Line profile of metal element fluorescent X-rays based on the effect of metal

X-ray fluorescence profile

Recently, the central energy, intensity ratio, type, and natural width of fluorescent X-rays can be easily obtained by using xraylib. However, in practice, it is necessary to take into account the effect of widening the line profile due to the effect of metal. For example, the Kalpha line of Mn should end with the intensity ratio of Kalpha1 and 2 being 1: 2, but 7 or 8 Voigts The resolution of the device cannot be evaluated correctly unless the function is inserted and fitted. Some typical line profile parameters and simple generation methods are summarized in python.

If you want to model fit the measured spectrum, please refer to How to fit multiple voigt functions with python by inputting responses.

code

The Lineprofile class generates the Voigt function by taking into account the energy, strength, natural width, and device resolution of the line. In the mymodel class, it is made into a function called rawfunc so that the tail on the low energy side can be calculated if necessary.

plot_manylines_wletail_qiita.py

#!/usr/bin/env python
__version__= '1.0'

import matplotlib.pyplot as plt
plt.rcParams['font.family'] = 'serif'
import numpy as np
import scipy.special

def mymodel(x,params, consts=[], tailonly = False):
    norm,gw,gain,P_tailfrac,P_tailtau,bkg1,bkg2 = params    
    # norm : nomarlizaion
    # gw : sigma of gaussian
    # gain : gain of the spectrum 
    # P_tailfrac : fraction of tail 
    # P_tailtau : width of the low energy tail
    # bkg1 : constant of background
    # bkg2 : linearity of background    
    initparams = [norm,gw,gain,bkg1,bkg2]
    def rawfunc(x): # local function, updated when mymodel is called 
        return Lineprofile(x,initparams,consts=consts)               
    model_y = smear(rawfunc, x, P_tailfrac, P_tailtau, tailonly=tailonly)
    return model_y

def Lineprofile(xval,params,consts=[]):
    norm,gw,gain,bkg1,bkg2 = params
    # norm : normalization 
    # gw : sigma of the gaussian 
    # gain : if gain changes
    # consttant facter if needed 
    prob = (amp * lgamma) / np.sum(amp * lgamma) # probabilites for each lines. 
    model_y = 0 
    if len(consts) == 0:
        consts = np.ones(len(energy))
    else:
        consts = consts

    for i, (ene,lg,pr,con) in enumerate(zip(energy,lgamma,prob,consts)):
        voi = voigt(xval,[ene*gain,lg*0.5,gw])
        model_y += norm * con * pr * voi

    background = bkg1 * np.ones(len(xval)) + (xval - np.mean(xval)) * bkg2
    model_y = model_y + background
    # print "bkg1,bkg2 = ", bkg1,bkg2, background
    return model_y

def voigt(xval,params):
    center,lw,gw = params
    # center : center of Lorentzian line
    # lw : HWFM of Lorentzian (half-width at half-maximum (HWHM))
    # gw : sigma of the gaussian 
    z = (xval - center + 1j*lw)/(gw * np.sqrt(2.0))
    w = scipy.special.wofz(z)
    model_y = (w.real)/(gw * np.sqrt(2.0*np.pi))
    return model_y

def smear(rawfunc, x, P_tailfrac, P_tailtau, tailonly = False):
    if P_tailfrac <= 1e-5:
        return rawfunc(x)

    dx = x[1] - x[0]
    freq = np.fft.rfftfreq(len(x), d=dx)
    rawspectrum = rawfunc(x)
    ft = np.fft.rfft(rawspectrum)
    if tailonly:
        ft *= P_tailfrac * (1.0 / (1 - 2j * np.pi * freq * P_tailtau) - 0)
    else:
        ft += ft * P_tailfrac * (1.0 / (1 - 2j * np.pi * freq * P_tailtau) - 1)

    smoothspectrum = np.fft.irfft(ft, n=len(x))
    if tailonly:
        pass
    else:
        smoothspectrum[smoothspectrum < 0] = 0
    return smoothspectrum    

class aline:
    def __init__(self,x,y,name):
        self.x = x
        self.y = y
        self.name = name

# global variables
gfwhm = 2
gw = gfwhm / 2.35
norm = 500000.0
gain = 1.0
bkg1 = 1.0
bkg2 = 0.0
P_tailfrac = 1e-6 # no tail 
P_tailtau = 10
nbin=1000
ewidth=500
init_params=[norm,gw,gain,P_tailfrac,P_tailtau,bkg1,bkg2]

linelist = []


#################################################################
name="Ti Kalpha"
energy = np.array((4510.918, 4509.954, 4507.763, 4514.002, 4504.910, 4503.088))
lgamma = np.array((1.37, 2.22, 3.75, 1.70, 1.88, 4.49))
amp = np.array((4549, 626, 236, 143, 2034, 54))
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))

#################################################################
name="Ti KBeta"
energy = np.array((25.37, 30.096, 31.967, 35.59)) + 4900
lgamma = np.array((16.3, 4.25, 0.42, 0.47))
amp = np.array((199, 455, 326, 19.2))
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))


#################################################################
name="V Kalpha"
energy = np.array((4952.237, 4950.656, 4948.266, 4955.269, 4944.672, 4943.014))
lgamma = np.array((1.45, 2.00, 1.81, 1.76, 2.94, 3.09))
amp = np.array((25832, 5410, 1536, 956, 12971, 603))
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))

#################################################################
name="V KBeta"
energy = np.array((18.19, 24.50, 26.992)) + 5400
lgamma = np.array((18.86, 5.48, 2.499))
amp = np.array((258, 236, 507))
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))


#################################################################
name="Cr Kalpha"
energy = 5400 + np.array([14.874, 14.099, 12.745, 10.583, 18.304, 5.551, 3.986])
lgamma = np.array([1.457, 1.760, 3.138, 5.149, 1.988, 2.224, 4.4740])
amp = np.array([882, 237, 85, 45, 15, 386, 36])
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))

#################################################################
name="Cr KBeta"
energy = 5900 + np.array((47.00, 35.31, 46.24, 42.04, 44.93))
lgamma = np.array([1.70, 15.98, 1.90, 6.69, 3.37])
amp = np.array([670, 55, 337, 82, 151])
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))


#################################################################
name="Mn Kalpha"
energy = 5800 + np.array((98.853, 97.867, 94.829, 96.532, 99.417, 102.712, 87.743, 86.495))
lgamma = np.array([1.715, 2.043, 4.499, 2.663, 0.969, 1.553, 2.361, 4.216])
amp = np.array([790, 264, 68, 96, 71, 10, 372, 100])
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))

#################################################################
name="Mn KBeta"
energy = 6400 + np.array((90.89, 86.31, 77.73, 90.06, 88.83))
lgamma = np.array((1.83, 9.40, 13.22, 1.81, 2.81))
amp = np.array([608, 109, 77, 397, 176])
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))


#################################################################
name="Fe Kalpha"
energy = np.array((6404.148, 6403.295, 6400.653, 6402.077, 6391.190, 6389.106, 6390.275))
lgamma = np.array((1.613, 1.965, 4.833, 2.803, 2.487, 2.339, 4.433))
amp = np.array([697, 376, 88, 136, 339, 60, 102])
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))

#################################################################
name="Fe Kbeta"
energy = np.array((7046.90, 7057.21, 7058.36, 7054.75))
lgamma = np.array((14.17, 3.12, 1.97, 6.38))
amp = np.array([107, 448, 615, 141])
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))


#################################################################
name="Co Kalpha"
energy = np.array((6930.425, 6929.388, 6927.676, 6930.941, 6915.713, 6914.659, 6913.078))
lgamma = np.array((1.795, 2.695, 4.555, 0.808, 2.406, 2.773, 4.463))
amp = np.array((809, 205, 107, 41, 314, 131, 43))
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))

#################################################################
name="Co Kbeta"
energy = np.array((7649.60, 7647.83, 7639.87, 7645.49, 7636.21, 7654.13))
lgamma = np.array((3.05, 3.58, 9.78, 4.89, 13.59, 3.79))
amp = np.array((798, 286, 85, 114, 33, 35))
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))


#################################################################
name="Ni Kalpha"
energy = np.array((7478.281, 7476.529, 7461.131, 7459.874, 7458.029))
lgamma = np.array((2.013, 4.711, 2.674, 3.039, 4.476))
amp = np.array((909, 136, 351, 79, 24))
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))

#################################################################
name="Ni Kbeta"
energy = np.array((8265.01, 8263.01, 8256.67, 8268.70))
lgamma = np.array((3.76, 4.34, 13.70, 5.18))
amp = np.array((722, 358, 89, 104))
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))


#################################################################
name="Cu Kalpha"
energy = np.array([8047.8372, 8045.3672, 8027.9935, 8026.5041])
lgamma = np.array([2.285, 3.358, 2.667, 3.571])
amp = np.array([957, 90, 334, 111])
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))

#################################################################
name="Cu KBeta"
energy = np.array([8905.532, 8903.109, 8908.462, 8897.387, 8911.39])
lgamma = np.array([3.52, 3.52, 3.55, 8.08, 5.31])
amp = np.array([757, 388, 171, 68, 55])
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))


#################################################################
name="As Kalpha"
energy = np.array((10543.2674,10507.50))
lgamma = np.array((3.08, 3.17))
amp = np.array((1.00, 0.51))
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))

#################################################################
name="As KBeta"
energy = np.array((11725.73,11719.86))
lgamma = np.array((2.09+2.25, 2.09+2.25))
amp = np.array((0.13, 0.06))
xmin=np.mean(energy) - ewidth
xmax=np.mean(energy) + ewidth
x = np.linspace(xmin,xmax,nbin)
model_y = mymodel(x,init_params)
linelist.append(aline(x,model_y,name))


plt.figure(figsize=(10,8))
plt.title("Line Profiles")

for oneline in linelist:

    plt.xlabel("Energy (eV)")
    plt.plot(oneline.x, oneline.y, '-', label = oneline.name)
    plt.legend(numpoints=1, frameon=False, loc="best")
    plt.grid(linestyle='dotted',alpha=0.5)

plt.savefig("linelist.png ")
plt.show()
 

How to use

If you simply execute it, such a figure will be generated.

If you want to insert the tail on the low energy side of the device, make the part of P_tailfrac = 1e-6 # no tail a large number.

The global variables energy, lgamma, and amp are prepared and rewritten in an evil way, but to be correct, it is better not to manage the information that becomes the database integrally with the code, and it is proper for each line. It is better to create an object.