def modelCalibratedD2O(q, name="$D_2O$", volFraction=1, temp=300, **kwargs):
"""A model for D2O background containing a single Lorentzian.
Parameters
----------
q : np.ndarray
Array of values for momentum transfer q.
name : str
Name for the model
kwargs : dict
Additional arguments to pass to Parameters.
Can override default parameter attributes.
"""
p = Parameters(amplitude={
"value": np.ones_like(q),
"bounds": (0.0, np.inf)
}, )
p.update(**kwargs)
m = Model(p, name)
m.addComponent(
Component(
"$D_2O$ background",
calibratedD2O,
volFraction=volFraction,
temp=temp,
skip_convolve=True,
))
return m
def modelD2OBackground(q, name="$D_2O$", **kwargs):
"""A model for D2O background containing a single Lorentzian.
Parameters
----------
q : np.ndarray
Array of values for momentum transfer q.
name : str
Name for the model
kwargs : dict
Additional arguments to pass to Parameters.
Can override default parameter attributes.
"""
p = Parameters(
scale={
"value": np.ones_like(q),
"bounds": (0.0, np.inf)
},
width={
"value": np.ones_like(q),
"bounds": (0.0, np.inf)
},
center={"value": 0.0},
)
p.update(**kwargs)
m = Model(p, name)
m.addComponent(
Component("$D_2O$ background", lorentzian, skip_convolve=True))
return m
def modelPVoigtBkgd(q, name="PVoigtBkgd", **kwargs):
"""A model containing a pseudo-Voigt profile with a background term.
Parameters
----------
q : np.ndarray
Array of values for momentum transfer q.
name : str
Name for the model
kwargs : dict
Additional arguments to pass to Parameters.
Can override default parameter attributes.
"""
p = Parameters(
scale={
"value": np.zeros_like(q) + 1.0,
"bounds": (0.0, np.inf)
},
frac={
"value": np.zeros_like(q) + 0.5,
"bounds": (0.0, 1.0)
},
width={
"value": np.zeros_like(q) + 1.0,
"bounds": (0.0, np.inf)
},
center={
"value": np.zeros_like(q) + 0.0,
"bounds": (-np.inf, np.inf)
},
bkgd={
"value": np.zeros_like(q) + 0.001,
"bounds": (0.0, np.inf)
},
)
p.update(**kwargs)
m = Model(p, name)
m.addComponent(
Component("lorentzian", lorentzian, scale="scale * (1 - frac)"))
m.addComponent(
Component(
"gaussian",
gaussian,
scale="scale * frac",
width="width / np.sqrt(2 * np.log(2))",
))
m.addComponent(Component("background", linear, True, a=0.0, b="bkgd"))
return m
def modelLorentzianSum(q, name="LorentzianSum", nLor=2, qWise=True, **kwargs):
"""A model containing a delta and a sum of Lorentzians.
Parameters
----------
q : np.ndarray
Array of values for momentum transfer q.
name : str
Name for the model
nLor : 2
Number of Lorentzian to be used.
qWise : bool
If True, no q dependence is imposed on the parameters and
the each spectrum is fitted independently.
kwargs : dict
Additional arguments to pass to Parameters.
Can override default parameter attributes.
"""
p = Parameters(
scale={
"value": np.zeros_like(q) + 1,
"bounds": (0.0, np.inf)
},
a0={
"value": np.zeros_like(q) + 0.5,
"bounds": (0.0, 1)
},
center={
"value": 0.0,
"fixed": True
},
)
aDefault = np.zeros_like(q) + 0.5
if qWise:
widthDefault = np.zeros_like(q) + 10
widthStr = "w%i"
else:
widthDefault = 10
widthStr = "w%i * q ** 2"
for idx in range(nLor):
p.set("a%i" % (idx + 1), value=aDefault, bounds=(0.0, 1.0))
p.set("w%i" % (idx + 1), value=widthDefault, bounds=(0.0, np.inf))
p.update(**kwargs)
m = Model(p, name)
m.addComponent(Component("EISF", delta, scale="a0 * scale"))
for idx in range(nLor):
m.addComponent(
Component(
r"$\mathcal{L}_{%i}$" % (idx + 1),
lorentzian,
scale="a%i * scale" % (idx + 1),
width=widthStr % (idx + 1),
))
return m
def modelGaussBkgd(q, name="GaussBkgd", **kwargs):
"""A model containing a Gaussian with a background term.
Parameters
----------
q : np.ndarray
Array of values for momentum transfer q.
name : str
Name for the model
kwargs : dict
Additional arguments to pass to Parameters.
Can override default parameter attributes.
"""
p = Parameters(
scale={
"value": np.zeros_like(q) + 1.0,
"bounds": (0.0, np.inf)
},
width={
"value": np.zeros_like(q) + 1.0,
"bounds": (0.0, np.inf)
},
center={
"value": np.zeros_like(q) + 0.0,
"bounds": (-np.inf, np.inf)
},
bkgd={
"value": np.zeros_like(q) + 0.001,
"bounds": (0.0, np.inf)
},
)
p.update(**kwargs)
m = Model(p, name)
m.addComponent(Component("gaussian", gaussian))
m.addComponent(Component("background", linear, True, a=0.0, b="bkgd"))
return m
def modelTwoStatesSwitchDiff(q, name="TwoStatesSwitch", **kwargs):
"""A model for protein in liquid state.
This model implements the two states switching diffusion model for
nPDyn [#]_.
Parameters
----------
q : np.ndarray
Array of values for momentum transfer q.
name : str
Name for the model
kwargs : dict
Additional arguments to pass to Parameters.
Can override default parameter attributes.
References
----------
.. [#] https://doi.org/10.1103/PhysRev.119.863
"""
p = Parameters(
beta={
"value": np.zeros_like(q) + 1,
"bounds": (0.0, np.inf)
},
a0={
"value": np.zeros_like(q) + 0.5,
"bounds": (0.0, 1)
},
tau1={
"value": 1,
"bounds": (0.0, np.inf)
},
tau2={
"value": 1,
"bounds": (0.0, np.inf)
},
g0={
"value": 5,
"bounds": (0.0, np.inf)
},
g1={
"value": 10,
"bounds": (0.0, np.inf)
},
g2={
"value": 100,
"bounds": (0.0, np.inf)
},
center={
"value": 0,
"fixed": True
},
)
w0 = "g0 * q**2"
w1 = ("(g1 * q**2 + g2 * q**2 + tau1 + tau2 + "
"np.sqrt(((g1 - g2) * q**2 + tau1 - tau2)**2 + 4 * "
"tau1 * tau2)) / 2")
w2 = ("(g1 * q**2 + g2 * q**2 + tau1 + tau2 - "
"np.sqrt(((g1 - g2) * q**2 + tau1 - tau2)**2 + 4 * "
"tau1 * tau2)) / 2")
s1 = ("beta * (1 - a0) * (tau1 / (tau1 + tau2) * g2 * q**2 + "
"tau2 / (tau1 + tau2) * g1 * q**2 + "
"tau1 + tau2 - %s) / (%s - %s)" % (w1, w2, w1))
s2 = ("beta * (1 - a0) * (tau1 / (tau1 + tau2) * g2 * q**2 + "
"tau2 / (tau1 + tau2) * g1 * q**2 + "
"tau1 + tau2 - %s) / (%s - %s)" % (w2, w1, w2))
p.update(**kwargs)
m = Model(p, name)
m.addComponent(
Component(r"$\mathcal{L}_g$", lorentzian, scale="beta * a0", width=w0))
m.addComponent(
Component(r"$\mathcal{L}_1$",
lorentzian,
scale=s1,
width=w0 + " + " + w1))
m.addComponent(
Component(r"$\mathcal{L}_2$",
lorentzian,
scale=s2,
width=w0 + " + " + w2))
return m
def modelProteinJumpDiff(q, name="proteinJumpDiff", qWise=False, **kwargs):
"""A model for protein in liquid state.
The model contains a Lorentzian of Fickian-type
diffusion accounting for
center-of-mass motions, a Lorentzian of width that
obeys the jump diffusion model [#]_ accounting for
internal dynamics.
Parameters
----------
q : np.ndarray
Array of values for momentum transfer q.
name : str
Name for the model
qWise : bool
If True, no q dependence is imposed on the parameters and
the each spectrum is fitted independently.
kwargs : dict
Additional arguments to pass to Parameters.
Can override default parameter attributes.
References
----------
.. [#] https://doi.org/10.1103/PhysRev.119.863
"""
if qWise:
p = Parameters(
beta={
"value": np.zeros_like(q) + 1,
"bounds": (0.0, np.inf)
},
a0={
"value": np.zeros_like(q) + 0.5,
"bounds": (0.0, 1)
},
wg={
"value": np.zeros_like(q) + 5,
"bounds": (0.0, np.inf)
},
wi={
"value": np.zeros_like(q) + 30,
"bounds": (0.0, np.inf)
},
center={
"value": 0.0,
"fixed": True
},
)
widthG = "wg"
widthI = "wg + wi"
else:
p = Parameters(
beta={
"value": np.zeros_like(q) + 1,
"bounds": (0.0, np.inf)
},
a0={
"value": np.zeros_like(q) + 0.5,
"bounds": (0.0, 1)
},
wg={
"value": 5,
"bounds": (0.0, np.inf)
},
wi={
"value": 30,
"bounds": (0.0, np.inf)
},
tau={
"value": 1e-2,
"bounds": (0.0, np.inf)
},
center={
"value": 0.0,
"fixed": True
},
)
widthG = "wg * q**2"
widthI = "wg * q**2 + wi * q**2 / (1 + wi * q**2 * tau)"
p.update(**kwargs)
m = Model(p, name)
m.addComponent(
Component(r"$\mathcal{L}_g$",
lorentzian,
scale="beta * a0",
width=widthG))
m.addComponent(
Component(
r"$\mathcal{L}_i$",
lorentzian,
scale="beta * (1 - a0)",
width=widthI,
))
return m
def modelWater(q, name="waterDynamics", **kwargs):
"""A model containing a delta, a Lorentzian for translational
motions, a Lorentzian for rotational motions, and a
background term.
Parameters
----------
q : np.ndarray
Array of values for momentum transfer q.
name : str
Name for the model
kwargs : dict
Additional arguments to pass to Parameters.
Can override default parameter attributes.
"""
p = Parameters(
a0={
"value": 0.33,
"bounds": (0.0, np.inf)
},
at={
"value": 0.33,
"bounds": (0.0, np.inf)
},
ar={
"value": 0.33,
"bounds": (0.0, np.inf)
},
wt={
"value": 5,
"bounds": (0.0, np.inf)
},
wr={
"value": 15,
"bounds": (0.0, np.inf)
},
center={
"value": 0.0,
"fixed": True
},
msd={
"value": 1.0,
"bounds": (0.0, np.inf)
},
bkgd={
"value": np.zeros_like(q) + 0.001,
"bounds": (0.0, np.inf)
},
)
p.update(**kwargs)
m = Model(p, name)
m.addComponent(
Component(
"EISF",
delta,
scale="np.exp(-q**2 * msd) * "
"(a0 + ar * spherical_jn(0, 0.96 * q)**2)",
))
m.addComponent(
Component(
r"$\mathcal{L}_r$",
rotations,
scale="np.exp(-q**2 * msd) * ar",
width="wr",
))
m.addComponent(
Component(
r"$\mathcal{L}_t$",
lorentzian,
scale="np.exp(-q**2 * msd) * at",
width="wt * q**2",
))
m.addComponent(Component("background", linear, True, a=0.0, b="bkgd"))
return m
def modelGeneralizedLorentzian(q,
name="GeneralizedLorentzian",
qWise=True,
**kwargs):
"""A model containing a delta and a generalized lorentzian.
This model has been described elsewhere [#]_.
Parameters
----------
q : np.ndarray
Array of values for momentum transfer q.
name : str
Name for the model
kwargs : dict
Additional arguments to pass to Parameters.
Can override default parameter attributes.
References
----------
.. [#] https://doi.org/10.1063/1.5121703
"""
if qWise:
alpha = {"value": np.zeros_like(q) + 0.5, "bounds": (0.0, 1)}
tau = {"value": np.zeros_like(q) + 0.1, "bounds": (0.0, np.inf)}
tauExpr = "tau"
else:
alpha = {"value": 0.5, "bounds": (0.0, 1)}
tau = {"value": 0.1, "bounds": (0.0, np.inf)}
tauExpr = "(tau * q ** 2)**(-1 / alpha)"
p = Parameters(
scale={
"value": np.zeros_like(q) + 1,
"bounds": (0.0, np.inf)
},
a={
"value": np.zeros_like(q) + 0.1,
"bounds": (0.0, 1)
},
center={
"value": 0.0,
"fixed": True
},
alpha=alpha,
tau=tau,
)
p.update(**kwargs)
m = Model(p, name)
m.addComponent(Component("EISF", delta, scale="scale * a"))
m.addComponent(
Component(
"$\\mathcal{L}_{\\alpha, \\tau}$",
generalizedLorentzian,
scale="scale * (1 - a)",
tau=tauExpr,
))
return m