Important Formulas¶

pyTEMlib

a pycroscopy ecosystem package

Notebook by

Gerd Duscher and Darel Pates

Materials Science & Engineering
Joint Institute of Advanced Materials
The University of Tennessee, Knoxville

Collection of usefull Formulas

%matplotlib widget
import matplotlib.pylab as plt
import numpy  as np
import scipy 

import pyTEMlib
pyTEMlib.__version__

'0.2025.12.0'

The formula for relativistic corrected wavelength is: $\lambda = \frac{h}{\sqrt{2m_e E_{kin} *(1+\frac{E_{kin}}{2 m_e c^2})}}$

with

h: Planck constant
m_e: rest mass of electron
$E_{kin}$ : kinetic energy of electron (after acceleration)
c: speed of light

# --- Input  ----
e0 = 60000  # acceleration voltage in eV
# ---------------

def get_wavelength(e0: float) -> float:
    """get deBroglie wavelength of electron accelerated by energy e0 (in eV) 

    Parameters
    ----------
    e0 : float
        acceleration voltage in eV
    Returns
    -------
    float
        wavelength in meters
    
    """
    e_kin = scipy.constants.e * e0
    m_e = scipy.constants.m_e
    c = scipy.constants.c
    h = scipy.constants.h
    return h / np.sqrt(2 * m_e * e_kin * (1 + e_kin / (2 * m_e * c**2)))

h = get_wavelength(e0)  
print(f"wavelength is {h*1e12:.2f} pm for  {e0/1000:.0f} keV")

wavelength is 4.87 pm for  60 keV

help (get_wavelength)

Help on function get_wavelength in module __main__:

get_wavelength(e0: float) -> float
    get deBroglie wavelength of electron accelerated by energy e0 (in eV)

    Parameters
    ----------
    e0 : float
        acceleration voltage in eV
    Returns
    -------
    float
        wavelength in meters

Depth of Focus¶

The depth of focus is

f_{depth} \approx \frac{\lambda}{\alpha^2}

(1)

wavelength of the electron $\lambda$
convergence angle $\alpha$

# --- Input  ----
e0 = 200000   # acceleration voltage in eV
convergence_angle = 10 / 1000  # convergence angle in radians
# ---------------

def depth_of_focus(acceleration_voltage: float, convergence_angle: float) -> float:
    """calculate depth of focus

    Parameters
    ----------
    acceleration_voltage : float
        acceleration voltage in eV
    convergence_angle : float
        convergence angle in radians

    Returns
    -------
    float
        depth of focus in meters
    """

    wavelength = get_wavelength(acceleration_voltage)
    return wavelength / convergence_angle**2

focus_depth = depth_of_focus(e0, convergence_angle)  
print(f"Depth of focus is {focus_depth*1e9:.2f} nm for convergence angle {convergence_angle*1000:.1f} mrad at {e0/1000:.0f} keV")

Depth of focus is 25.08 nm for convergence angle 10.0 mrad at 200 keV

Particle Flux and Current¶

It is important todetermine the order of magitude of how many electrons are hitting the sample.

The defition of an Ampere:

A = \frac{C}{s}

(2)

That definition is enough to calculate the number of electrons per time unit (flux).

# ------- Input  ----
current = 150e-12  # current in Ampere
# ---------------


def current_to_number_of_electrons(current: float) -> float:
    """convert current in Ampere to number of electrons per second

    Parameters
    ----------
    current : float
        current in Ampere

    Returns
    -------
    float
        number of electrons per second
    """
    return current / scipy.constants.elementary_charge

number_of_electrons = current_to_number_of_electrons(current)

print(f"Number of electrons per second for {current*1e12:.1f} pA is {number_of_electrons:.2e} electrons / second")

print(f'\n at {current*1e12:.1f} pAan electron will hit the sample every {scipy.constants.elementary_charge/current * 1e9:.2f} ns ')

Number of electrons per second for 150.0 pA is 9.36e+08 electrons / second

 at 150.0 pAan electron will hit the sample every 1.07 ns