Electron de Broglie wavelength

In an experiment of thoughts the 3D piece of metal shall be reduced in the z-direction to only a few nanometres, comparable with the electronic de Broglie wavelength (t4 = k), whereas in x- and y-direction it is kept infinite a 2D quantum well is formed. Compared with the former 3D situation, the electrons in x-and y-direction can still freely move in these directions, but not in the z-direction. Electrons in this direction are confined like in a box. The states are quantized, whereas in x- and y-direction the situation does not differ from that in the 3D case. Figure 5a shows the quantized situation in z-direction with well-defined Ak values n— I, 2, etc., and Figure 5b indicates the Ak values close to 0. 

The circumference of the orbit, 2 nr for a particular state n, must be equal to some integer number n of electron de Broglie wavelengths ... 

Band gap engineetring confined hetetrostruciutres. When the thickness of a crystalline film is comparable with the de Broglie wavelength, the conduction and valence bands will break into subbands and as the thickness increases, the Fermi energy of the electrons oscillates. This leads to the so-called quantum size effects, which had been precociously predicted in Russia by Lifshitz and Kosevich (1953). A piece of semiconductor which is very small in one, two or three dimensions - a confined structure - is called a quantum well, quantum wire or quantum dot, respectively, and much fundamental physics research has been devoted to these in the last two decades. However, the world of MSE only became involved when several quantum wells were combined into what is now termed a heterostructure. 

So, despite the very small diameter of the MWCNT with respeet to the de Broglie wavelengths of the charge carriers, the cylindrical structure of the honeycomb lattice gives rise to a 2D electron gas for both weak localisation and UCF effects. Indeed, both the amplitude and the temperature dependence of the conductance fluctuations were found to be consistent with the universal conductance fluctuations models for mesoscopic 2D systems applied to the particular cylindrical structure of MWCNTs [10]. 

The surface sensitivity of most electron probe techniques is due to the fact that the penetration depth of electrons into metals falls to a minimum of 4 to 20 A when their kinetic energy is between 10 and 500 eV. It is also convenient that electrons at these energies have de Broglie wavelengths on the order of angstroms. With a monochromatic beam, it is possible to do LEED. 

Figure 5. Electrons in a two-dimensional system, (a) The z-direction is reduced to an extension corresponding to the de Broglie wavelength of free electrons resulting in discrete AA values, (b) Instead of a quasi-continuous situation, a so-called staircase results, (c) In X- and y-direction the situation corresponds to that in Figure 4a. (Reprinted from Ref [5], 2004, with permission from Wiley-VCH.)...

The only size-related quantity for the incident electron is its de Broglie wavelength,... 

Intermolecular vibration involving nearest-neighbor molecules are easily excited in liquid hydrocarbons, since the de Broglie wavelength of the electron at a few tenths of electron-volt energy is comparable to the intermolecular separation. The quantum for this vibration lies in the far IR and can be observed indirectly by Raman spectra. Raman shifts in many hydrocarbon liquids have... 

Although the De Broglie wavelength of free electrons is 0.1 nm, the value of an electron in a small crystallite can be much larger because the effective mass of electrons in a small particle is considerably smaller. Energy levels evolve from HOMO and LUMO to those of clusters, Q-sized particles, and finally bulk semiconductor. Figure 7.7 shows the energy levels in bulk- and Q-sized particulate semiconductors. 

The electronic structure and hence optical properties of nanomaterials depend on the core size. For example, nanoparticles of core size >3 nm show surface plasmon resonance, which is due to the excitation of surface plasmons of nanoparticles by light. When the size of gold nanoparticles comes down to around 1 nm, which is equal to the de Broglie wavelength of the conduction electrons, the electronic bands... 

The second view is based on the uncertainty relation between coordinate and momentum. In the region of a classically allowed barrier, the de Broglie wavelength of an electron is... 

The coherence length of the electron beam is determined by the de Broglie wavelength of the electrons, XdBr and the degree of velocity monochromaticity of the electron beam, Av/v ... 

See also de Broglie Wavelength Electron Theory and Quantum Number... 

---