.. index:: IsoRotDiff
This fitting function models the dynamic structure factor for a particle undergoing continuous and isotropic rotational diffusion [1].
S(Q,E) = Height \cdot \big(j_0(Q\cdot Radius)^2 \delta (E-Centre) + \sum_{l=1}^N (2l+1)j_l(Q\cdot Radius)^2 \frac{1}{\pi} \frac{\Gamma_l}{\Gamma_l^2+(E-Centre)^2}\big)
\Gamma_l = l(l+1)\hbar/Tau
where:
- Height - Intensity scaling, a fit parameter
- N - Maximum number of components, an attribute (non-fitting)
- Q - Momentum transfer, an attribute (non-fitting)
- Radius - Radius of rotation, a fit parameter
- Centre - Centre of peak, a fit parameter
- Tau - Relaxation time, inverse of the rotational diffusion coefficient, a fit parameter
Because of the spherical symmetry of the problem, the structure factor is expressed in terms of the j_l(z) spherical Bessel functions.
.. attributes::
Q (double, default=0.3) Momentum transfer
N (integer, default=25) The default N=25 assures normalization condition
\int_{-\infty}^{\infty}S(Q,E)dE \equiv 1 with three significant digits for Q\cdot Radius<20, a comfortable upper bound for the vast majority of QENS data.
[1] |
|
Example - Global fit to a synthetic signal:
The signal is modeled by the convolution of a resolution function with the structure factor of a rotator. The resolution is modeled as a normal distribution. We insert a random noise in the rotator. Finally, we choose a linear background noise. The goal is to find out the radius of the rotator. the relaxation time, and the overal intensity of the signal with a fit to the following model:
S(Q,E) = R(Q,E) \otimes IsoRotDiff(Q,E) + (a+bE)
.. testcode:: ExampleIsoRotDiff import numpy as np try: from scipy.special import spherical_jn def sjn(n, z): return spherical_jn(range(n+1), z) except ImportError: from scipy.special import sph_jn def sjn(n, z): return sph_jn(n, z)[0] """Generate resolution function with the following properties: 1. Gaussian in Energy 2. Dynamic range = [-0.1, 0.1] meV with spacing 0.0004 meV 3. FWHM = 0.005 meV """ E0=0.0012 dE=0.0004 FWHM=0.005 sigma = FWHM/(2*np.sqrt(2*np.log(2))) dataX = np.arange(-0.1,0.1,dE) nE=len(dataX) rdataY = np.exp(-0.5*(dataX/sigma)**2) # the resolution function Qs = np.array([0.3, 0.5, 0.7, 0.9, 1.1, 1.3, 1.5, 1.9]) # Q-values nQ = len(Qs) resolution=CreateWorkspace(np.tile(dataX,nQ), np.tile(rdataY,nQ), NSpec=nQ, UnitX="deltaE", VerticalAxisUnit="MomentumTransfer", VerticalAxisValues=Qs) """Generate the signal of a particle undergoing isotropic rotational diffusion. 1. Radius of rotation = 0.45 Angstroms 2. Relaxation time = 26 ps 3. linear background noise, up to 10% of the inelastic intensity 4. Up to 10% of noise in the quasi-elastic signal """ R=0.45; tau=26; hbar=0.658211626 # units of hbar are ps*meV N=25 # number of harmonics in the inelastic signal qdataY=np.empty(0) # will hold all Q-values (all spectra) H=2-np.random.random() # global intensity for Q in Qs: centre=0 # some shift along the energy axis dataY=np.zeros(nE) js=sjn(N,Q*R) # spherical bessel functions from L=0 to L=N for L in range(1,N+1): HWHM = L*(L+1)*hbar/tau aL = (2*L+1)*js[L]**2 dataY += H*aL/np.pi * HWHM/(HWHM**2+(dataX-centre)**2) # add inelastic component dataY = dE*np.convolve(rdataY, dataY, mode="same") # convolve with resolution noise = dataY*np.random.random(nE)*0.1 # noise is up to 10% of the elastic signal background = np.random.random()+np.random.random()*dataX # linear background background = (0.1*H*max(dataY)) * (background/max(np.abs(background))) # up to 1% dataY += background dataY += H*(js[0]**2)*np.exp(-0.5*((dataX-centre)/sigma)**2) # the elastic component qdataY=np.append(qdataY, dataY) data=CreateWorkspace(np.tile(dataX,nQ), qdataY, NSpec=nQ, UnitX="deltaE", VerticalAxisUnit="MomentumTransfer", VerticalAxisValues=Qs) """Our model is: S(Q,E) = Convolution(resolution, IsoRotDiff) + LinearBackground We do a global fit (all spectra) to find out the radius, relaxation time, and intensity """ # This is the template fitting model for each spectrum (each Q-value): single_model_template="""(composite=Convolution,FixResolution=true,NumDeriv=true; name=TabulatedFunction,Workspace=resolution,WorkspaceIndex=_WI_,Scaling=1,Shift=0,XScaling=1; (name=IsoRotDiff,NumDeriv=true,Q=_Q_,f0.Height=1,f0.Centre=0,f0.Radius=0.98,f1.Tau=10)); name=LinearBackground,A0=0,A1=0""" # Now create the string representation of the global model (all spectra, all Q-values): global_model="composite=MultiDomainFunction,NumDeriv=true;" wi=0 for Q in Qs: single_model = single_model_template.replace("_Q_", str(Q)) # insert Q-value single_model = single_model.replace("_WI_", str(wi)) # workspace index global_model += "(composite=CompositeFunction,NumDeriv=true,$domains=i;{0});\n".format(single_model) wi+=1 # The Height, Radius, and Tau are the same for all spectra, thus tie them: ties=['='.join(["f{0}.f0.f1.f0.Height".format(wi) for wi in reversed(range(nQ))]), '='.join(["f{0}.f0.f1.f0.Radius".format(wi) for wi in reversed(range(nQ))]), '='.join(["f{0}.f0.f1.f1.Tau".format(wi) for wi in reversed(range(nQ))]) ] global_model += "ties=("+','.join(ties)+')' # tie Radius # Now relate each domain(i.e. spectrum) to each single model domain_model=dict() for wi in range(nQ): if wi == 0: domain_model.update({"InputWorkspace": data.name(), "WorkspaceIndex": str(wi), "StartX": "-0.09", "EndX": "0.09"}) else: domain_model.update({"InputWorkspace_"+str(wi): data.name(), "WorkspaceIndex_"+str(wi): str(wi), "StartX_"+str(wi): "-0.09", "EndX_"+str(wi): "0.09"}) # Invoke the Fit algorithm using global_model and domain_model: output_workspace = "glofit_"+data.name() Fit(Function=global_model, Output=output_workspace, CreateOutput=True, MaxIterations=500, **domain_model) # Extract Height, Radius, and Tau from workspace glofit_data_Parameters, the output of Fit: nparms=0 parameter_ws = mtd[output_workspace+"_Parameters"] for irow in range(parameter_ws.rowCount()): row = parameter_ws.row(irow) if row["Name"]=="f0.f0.f1.f1.Radius": Radius=row["Value"] nparms+=1 elif row["Name"]=="f0.f0.f1.f1.Height": Height=row["Value"] nparms+=1 elif row["Name"]=="f0.f0.f1.f1.Tau": Tau=row["Value"] nparms+=1 if nparms==3: break # We got the three parameters we are interested in # Check nominal and optimal values are within error ranges: if abs(H-Height)/H < 0.1: print("Optimal Height within 10% of nominal value") else: print("Error. Obtained Height= {0} instead of {1}".format(Height,H)) if abs(R-Radius)/R < 0.05: print("Optimal Radius within 5% of nominal value") else: print("Error. Obtained Radius= {0} instead of {1}".format(Radius,R)) if abs(tau-Tau)/tau < 1.0: print("Optimal Tau within 100% of nominal value") else: print("Error. Obtained Tau= {0} instead of {1}".format(Tau,tau))
Output:
.. testoutput:: ExampleIsoRotDiff Optimal Height within 10% of nominal value Optimal Radius within 5% of nominal value Optimal Tau within 100% of nominal value
.. categories::
.. sourcelink::