Orbital angular momentum of free electrons

Last updated February 08, 2024

$Phase (color) and amplitude (brightness) of electron wavefunctions with several values of the orbital angular momentum quantum number m {\displaystyle m} and a Laguerre-Gauss amplitude profile. l = + 1 {\displaystyle \ell =+1} (top left), l = - 1 {\displaystyle \ell =-1} (top right), l = 0 {\displaystyle \ell =0} (lower left) are all eigenstates of the orbital angular momentum operator, while the superposition of l = + 1 {\displaystyle \ell =+1} and l = - 1 {\displaystyle \ell =-1} (lower right) is not. Both of the upper wavefunctions have < L z > [?] 0 {\displaystyle \langle L_{z}\rangle \neq 0} , while the lower wavefunctions have < L z > = 0 {\displaystyle \langle L_{z}\rangle =0} . Laguerre-gauss modes larger text.svg$

Phase (color) and amplitude (brightness) of electron wavefunctions with several values of the orbital angular momentum quantum number

m

and a Laguerre-Gauss amplitude profile.

\ell =+1

(top left),

\ell =-1

(top right),

\ell =0

(lower left) are all eigenstates of the orbital angular momentum operator, while the superposition of

\ell =+1

and

\ell =-1

(lower right) is not. Both of the upper wavefunctions have

\langle L_{z}\rangle \neq 0

, while the lower wavefunctions have

\langle L_{z}\rangle =0

.

Electrons in free space can carry quantized orbital angular momentum (OAM) projected along the direction of propagation.^[1] This orbital angular momentum corresponds to helical wavefronts, or, equivalently, a phase proportional to the azimuthal angle.^[2] Electron beams with quantized orbital angular momentum are also called electron vortex beams.

Theory

An electron in free space travelling at non-relativistic speeds, follows the Schrödinger equation for a free particle, that is

i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,t)={\frac {-\hbar ^{2}}{2m}}\nabla ^{2}\Psi (\mathbf {r} ,t),

where ${\textstyle \hbar }$ is the reduced Planck constant, ${\textstyle \Psi (\mathbf {r} ,t)}$ is the single-electron wave function, ${\textstyle m}$ its mass, ${\textstyle \mathbf {r} }$ the position vector, and ${\textstyle t}$ is time. This equation is a type of wave equation and when written in the Cartesian coordinate system ( ${\textstyle x}$ , ${\textstyle y}$ , ${\textstyle z}$ ), the solutions are given by a linear combination of plane waves, in the form of

\Psi _{\mathbf {p} }(\mathbf {r} ,t)\propto e^{i(\mathbf {p} \cdot \mathbf {r} -E(\mathbf {p} )t)/\hbar }

where ${\textstyle \mathbf {p} }$ is the linear momentum and ${\textstyle E(\mathbf {p} )}$ is the electron energy, given by the usual dispersion relation ${\textstyle E(\mathbf {p} )={\frac {p^{2}}{2m}}}$ . By measuring the momentum of the electron, its wave function must collapse and give a particular value. If the energy of the electron beam is selected beforehand, the total momentum (not its directional components) of the electrons is fixed to a certain degree of precision. When the Schrödinger equation is written in the cylindrical coordinate system ( ${\textstyle \rho }$ , ${\textstyle \theta }$ , ${\textstyle z}$ ), the solutions are no longer plane waves, but instead are given by Bessel beams,^[2] solutions that are a linear combination of

\Psi _{p_{\rho },\,p_{z},\,\ell }(\rho ,\theta ,z)\propto J_{|\ell |}\left({\frac {p_{\rho }\rho }{\hbar }}\right)e^{i(p_{z}z-Et)/\hbar }e^{i\ell \theta },

that is, the product of three types of functions: a plane wave with momentum ${\textstyle p_{z}}$ in the ${\textstyle z}$ -direction, a radial component written as a Bessel function of the first kind ${\textstyle J_{|\ell |}}$ , where ${\textstyle p_{\rho }}$ is the linear momentum in the radial direction, and finally an azimuthal component written as ${\textstyle e^{i\ell \theta }}$ where ${\textstyle \ell }$ (sometimes written ${\textstyle m_{z}}$ ) is the magnetic quantum number related to the angular momentum ${\textstyle L_{z}}$ in the ${\textstyle z}$ -direction. Thus, the dispersion relation reads ${\textstyle E=(p_{z}^{2}+p_{\rho }^{2})/2m}$ . By azimuthal symmetry, the wave function has the property that ${\textstyle \ell =0,\pm 1,\pm 2,\cdots }$ is necessarily an integer, thus ${\textstyle L_{z}=\hbar \ell }$ is quantized. If a measurement of ${\textstyle L_{z}}$ is performed on an electron with selected energy, as ${\textstyle E}$ does not depend on ${\textstyle \ell }$ , it can give any integer value. It is possible to experimentally prepare states with non-zero ${\textstyle \ell }$ by adding an azimuthal phase to an initial state with ${\textstyle \ell =0}$ ; experimental techniques designed to measure the orbital angular momentum of a single electron are under development. Simultaneous measurement of electron energy and orbital angular momentum is allowed because the Hamiltonian commutes with the angular momentum operator related to ${\textstyle L_{z}}$ .

Note that the equations above follow for any free quantum particle with mass, not necessarily electrons. The quantization of $L_{z}$ can also be shown in the spherical coordinate system, where the wave function reduces to a product of spherical Bessel functions and spherical harmonics.

Preparation

There are a variety of methods to prepare an electron in an orbital angular momentum state. All methods involve an interaction with an optical element such that the electron acquires an azimuthal phase. The optical element can be material,^[3]^[4]^[5] magnetostatic,^[6] or electrostatic.^[7] It is possible to either directly imprint an azimuthal phase, or to imprint an azimuthal phase with a holographic diffraction grating, where grating pattern is defined by the interference of the azimuthal phase and a planar^[8] or spherical^[9] carrier wave.

Applications

Electron vortex beams have a variety of proposed and demonstrated applications, including for mapping magnetization,^[4]^[10]^[11]^[12] studying chiral molecules and chiral plasmon resonances,^[13] and identification of crystal chirality.^[14]

Measurement

Interferometric methods borrowed from light optics also work to determine the orbital angular momentum of free electrons in pure states. Interference with a planar reference wave,^[5] diffractive filtering and self-interference^[15]^[16]^[17] can serve to characterize a prepared electron orbital angular momentum state. In order to measure the orbital angular momentum of a superposition or of the mixed state that results from interaction with an atom or material, a non-interferometric method is necessary. Wavefront flattening,^[17]^[18] transformation of an orbital angular momentum state into a planar wave,^[19] or cylindrically symmetric Stern-Gerlach-like measurement^[20] is necessary to measure the orbital angular momentum mixed or superposition state.

Related Research Articles

The quantum Hall effect is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance $R xy$ exhibits steps that take on the quantized values

Matter waves are a central part of the theory of quantum mechanics, being half of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave.

In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the nucleus and electron clouds.

<span class="mw-page-title-main">Azimuthal quantum number</span> Quantum number denoting orbital angular momentum

In quantum mechanics, the azimuthal quantum number is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers that describe the unique quantum state of an electron. It is also known as the orbital angular momentum quantum number, orbital quantum number, subsidiary quantum number, or second quantum number, and is symbolized as $ℓ$ .

In atomic physics, a magnetic quantum number is a quantum number used to distinguish quantum states of an electron or other particle according to its angular momentum along a given axis in space. The orbital magnetic quantum number distinguishes the orbitals available within a given subshell of an atom. It specifies the component of the orbital angular momentum that lies along a given axis, conventionally called the z-axis, so it describes the orientation of the orbital in space. The spin magnetic quantum number $m s$ specifies the z-axis component of the spin angular momentum for a particle having spin quantum number $s$ . For an electron, $s$ is 1⁄2, and $m s$ is either +1⁄2 or −1⁄2, often called "spin-up" and "spin-down", or α and β. The term magnetic in the name refers to the magnetic dipole moment associated with each type of angular momentum, so states having different magnetic quantum numbers shift in energy in a magnetic field according to the Zeeman effect.

A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, n. The higher the value of n, the farther the electron is from the nucleus, on average. Rydberg atoms have a number of peculiar properties including an exaggerated response to electric and magnetic fields, long decay periods and electron wavefunctions that approximate, under some conditions, classical orbits of electrons about the nuclei. The core electrons shield the outer electron from the electric field of the nucleus such that, from a distance, the electric potential looks identical to that experienced by the electron in a hydrogen atom.

In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron resulting from its intrinsic properties of spin and electric charge. The value of the electron magnetic moment is −9.2847647043(28)×10⁻²⁴ J⋅T⁻¹. In units of the Bohr magneton (μ_B), it is −1.00115965218059(13) μ_B, a value that was measured with a relative accuracy of 1.3×10⁻¹³.

In physics, the zitterbewegung (German pronunciation:[ˈtsɪtɐ.bəˌveːɡʊŋ], from German zittern 'to tremble, jitter', and Bewegung 'motion') is the theoretical prediction of a rapid oscillatory motion of elementary particles that obey relativistic wave equations. This prediction was first discussed by Gregory Breit in 1928 and later by Erwin Schrödinger in 1930 as a result of analysis of the wave packet solutions of the Dirac equation for relativistic electrons in free space, in which an interference between positive and negative energy states produces an apparent fluctuation (up to the speed of light) of the position of an electron around the median, with an angular frequency of $2 mc 2 / ℏ$ , or approximately 1.6×10²¹ radians per second.

In quantum physics, the spin–orbit interaction is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is at the origin of magnetocrystalline anisotropy and the spin Hall effect.

An optical vortex is a zero of an optical field; a point of zero intensity. The term is also used to describe a beam of light that has such a zero in it. The study of these phenomena is known as singular optics.

<span class="mw-page-title-main">Optical lattice</span> Atomic-scale structure formed through the Stark shift by opposing beams of light

An optical lattice is formed by the interference of counter-propagating laser beams, creating a spatially periodic polarization pattern. The resulting periodic potential may trap neutral atoms via the Stark shift. Atoms are cooled and congregate at the potential extrema. The resulting arrangement of trapped atoms resembles a crystal lattice and can be used for quantum simulation.

A g-factor is a dimensionless quantity that characterizes the magnetic moment and angular momentum of an atom, a particle or the nucleus. It is essentially a proportionality constant that relates the different observed magnetic moments μ of a particle to their angular momentum quantum numbers and a unit of magnetic moment, usually the Bohr magneton or nuclear magneton. Its value is proportional to the gyromagnetic ratio.

Angle-resolved photoemission spectroscopy (ARPES) is an experimental technique used in condensed matter physics to probe the allowed energies and momenta of the electrons in a material, usually a crystalline solid. It is based on the photoelectric effect, in which an incoming photon of sufficient energy ejects an electron from the surface of a material. By directly measuring the kinetic energy and emission angle distributions of the emitted photoelectrons, the technique can map the electronic band structure and Fermi surfaces. ARPES is best suited for the study of one- or two-dimensional materials. It has been used by physicists to investigate high-temperature superconductors, graphene, topological materials, quantum well states, and materials exhibiting charge density waves.

In physics, a quantum vortex represents a quantized flux circulation of some physical quantity. In most cases, quantum vortices are a type of topological defect exhibited in superfluids and superconductors. The existence of quantum vortices was first predicted by Lars Onsager in 1949 in connection with superfluid helium. Onsager reasoned that quantisation of vorticity is a direct consequence of the existence of a superfluid order parameter as a spatially continuous wavefunction. Onsager also pointed out that quantum vortices describe the circulation of superfluid and conjectured that their excitations are responsible for superfluid phase transitions. These ideas of Onsager were further developed by Richard Feynman in 1955 and in 1957 were applied to describe the magnetic phase diagram of type-II superconductors by Alexei Alexeyevich Abrikosov. In 1935 Fritz London published a very closely related work on magnetic flux quantization in superconductors. London's fluxoid can also be viewed as a quantum vortex.

The Gross–Pitaevskii equation describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.

In quantum mechanics, the Pauli equation or Schrödinger–Pauli equation is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927.

In quantum mechanics, orbital magnetization, M_orb, refers to the magnetization induced by orbital motion of charged particles, usually electrons in solids. The term "orbital" distinguishes it from the contribution of spin degrees of freedom, M_spin, to the total magnetization. A nonzero orbital magnetization requires broken time-reversal symmetry, which can occur spontaneously in ferromagnetic and ferrimagnetic materials, or can be induced in a non-magnetic material by an applied magnetic field.

The angular momentum of light is a vector quantity that expresses the amount of dynamical rotation present in the electromagnetic field of the light. While traveling approximately in a straight line, a beam of light can also be rotating around its own axis. This rotation, while not visible to the naked eye, can be revealed by the interaction of the light beam with matter.

The orbital angular momentum of light (OAM) is the component of angular momentum of a light beam that is dependent on the field spatial distribution, and not on the polarization. It can be further split into an internal and an external OAM. The internal OAM is an origin-independent angular momentum of a light beam that can be associated with a helical or twisted wavefront. The external OAM is the origin-dependent angular momentum that can be obtained as cross product of the light beam position and its total linear momentum.

Electric dipole spin resonance (EDSR) is a method to control the magnetic moments inside a material using quantum mechanical effects like the spin–orbit interaction. Mainly, EDSR allows to flip the orientation of the magnetic moments through the use of electromagnetic radiation at resonant frequencies. EDSR was first proposed by Emmanuel Rashba.

References

↑ Bliokh, Konstantin; Bliokh, Yury; Savel’ev, Sergey; Nori, Franco (November 2007). "Semiclassical Dynamics of Electron Wave Packet States with Phase Vortices". Physical Review Letters. 99 (19): 190404. arXiv: 0706.2486 . Bibcode:2007PhRvL..99s0404B. doi:10.1103/PhysRevLett.99.190404. ISSN 0031-9007. PMID 18233051. S2CID 17918457.
1 2 Bliokh, K. Y.; Ivanov, I. P.; Guzzinati, G.; Clark, L.; Van Boxem, R.; Béché, A.; Juchtmans, R.; Alonso, M. A.; Schattschneider, P.; Nori, F.; Verbeeck, J. (2017-05-24). "Theory and applications of free-electron vortex states". Physics Reports. 690: 1–70. arXiv: 1703.06879 . Bibcode:2017PhR...690....1B. doi:10.1016/j.physrep.2017.05.006. ISSN 0370-1573. S2CID 119067068.Lloyd, S. M.; Babiker, M.; Thirunavukkarasu, G.; Yuan, J. (2017-08-16). "Electron vortices: Beams with orbital angular momentum" (PDF). Reviews of Modern Physics. 89 (3): 035004. Bibcode:2017RvMP...89c5004L. doi:10.1103/RevModPhys.89.035004. S2CID 125753983.
↑ Uchida, Masaya; Tonomura, Akira (2010-04-01). "Generation of electron beams carrying orbital angular momentum". Nature. 464 (7289): 737–9. Bibcode:2010Natur.464..737U. doi:10.1038/nature08904. PMID 20360737. S2CID 4423382.
1 2 Verbeeck, J.; Tian, H.; Schattschneider, P. (2010). "Production and application of electron vortex beams". Nature. 467 (7313): 301–4. Bibcode:2010Natur.467..301V. doi:10.1038/nature09366. PMID 20844532. S2CID 2970408.
1 2 McMorran, Benjamin J.; Agrawal, Amit; Anderson, Ian M.; Herzing, Andrew A.; Lezec, Henri J.; McClelland, Jabez J.; Unguris, John (2011-01-14). "Electron Vortex Beams with High Quanta of Orbital Angular Momentum". Science. 331 (6014): 192–195. Bibcode:2011Sci...331..192M. doi:10.1126/science.1198804. PMID 21233382. S2CID 37753036.
↑ Blackburn, A. M.; Loudon, J. C. (January 2014). "Vortex beam production and contrast enhancement from a magnetic spiral phase plate". Ultramicroscopy. 136: 127–143. doi:10.1016/j.ultramic.2013.08.009. PMID 24128851.Béché, Armand; Van Boxem, Ruben; Van Tendeloo, Gustaaf; Verbeeck, Jo (January 2014). "Magnetic monopole field exposed by electrons". Nature Physics. 10 (1): 26–29. arXiv: 1305.0570 . Bibcode:2014NatPh..10...26B. doi:10.1038/nphys2816. S2CID 17919153.
↑ Pozzi, Giulio; Lu, Peng-Han; Tavabi, Amir H.; Duchamp, Martial; Dunin-Borkowski, Rafal E. (2017-10-01). "Generation of electron vortex beams using line charges via the electrostatic Aharonov-Bohm effect". Ultramicroscopy. 181: 191–196. doi: 10.1016/j.ultramic.2017.06.001 . PMID 28609665.
↑ Grillo, Vincenzo; Gazzadi, Gian Carlo; Karimi, Ebrahim; Mafakheri, Erfan; Boyd, Robert W.; Frabboni, Stefano (2014-01-30). "Highly efficient electron vortex beams generated by nanofabricated phase holograms". Applied Physics Letters. 104 (4): 043109. Bibcode:2014ApPhL.104d3109G. doi:10.1063/1.4863564. S2CID 142215.Harvey, Tyler R.; Pierce, Jordan S.; Agrawal, Amit K.; Ercius, Peter; Linck, Martin; McMorran, Benjamin J. (2014-09-01). "Efficient diffractive phase optics for electrons". New Journal of Physics. 16 (9): 093039. Bibcode:2014NJPh...16i3039H. doi: 10.1088/1367-2630/16/9/093039 .
↑ Saitoh, Koh; Hasegawa, Yuya; Tanaka, Nobuo; Uchida, Masaya (2012-06-01). "Production of electron vortex beams carrying large orbital angular momentum using spiral zone plates". Journal of Electron Microscopy. 61 (3): 171–177. doi: 10.1093/jmicro/dfs036 . PMID 22394576.Verbeeck, J.; Tian, H.; Béché, A. (February 2012). "A new way of producing electron vortex probes for STEM". Ultramicroscopy. 113: 83–87. arXiv: 1405.7222 . doi:10.1016/j.ultramic.2011.10.008. S2CID 54728013.
↑ Idrobo, Juan C.; Pennycook, Stephen J. (2011-10-01). "Vortex beams for atomic resolution dichroism". Journal of Electron Microscopy. 60 (5): 295–300. Bibcode:2011MiMic..17S1296I. doi: 10.1093/jmicro/dfr069 . PMID 21949052.
↑ Lloyd, Sophia; Babiker, Mohamed; Yuan, Jun (2012-02-15). "Quantized Orbital Angular Momentum Transfer and Magnetic Dichroism in the Interaction of Electron Vortices with Matter". Physical Review Letters. 108 (7): 074802. arXiv: 1111.3259 . Bibcode:2012PhRvL.108g4802L. doi:10.1103/PhysRevLett.108.074802. PMID 22401214. S2CID 14016354.
↑ Rusz, Ján; Bhowmick, Somnath (2013-09-06). "Boundaries for Efficient Use of Electron Vortex Beams to Measure Magnetic Properties". Physical Review Letters. 111 (10): 105504. arXiv: 1304.5461 . Bibcode:2013PhRvL.111j5504R. doi:10.1103/PhysRevLett.111.105504. PMID 25166681. S2CID 42498494.
↑ Asenjo-Garcia, A.; García de Abajo, F. J. (2014-08-06). "Dichroism in the Interaction between Vortex Electron Beams, Plasmons, and Molecules". Physical Review Letters. 113 (6): 066102. Bibcode:2014PhRvL.113f6102A. doi:10.1103/PhysRevLett.113.066102. PMID 25148337.Harvey, Tyler R.; Pierce, Jordan S.; Chess, Jordan J.; McMorran, Benjamin J. (2015-07-05). "Demonstration of electron helical dichroism as a local probe of chirality". arXiv: 1507.01810 [cond-mat.mtrl-sci].Guzzinati, Giulio; Béché, Armand; Lourenço-Martins, Hugo; Martin, Jérôme; Kociak, Mathieu; Verbeeck, Jo (2017-04-12). "Probing the symmetry of the potential of localized surface plasmon resonances with phase-shaped electron beams". Nature Communications. 8: 14999. arXiv: 1608.07449 . Bibcode:2017NatCo...814999G. doi:10.1038/ncomms14999. PMC 5394338 . PMID 28401942.
↑ Juchtmans, Roeland; Béché, Armand; Abakumov, Artem; Batuk, Maria; Verbeeck, Jo (2015-03-26). "Using electron vortex beams to determine chirality of crystals in transmission electron microscopy". Physical Review B. 91 (9): 094112. arXiv: 1410.2155 . Bibcode:2015PhRvB..91i4112J. doi:10.1103/PhysRevB.91.094112. S2CID 19753751.
↑ Shiloh, Roy; Tsur, Yuval; Remez, Roei; Lereah, Yossi; Malomed, Boris A.; Shvedov, Vladlen; Hnatovsky, Cyril; Krolikowski, Wieslaw; Arie, Ady (2015-03-04). "Unveiling the Orbital Angular Momentum and Acceleration of Electron Beams". Physical Review Letters. 114 (9): 096102. arXiv: 1402.3133 . Bibcode:2015PhRvL.114i6102S. doi:10.1103/PhysRevLett.114.096102. PMID 25793830. S2CID 6396731.
↑ Clark, L.; Béché, A.; Guzzinati, G.; Verbeeck, J. (2014-05-13). "Quantitative measurement of orbital angular momentum in electron microscopy". Physical Review A. 89 (5): 053818. arXiv: 1403.4398 . Bibcode:2014PhRvA..89e3818C. doi:10.1103/PhysRevA.89.053818. S2CID 45042167.
1 2 Guzzinati, Giulio; Clark, Laura; Béché, Armand; Verbeeck, Jo (2014-02-13). "Measuring the orbital angular momentum of electron beams". Physical Review A. 89 (2): 025803. arXiv: 1401.7211 . Bibcode:2014PhRvA..89b5803G. doi:10.1103/PhysRevA.89.025803. S2CID 19593282.
↑ Saitoh, Koh; Hasegawa, Yuya; Hirakawa, Kazuma; Tanaka, Nobuo; Uchida, Masaya (2013-08-14). "Measuring the Orbital Angular Momentum of Electron Vortex Beams Using a Forked Grating". Physical Review Letters. 111 (7): 074801. arXiv: 1307.6304 . Bibcode:2013PhRvL.111g4801S. doi:10.1103/PhysRevLett.111.074801. PMID 23992070. S2CID 37702862.
↑ McMorran, Benjamin J.; Harvey, Tyler R.; Lavery, Martin P. J. (2017). "Efficient sorting of free electron orbital angular momentum". New Journal of Physics. 19 (2): 023053. arXiv: 1609.09124 . Bibcode:2017NJPh...19b3053M. doi:10.1088/1367-2630/aa5f6f. S2CID 119192171.Grillo, Vincenzo; Tavabi, Amir H.; Venturi, Federico; Larocque, Hugo; Balboni, Roberto; Gazzadi, Gian Carlo; Frabboni, Stefano; Lu, Peng-Han; Mafakheri, Erfan; Bouchard, Frédéric; Dunin-Borkowski, Rafal E.; Boyd, Robert W.; Lavery, Martin P. J.; Padgett, Miles J.; Karimi, Ebrahim (2017-05-24). "Measuring the orbital angular momentum spectrum of an electron beam". Nature Communications. 8: 15536. Bibcode:2017NatCo...815536G. doi:10.1038/ncomms15536. PMC 5458084 . PMID 28537248.
↑ Harvey, Tyler R.; Grillo, Vincenzo; McMorran, Benjamin J. (2017-02-28). "Stern-Gerlach-like approach to electron orbital angular momentum measurement". Physical Review A. 95 (2): 021801. arXiv: 1606.03631 . Bibcode:2017PhRvA..95b1801H. doi:10.1103/PhysRevA.95.021801. S2CID 119086719.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Bliokh, Konstantin; Bliokh, Yury; Savel’ev, Sergey; Nori, Franco (November 2007). "Semiclassical Dynamics of Electron Wave Packet States with Phase Vortices". Physical Review Letters. 99 (19): 190404. arXiv: 0706.2486 . Bibcode:2007PhRvL..99s0404B. doi:10.1103/PhysRevLett.99.190404. ISSN 0031-9007. PMID 18233051. S2CID 17918457.

[Bliokh_2017-2] 1 2 Bliokh, K. Y.; Ivanov, I. P.; Guzzinati, G.; Clark, L.; Van Boxem, R.; Béché, A.; Juchtmans, R.; Alonso, M. A.; Schattschneider, P.; Nori, F.; Verbeeck, J. (2017-05-24). "Theory and applications of free-electron vortex states". Physics Reports. 690: 1–70. arXiv: 1703.06879 . Bibcode:2017PhR...690....1B. doi:10.1016/j.physrep.2017.05.006. ISSN 0370-1573. S2CID 119067068.Lloyd, S. M.; Babiker, M.; Thirunavukkarasu, G.; Yuan, J. (2017-08-16). "Electron vortices: Beams with orbital angular momentum" (PDF). Reviews of Modern Physics. 89 (3): 035004. Bibcode:2017RvMP...89c5004L. doi:10.1103/RevModPhys.89.035004. S2CID 125753983.

[3] Uchida, Masaya; Tonomura, Akira (2010-04-01). "Generation of electron beams carrying orbital angular momentum". Nature. 464 (7289): 737–9. Bibcode:2010Natur.464..737U. doi:10.1038/nature08904. PMID 20360737. S2CID 4423382.

[production_and_application-4] 1 2 Verbeeck, J.; Tian, H.; Schattschneider, P. (2010). "Production and application of electron vortex beams". Nature. 467 (7313): 301–4. Bibcode:2010Natur.467..301V. doi:10.1038/nature09366. PMID 20844532. S2CID 2970408.

[sciencemag.org-5] 1 2 McMorran, Benjamin J.; Agrawal, Amit; Anderson, Ian M.; Herzing, Andrew A.; Lezec, Henri J.; McClelland, Jabez J.; Unguris, John (2011-01-14). "Electron Vortex Beams with High Quanta of Orbital Angular Momentum". Science. 331 (6014): 192–195. Bibcode:2011Sci...331..192M. doi:10.1126/science.1198804. PMID 21233382. S2CID 37753036.

[6] Blackburn, A. M.; Loudon, J. C. (January 2014). "Vortex beam production and contrast enhancement from a magnetic spiral phase plate". Ultramicroscopy. 136: 127–143. doi:10.1016/j.ultramic.2013.08.009. PMID 24128851.Béché, Armand; Van Boxem, Ruben; Van Tendeloo, Gustaaf; Verbeeck, Jo (January 2014). "Magnetic monopole field exposed by electrons". Nature Physics. 10 (1): 26–29. arXiv: 1305.0570 . Bibcode:2014NatPh..10...26B. doi:10.1038/nphys2816. S2CID 17919153.

[7] Pozzi, Giulio; Lu, Peng-Han; Tavabi, Amir H.; Duchamp, Martial; Dunin-Borkowski, Rafal E. (2017-10-01). "Generation of electron vortex beams using line charges via the electrostatic Aharonov-Bohm effect". Ultramicroscopy. 181: 191–196. doi: 10.1016/j.ultramic.2017.06.001 . PMID 28609665.

[8] Grillo, Vincenzo; Gazzadi, Gian Carlo; Karimi, Ebrahim; Mafakheri, Erfan; Boyd, Robert W.; Frabboni, Stefano (2014-01-30). "Highly efficient electron vortex beams generated by nanofabricated phase holograms". Applied Physics Letters. 104 (4): 043109. Bibcode:2014ApPhL.104d3109G. doi:10.1063/1.4863564. S2CID 142215.Harvey, Tyler R.; Pierce, Jordan S.; Agrawal, Amit K.; Ercius, Peter; Linck, Martin; McMorran, Benjamin J. (2014-09-01). "Efficient diffractive phase optics for electrons". New Journal of Physics. 16 (9): 093039. Bibcode:2014NJPh...16i3039H. doi: 10.1088/1367-2630/16/9/093039 .

[9] Saitoh, Koh; Hasegawa, Yuya; Tanaka, Nobuo; Uchida, Masaya (2012-06-01). "Production of electron vortex beams carrying large orbital angular momentum using spiral zone plates". Journal of Electron Microscopy. 61 (3): 171–177. doi: 10.1093/jmicro/dfs036 . PMID 22394576.Verbeeck, J.; Tian, H.; Béché, A. (February 2012). "A new way of producing electron vortex probes for STEM". Ultramicroscopy. 113: 83–87. arXiv: 1405.7222 . doi:10.1016/j.ultramic.2011.10.008. S2CID 54728013.

[10] Idrobo, Juan C.; Pennycook, Stephen J. (2011-10-01). "Vortex beams for atomic resolution dichroism". Journal of Electron Microscopy. 60 (5): 295–300. Bibcode:2011MiMic..17S1296I. doi: 10.1093/jmicro/dfr069 . PMID 21949052.

[11] Lloyd, Sophia; Babiker, Mohamed; Yuan, Jun (2012-02-15). "Quantized Orbital Angular Momentum Transfer and Magnetic Dichroism in the Interaction of Electron Vortices with Matter". Physical Review Letters. 108 (7): 074802. arXiv: 1111.3259 . Bibcode:2012PhRvL.108g4802L. doi:10.1103/PhysRevLett.108.074802. PMID 22401214. S2CID 14016354.

[12] Rusz, Ján; Bhowmick, Somnath (2013-09-06). "Boundaries for Efficient Use of Electron Vortex Beams to Measure Magnetic Properties". Physical Review Letters. 111 (10): 105504. arXiv: 1304.5461 . Bibcode:2013PhRvL.111j5504R. doi:10.1103/PhysRevLett.111.105504. PMID 25166681. S2CID 42498494.

[plasmon_dichroism-13] Asenjo-Garcia, A.; García de Abajo, F. J. (2014-08-06). "Dichroism in the Interaction between Vortex Electron Beams, Plasmons, and Molecules". Physical Review Letters. 113 (6): 066102. Bibcode:2014PhRvL.113f6102A. doi:10.1103/PhysRevLett.113.066102. PMID 25148337.Harvey, Tyler R.; Pierce, Jordan S.; Chess, Jordan J.; McMorran, Benjamin J. (2015-07-05). "Demonstration of electron helical dichroism as a local probe of chirality". arXiv: 1507.01810 [cond-mat.mtrl-sci].Guzzinati, Giulio; Béché, Armand; Lourenço-Martins, Hugo; Martin, Jérôme; Kociak, Mathieu; Verbeeck, Jo (2017-04-12). "Probing the symmetry of the potential of localized surface plasmon resonances with phase-shaped electron beams". Nature Communications. 8: 14999. arXiv: 1608.07449 . Bibcode:2017NatCo...814999G. doi:10.1038/ncomms14999. PMC 5394338 . PMID 28401942.

[14] Juchtmans, Roeland; Béché, Armand; Abakumov, Artem; Batuk, Maria; Verbeeck, Jo (2015-03-26). "Using electron vortex beams to determine chirality of crystals in transmission electron microscopy". Physical Review B. 91 (9): 094112. arXiv: 1410.2155 . Bibcode:2015PhRvB..91i4112J. doi:10.1103/PhysRevB.91.094112. S2CID 19753751.

[15] Shiloh, Roy; Tsur, Yuval; Remez, Roei; Lereah, Yossi; Malomed, Boris A.; Shvedov, Vladlen; Hnatovsky, Cyril; Krolikowski, Wieslaw; Arie, Ady (2015-03-04). "Unveiling the Orbital Angular Momentum and Acceleration of Electron Beams". Physical Review Letters. 114 (9): 096102. arXiv: 1402.3133 . Bibcode:2015PhRvL.114i6102S. doi:10.1103/PhysRevLett.114.096102. PMID 25793830. S2CID 6396731.

[16] Clark, L.; Béché, A.; Guzzinati, G.; Verbeeck, J. (2014-05-13). "Quantitative measurement of orbital angular momentum in electron microscopy". Physical Review A. 89 (5): 053818. arXiv: 1403.4398 . Bibcode:2014PhRvA..89e3818C. doi:10.1103/PhysRevA.89.053818. S2CID 45042167.

[meas._diff._and_wf-17] 1 2 Guzzinati, Giulio; Clark, Laura; Béché, Armand; Verbeeck, Jo (2014-02-13). "Measuring the orbital angular momentum of electron beams". Physical Review A. 89 (2): 025803. arXiv: 1401.7211 . Bibcode:2014PhRvA..89b5803G. doi:10.1103/PhysRevA.89.025803. S2CID 19593282.

[18] Saitoh, Koh; Hasegawa, Yuya; Hirakawa, Kazuma; Tanaka, Nobuo; Uchida, Masaya (2013-08-14). "Measuring the Orbital Angular Momentum of Electron Vortex Beams Using a Forked Grating". Physical Review Letters. 111 (7): 074801. arXiv: 1307.6304 . Bibcode:2013PhRvL.111g4801S. doi:10.1103/PhysRevLett.111.074801. PMID 23992070. S2CID 37702862.

[19] McMorran, Benjamin J.; Harvey, Tyler R.; Lavery, Martin P. J. (2017). "Efficient sorting of free electron orbital angular momentum". New Journal of Physics. 19 (2): 023053. arXiv: 1609.09124 . Bibcode:2017NJPh...19b3053M. doi:10.1088/1367-2630/aa5f6f. S2CID 119192171.Grillo, Vincenzo; Tavabi, Amir H.; Venturi, Federico; Larocque, Hugo; Balboni, Roberto; Gazzadi, Gian Carlo; Frabboni, Stefano; Lu, Peng-Han; Mafakheri, Erfan; Bouchard, Frédéric; Dunin-Borkowski, Rafal E.; Boyd, Robert W.; Lavery, Martin P. J.; Padgett, Miles J.; Karimi, Ebrahim (2017-05-24). "Measuring the orbital angular momentum spectrum of an electron beam". Nature Communications. 8: 15536. Bibcode:2017NatCo...815536G. doi:10.1038/ncomms15536. PMC 5458084 . PMID 28537248.

[20] Harvey, Tyler R.; Grillo, Vincenzo; McMorran, Benjamin J. (2017-02-28). "Stern-Gerlach-like approach to electron orbital angular momentum measurement". Physical Review A. 95 (2): 021801. arXiv: 1606.03631 . Bibcode:2017PhRvA..95b1801H. doi:10.1103/PhysRevA.95.021801. S2CID 119086719.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]