LOG#233. Electron microscopes.

Surprise! Second post today. It is a nice post, I believe.

Usually, we see the world using photons in certain wavelengths. Our eyes can see only a very limited width of the electromagnetic spectrum. The quantum revolution taught us that we can use other particles (and other wavelengths) to see the world and the Universe in ways we could have never ever imagined. This fact is even more general and can be thought valid even for gravitational waves (bunches of gravitons!).

Electron microscopy first from wikispaces… There are several types of electron microscopes:

1st. Transmission electrom microscopy. The original form of the electron microscope, the transmission electron microscope (TEM), uses a high voltage electron beam to illuminate the specimen and create an image. From wikipedia, The resolution of TEMs is limited primarily by spherical aberration, but a new generation of hardware correctors can reduce spherical aberration to increase the resolution in high-resolution transmission electron microscopy (HRTEM) to below 0.5 angstrom (50 picometres), enabling magnifications above 50 million times. The ability of HRTEM to determine the positions of atoms within materials is useful for nano-technologies research and development. TEM consists of an emission source or cathode, which may be a tungsten filament or needle, or a lanthanum hexaboride LaB_6. Cryo-TEM is the cryoscopic modification of TEM in order to do EM for biology and precision TEM imaging. Samples cooled to cryogenic temperatures and embedded in an environment of vitreous water allows useful biological studies, and it deserved a Nobel Prize in 2017, to Jacques Dubochet, Joachim Frank, and Richard Henderson “for developing cryo-electron microscopy for the high-resolution structure determination of biomolecules in solution.

2nd. Scanning electron microscopy (SEM). The scanning electron microscope (SEM) is a type of electron microscope that produces images of a sample by scanning the surface with a focused beam of electrons. The electrons interact with atoms in the sample, producing various signals that contain information about the surface topography and composition of the sample. The electron beam is scanned in a raster scan pattern, and the position of the beam is combined with the intensity of the detected signal to produce an image. In the most common SEM mode, secondary electrons emitted by atoms excited by the electron beam are detected using an Everhart-Thornley detector. The number of secondary electrons that can be detected, and thus the signal intensity, depends, among other things, on specimen topography. SEM can achieve resolution better than 1 nanometer. It can also made cryoscopic, as Wikipedia says: “(…)Scanning electron cryomicroscopy (CryoSEM) is a form of electron microscopy where a hydrated but cryogenically fixed sample is imaged on a scanning electron microscope‘s cold stage in a cryogenic chamber. The cooling is usually achieved with liquid nitrogen. CryoSEM of biological samples with a high moisture content can be done faster with fewer sample preparation steps than conventional SEM. In addition, the dehydration processes needed to prepare a biological sample for a conventional SEM chamber create numerous distortions in the tissue leading to structural artifacts during imaging(…)”.

3rd. Serial-section electron microscopy (ssEM). One application of TEM is serial-section electron microscopy (ssEM), for example in analyzing the connectivity in volumetric samples of brain tissue by imaging many thin sections in sequence.

4th. Reflection electron microscopy (REM). In the reflection electron microscope (REM) as in the TEM, an electron beam is incident on a surface but instead of using the transmission (TEM) or secondary electrons (SEM), the reflected beam of elastically scattered electrons is detected. This technique is typically coupled with reflection high energy electron diffraction (RHEED) and reflection high-energy loss spectroscopy (RHELS). Another variation is spin-polarized low-energy electron microscopy (SPLEEM), which is used for looking at the microstructure of magnetic domains.

Non-relativistic electrons have a kinetic energy

(1)   \begin{equation*} E_k=\dfrac{1}{2}mv^2\end{equation*}

where m=m_e=9.11\cdot 10^{-31}kg\sim 10^{-30}kg= 1 y\mu g. Any electron that is accelerated by a voltage \Delta V change its kinetic energy in a conservative way, so

(2)   \begin{equation*}\boxed{ \Delta E_k=-\Delta E_p=-q_e\Delta V}\end{equation*}

where the electron charge is

    \[q_e=e=-1.6\cdot 10^{-19}\]

Suppose that initially v_0=0m/s and V_0=0V, V_f=V. Then, the final kinetic energy reads

(3)   \begin{equation*}E_k(f)=\dfrac{1}{2}mv^2_f=\dfrac{p^2}{2m_2}=eV\end{equation*}

where p=mv is the non-relativistic linear momentum. In the quantum realm, any particle like the electron has an associated wave and wavelength. It is the de Broglie wavelength. And it reads

(4)   \begin{equation*}\boxed{\lambda_{db}=\dfrac{h}{mv}}\end{equation*}

Using the energy equation above, you can derive that

(5)   \begin{equation*} \boxed{p=\sqrt{2m_e eV}}\end{equation*}

and thus, you can derive the equation of the (non-relativistic) electron microscopy

(6)   \begin{equation*}\boxed{\lambda_e=\dfrac{h}{\sqrt{2m_eeV}}}\end{equation*}

Indeed, you can generalize this equation to microscope of X-particles, where X-particles are particles with mass m_X and electric charger q_X=Ze as follows:

(7)   \begin{equation*}\boxed{\lambda_X=\dfrac{h}{\sqrt{2m_XZeV}}}\end{equation*}

Good!!!! Now, some numerology. You can use the value of the Planck constant as roughly h\approx 6.63\cdot 10^{-34}J\cdot s, and then you can write

(8)   \begin{equation*}\boxed{\lambda_e=\dfrac{12.25\cdot 10^{-10}m}{\sqrt{V}}=\dfrac{1.225nm}{\sqrt{V}}}\end{equation*}

(9)   \begin{equation*}\boxed{\lambda_X=\dfrac{12.25\cdot 10^{-10}m}{\sqrt{N_XZ_XV}}=\dfrac{1.225nm}{\sqrt{N_XZ_XV}}}\end{equation*}

and where we wrote m_X=N_Xm_e and q_X=Z_Xe. If, in particular, the energy is given in eV (electron-volts), then you get

Example 1. \lambda_e=3.88pm at E=100keV.
Example 2. \lambda_e=2.74pm at E=200keV.
Example 3. \lambda_e=2.24pm at E=300keV.
Example 4. \lambda_e=1.23pm at E=1MeV.

Imagine muon microscopy, tau particle microscopy or W boson microscopy…

Now, you can enter the special relativistic electron microscope realm. Just as you have for photons (or any massless particle) E=pc and \lambda=\dfrac{hc}{E}, for any relativistic particle with mass M=m\gamma and energy E=Mc^2=m\gamma c^2 (rest mass E_0=mc^2), the kinetic energy reads E_k=T=E-mc^2=mc^2(\gamma-1), since E=T+mc^2. Again, for a conservative force set-up, \Delta E_c=-\Delta E_p=-q\Delta V=+eV. Taking into account that

    \[\gamma=\dfrac{1}{\sqrt{1-\frac{v^2}{c^2}}}\]

the special relativity theory generalizes the de Broglie relationship (indeed, de Broglie himself used SR in his wave-particle duality!)

    \[\lambda=\dfrac{h}{m\gamma v}=\dfrac{h}{mv}\sqrt{1-\frac{v^2}{c^2}}\]

From E=mc^2\gamma you get

    \[1-\dfrac{v^2}{c^2}=\left(\dfrac{mc^2}{E}\right)^2\]

From p^2=m^2\gamma^2v^2=\dfrac{m^2v^2}{1-\frac{v^2}{c^2}} you obtain algebraically

    \[p^2=\dfrac{1}{c^2}\left(E^2-(mc^2)^2\right)\]

and

(10)   \begin{equation*}\boxed{p=\sqrt{2Tm\left(1+\dfrac{T}{2mc^2}\right)}}\end{equation*}

Inserting this momentum into the relatistic de Broglie wavelength you finally derive the full relativistic electron microscope equation

(11)   \begin{equation*}\boxed{\lambda=\dfrac{h}{p}=\dfrac{h}{\sqrt{2Tm\left(1+\frac{T}{2mc^2}\right)}}}\end{equation*}

or inserting T as eV units, you also get equivalently

(12)   \begin{equation*}\boxed{\lambda=\dfrac{h}{p}=h\left[2m_eeV\left(1+\dfrac{eV}{2mc^2}\right)\right]^{-1/2}}\end{equation*}

Some numbers:

(13)   \begin{equation*}\begin{vmatrix}\mbox{Voltage}\; V(kV)& \lambda_{nr}(nm)& \lambda_r(nm)& mass (\cdot m_e)& v(\cdot 10^8ms^{-1})\\ 100& 0.00386& 0.00370& 1.196& 1.644\\ 200& 0.00274& 0.00251& 1.391& 2.086\\ 400& 0.00193& 0.00164& 1.783& 2.484\\ 1000& 0.00122& 0.00087& 2.957& 2.823\end{vmatrix}\end{equation*}

and where \lambda_{nr} is the non-relativistic wavelength and \lambda_r is the full relativistic wavelength. They are linked through the following expressions:

    \[\lambda_{nr}=\dfrac{h}{\sqrt{2m_eeV}}\]

    \[\Xi=\dfrac{1}{\sqrt{\left(1+\frac{T}{2mc^2}\right)}}\]

    \[\lambda_{r}=\lambda_{nr}\Xi\]

With simple scaling rules, you can extended the relativistic electron microscopes to relativistic X-particle microscopes as follows

(14)   \begin{equation*}\boxed{\lambda=\dfrac{h}{p}=h\left[2m_XZ_XeV\left(1+\dfrac{Z_XeV}{2m_Xc^2}\right)\right]^{-1/2}}\end{equation*}

Definition 1 (Electron microscope). \lambda_e=\dfrac{h}{\sqrt{2m_eeV}}.

Definition 2 (X-particle microscope). \lambda_X=\dfrac{h}{\sqrt{2m_XZeV}}.

Definition 3 (Electron microscope(II)). \lambda_e=\dfrac{12.25\cdot 10^{-10}m}{\sqrt{V}}=\dfrac{1.225nm}{\sqrt{V}}.

Definition 4 (X-particle microscope(II)). \lambda_e=\dfrac{12.25\cdot 10^{-10}m}{\sqrt{N_XZ_XV}}=\dfrac{1.225nm}{\sqrt{N_XZ_XV}}.

Definition 5 (Relativistic electron microscope). \lambda=\dfrac{h}{p}=h\left[2m_eeV\left(1+\dfrac{eV}{2mc^2}\right)\right]^{-1/2}.

Definition 6 (Relativistic X-particle microscope). \lambda=\dfrac{h}{p}=h\left[2m_XZ_XeV\left(1+\dfrac{Z_XeV}{2m_Xc^2}\right)\right]^{-1/2}.

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