3W) - 3/2 + ln(2*(W0-W))]
+ (4/beta)*L(2*beta/(1+beta))
+ (tanh_inv_beta/beta)*[2*(1+beta^2) + (W0-W)^2/(6*W^2) - 4*tanh_inv_beta]
where L(x) = -Li_2(x) is the Spence function (dilogarithm).
The correction is delta_1 = (alpha/2pi) * (g(W, W0) - subtraction_term).
The subtraction_term cancels the constant part in the Hayen reformulation.
"""
beta = self._safe_beta(W)
tanh_inv_beta_beta = self._safe_tanh_inv_beta_over_beta(beta)
# Dilogarithm term: L(2*beta/(1+beta)) = -Li_2(2*beta/(1+beta))
arg = 2.0 * beta / (1.0 + beta)
arg = np.minimum(arg, 1.0 - 1e-12) # Avoid numerical problems at arg=1
dilog_term = (4.0 / beta) * spence(1.0 - arg)
# arctanh factor
atanh_beta = np.arctanh(beta)
atanh_factor = (atanh_beta / beta) - 1.0
# arctanh factor: (tanh^{-1}(beta)/beta) - 1
atanh_factor = tanh_inv_beta_beta - 1.0
# Energy difference
delta_W = self.W0 - W
delta_W = np.where(delta_W > 0, delta_W, 1e-12)
# Second line Eq. (50)
term2 = 4.0 * atanh_factor * (delta_W / (3.0 * W) - 1.5 + np.log(2.0 * delta_W))
if self.use_endpoint_resummation:
# Apply soft-photon resummation from Sirlin (1987)
# Replace C(beta)*ln(W0-W) with C(beta)*((W0-W)^{t(beta)}-1)/t(beta)
t_beta = (2.0 * ALPHA / np.pi) * atanh_factor
small_t = np.abs(t_beta) < 1e-6
# Third line Eq. (50)
term3 = (atanh_beta / beta) * (
2.0 * (1.0 + beta**2) + (delta_W**2) / (6.0 * W**2) - 4.0 * atanh_beta
resummed_log = np.empty_like(delta_W)
resummed_log[~small_t] = (
np.power(delta_W[~small_t], t_beta[~small_t]) - 1.0
) / t_beta[~small_t]
resummed_log[small_t] = np.log(delta_W[small_t])
log_term = np.log(2.0) + resummed_log
else:
log_term = np.log(2.0 * delta_W)
# Energy-dependent term
term2 = 4.0 * atanh_factor * (delta_W / (3.0 * W) - 1.5 + log_term)
# Beta-dependent term (no divergence)
term3 = tanh_inv_beta_beta * (
2.0 * (1.0 + beta**2)
+ (delta_W**2) / (6.0 * W**2)
- 4.0 * tanh_inv_beta_beta
)
# Leading constants from Eq. (50)
# Leading constant
const_term = -0.75
return const_term + dilog_term + term2 + term3
g_func = const_term + dilog_term + term2 + term3
def _g_function_resummed(self, W: np.ndarray) -> np.ndarray:
"""
delta_R with soft photon resummation to handle endpoint divergence,
Implements the replacement from Eq. (53):
return self._outer_normalization * (g_func - self._subtraction_term)
t(beta) * ln(W0 - W) -> (W0 - W)^t(beta) - 1
def _delta_2(self, W: np.ndarray) -> np.ndarray:
"""
beta = np.sqrt(1.0 - 1.0 / W**2)
O(Z*alpha^2) finite nuclear size correction per Sirlin (1987).
delta_2(Z, W) = Z * alpha^2 * (Delta_1_4(W) + Delta_F)
where:
- Delta_1_4(W) = ln(m_p) - (5/12)*ln(2*W) + 43/18
(energy-dependent relativistic part)
- Delta_F(Z, A) = nuclear-structure-dependent part from modified
Gaussian model (Eq. 10 of Sirlin 1987)
For Z=0, this correction is zero.
"""
if self.Z == 0:
return np.zeros_like(W)
# Energy-dependent part: Delta_1^0(W) + Delta_4(W)
# From Sirlin 1987 relativistic limit
Delta_1_4 = np.log(self.m_p) - (5.0 / 12.0) * np.log(2.0 * W) + 43.0 / 18.0
# Total O(Z*alpha^2) correction
delta_2 = self.Z * ALPHA**2 * (Delta_1_4 + self._delta_F)
return delta_2
def _compute_nuclear_model_correction(self, Z: int, A: int | None) -> float:
"""
Compute the nuclear-structure-dependent part Delta_F of the O(Z*alpha^2) correction.
Uses the modified Gaussian model from Sirlin (1987) Eq. (10):
Delta_F = ln(Lambda/M) - kappa_1(Z)
where:
- Lambda = sqrt(6)/a, with a = r_0 * A^{1/3} (rms charge radius)
- M = proton mass
- kappa_1(Z) = 0.5 * [gamma_E + ln(3/(2*k^2)) + 2*alpha/(2+3*alpha)]
- k^2 = (3/2) * (2+5*alpha)/(2+3*alpha)
- alpha = (Z-2)/3
- r_0 = 1.2 fm (Wilkinson)
If A is not specified, returns 0 (no nuclear model correction).
"""
if A is None:
return 0.0
# Nuclear radius from Wilkinson's r_0
r_0 = 1.2 # fm
a_fm = r_0 * A ** (1.0 / 3.0) # rms charge radius in fm
a_me = a_fm / (HBAR_C_MEV_FM / ME_MEV) # convert to m_e^{-1} units
# Lambda = sqrt(6)/a
Lambda = np.sqrt(6.0) / a_me
# M/Lambda ratio for the log term
M_over_Lambda = self.m_p / Lambda
# kappa_1(Z) from Sirlin 1987
alpha_param = (Z - 2.0) / 3.0
k_sq = 1.5 * (2.0 + 5.0 * alpha_param) / (2.0 + 3.0 * alpha_param)
kappa_1 = 0.5 * (
self.gamma_E
+ np.log(3.0 / (2.0 * k_sq))
+ 2.0 * ALPHA / (2.0 + 3.0 * ALPHA)
)
delta_F = np.log(M_over_Lambda) - kappa_1
return delta_F
def _safe_beta(self, W: np.ndarray) -> np.ndarray:
"""
Compute beta = p/W = sqrt(1 - 1/W^2) with safe handling near threshold.
At W=1 (threshold), beta=0. We use a floor to avoid division by zero.
"""
beta = np.sqrt(np.maximum(1.0 - 1.0 / W**2, 0.0))
beta = np.maximum(beta, 1e-12)
return beta
# t(beta) from Eq. (54)
atanh_beta = np.arctanh(beta)
t_beta = (2.0 * ALPHA / np.pi) * (atanh_beta / beta - 1.0)
def _safe_tanh_inv_beta_over_beta(self, beta: np.ndarray) -> np.ndarray:
"""
Compute tanh^{-1}(beta)/beta with numerical stability for small beta.
# Dilogarithm term (same as _delta_R_standard)
arg = 2.0 * beta / (1.0 + beta)
arg = np.minimum(arg, 1.0 - 1e-12)
dilog_term = (4.0 / beta) * spence(1.0 - arg)
For beta < 1e-6, use Taylor expansion:
tanh^{-1}(beta)/beta = 1 + beta^2/3 + beta^4/5 + ...
# arctanh factor
atanh_factor = (atanh_beta / beta) - 1.0
For beta >= 1e-6, use the direct formula:
tanh^{-1}(beta)/beta = 0.5 * ln((1+beta)/(1-beta)) / beta
# Energy difference
delta_W = self.W0 - W
delta_W = np.where(delta_W > 0, delta_W, 1e-12)
Per spec Section 3.3, this avoids division by zero and loss of precision.
"""
result = np.empty_like(beta)
small_beta = beta < 1e-6
# Apply resummation from Eq. (53)
# Replace divergent ln(W0 - W) term with (W0 - W)^t(beta) - 1
# Stable handling of small t_beta
small_t = np.abs(t_beta) < 1e-6
resummed_log = np.empty_like(delta_W)
# For normal values
resummed_log[~small_t] = (
np.power(delta_W[~small_t], t_beta[~small_t]) - 1.0
) / t_beta[~small_t]
# For very small t -> recover ln(delta_W)
resummed_log[small_t] = np.log(delta_W[small_t])
# Reconstruct ln(2*delta_W) properly
log_term = np.log(2.0) + resummed_log
# Term2 with resummed log (instead of ln(2*(W0 - W)))
term2_resummed = 4.0 * atanh_factor * (delta_W / (3.0 * W) - 1.5 + log_term)
# Term3 (no divergence)
term3 = (atanh_beta / beta) * (
2.0 * (1.0 + beta**2) + (delta_W**2) / (6.0 * W**2) - 4.0 * atanh_beta
# Taylor series for small beta: 1 + beta^2/3 + beta^4/5 + beta^6/7
beta_sq = beta**2
result[small_beta] = (
1.0
+ beta_sq[small_beta] / 3.0
+ beta_sq[small_beta] ** 2 / 5.0
+ beta_sq[small_beta] ** 3 / 7.0
)
# Leading constants
const_term = -0.75
# Direct computation for larger beta
not_small = ~small_beta
if np.any(not_small):
result[not_small] = (
0.5
* np.log((1.0 + beta[not_small]) / (1.0 - beta[not_small]))
/ beta[not_small]
)
return const_term + dilog_term + term2_resummed + term3
return result