Nuclear magnetic resonance (NMR) is a nuclear spectroscopy that is widely used in all physical sciences and industry. In NMR for probing intrinsic spin properties of atomic nuclei using a large magnet. Like any spectroscopy, it uses electromagnetic radiation (radio frequency waves in the VHF range) to create a transition between energy levels (resonance). In chemistry, NMR helps determine the structure of small molecules. Nuclear magnetic resonance in medicine has found application in magnetic resonance imaging (MRI).

Opening

NMR was discovered in 1946 by Harvard University scientists Purcell, Pound, and Torrey, and Stanford's Bloch, Hansen, and Packard. They noticed that the 1 H and 31 P nuclei (proton and phosphorus-31) are able to absorb radio frequency energy when exposed to a magnetic field, the strength of which is specific to each atom. When absorbed, they began to resonate, each element at its own frequency. This observation allowed a detailed analysis of the structure of the molecule. Since then, NMR has found application in kinetic and structural studies of solids, liquids and gases, resulting in 6 Nobel Prizes.

Spin and magnetic properties

The nucleus is made up of elementary particles called neutrons and protons. They have their own angular momentum, called spin. Like electrons, the spin of a nucleus can be described by quantum numbers I and m in a magnetic field. Atomic nuclei with an even number of protons and neutrons have zero spin, while all others have non-zero. In addition, molecules with non-zero spin have a magnetic moment μ = γ I, where γ is the gyromagnetic ratio, the constant of proportionality between the magnetic dipole moment and the angular moment, which is different for each atom.

The magnetic moment of the core makes it behave like a tiny magnet. In the absence of an external magnetic field, each magnet is randomly oriented. During the NMR experiment, the sample is placed in an external magnetic field B 0 , which causes the low energy bar magnets to align in the direction of B 0 and the high energy in the opposite direction. In this case, the orientation of the spin of the magnets changes. To understand this rather abstract concept, one must consider the energy levels of the nucleus during an NMR experiment.

Energy levels

A spin flip requires an integer number of quanta. For any m, there are 2m + 1 energy levels. For a nucleus with spin 1/2 there are only 2 of them - low, occupied by spins aligned with B 0 , and high, occupied by spins directed against B 0 . Each energy level is defined by E = -mℏγВ 0 , where m is the magnetic quantum number, in this case +/- 1/2. The energy levels for m > 1/2, known as quadrupole nuclei, are more complex.

The energy difference between the levels is: ΔE = ℏγB 0 , where ℏ is Planck's constant.

As can be seen, the strength of the magnetic field is of great importance, since in its absence the levels degenerate.

Energy transitions

For nuclear magnetic resonance to occur, a spin flip must occur between energy levels. The energy difference between the two states corresponds to the energy of electromagnetic radiation, which causes the nuclei to change their energy levels. For most NMR spectrometers At 0 it has the order of 1 Tesla (T), and γ - 10 7 . Therefore, the required electromagnetic radiation is of the order of 10 7 Hz. The photon energy is represented by the formula E = hν. Therefore, the frequency required for absorption is: ν= γВ 0 /2π.

Nuclear shielding

The physics of NMR is based on the concept of nuclear shielding, which makes it possible to determine the structure of matter. Each atom is surrounded by electrons that revolve around the nucleus and act on its magnetic field, which in turn causes small changes in energy levels. This is called shielding. Nuclei that experience different magnetic fields associated with local electronic interactions are called non-equivalent. Changing the energy levels for a spin flip requires a different frequency, which creates a new peak in the NMR spectrum. Screening allows the structural determination of molecules by analyzing the NMR signal using the Fourier transform. The result is a spectrum consisting of a set of peaks, each corresponding to a different chemical environment. The peak area is directly proportional to the number of nuclei. Detailed structure information is retrieved by NMR interactions, which change the spectrum in different ways.

Relaxation

Relaxation refers to the phenomenon of returning nuclei to their thermodynamically stable after excitation to higher energy levels of the state. In this case, the energy absorbed during the transition from a lower level to a higher one is released. This is a rather complex process that takes place in different time frames. The two most widespread relaxation types are spin-lattice and spin-spin.

To understand relaxation, it is necessary to consider the entire sample. If the nuclei are placed in an external magnetic field, they will create bulk magnetization along the Z axis. Their spins are also coherent and allow the signal to be detected. NMR shifts the bulk magnetization from the Z axis to the XY plane, where it manifests itself.

Spin-lattice relaxation is characterized by the time T 1 required to recover 37% of the bulk magnetization along the Z axis. The more efficient the relaxation process, the smaller T 1 . In solids, since the movement between molecules is limited, the relaxation time is long. Measurements are usually carried out by pulse methods.

Spin-spin relaxation is characterized by the loss of mutual coherence T 2 . It may be less than or equal to T 1 .

Nuclear magnetic resonance and its applications

The two main areas in which NMR has proved extremely important are medicine and chemistry, but new applications are being developed every day.

Nuclear magnetic resonance imaging, more commonly known as magnetic resonance imaging (MRI), is important medical diagnostic tool used to study the functions and structure of the human body. It allows you to get detailed images of any organ, especially soft tissues, in all possible planes. Used in the areas of cardiovascular, neurological, musculoskeletal and oncological imaging. Unlike alternative computed tomography, magnetic resonance imaging does not use ionizing radiation, therefore it is completely safe.

MRI can detect subtle changes that occur over time. MRI imaging can be used to identify structural abnormalities that occur during the course of the disease, how they affect subsequent development, and how their progression correlates with the mental and emotional aspects of the disorder. Since MRI does not visualize bone well, excellent intracranial and intravertebral content.

Principles of using nuclear magnetic resonance in diagnostics

During an MRI procedure, the patient lies inside a massive hollow cylindrical magnet and is exposed to a powerful, stable magnetic field. Different atoms in the scanned part of the body resonate at different frequencies of the field. MRI is used primarily to detect vibrations of hydrogen atoms, which contain a rotating proton nucleus with a small magnetic field. In MRI, the background magnetic field lines up all the hydrogen atoms in the tissue. The second magnetic field, whose orientation differs from that of the background, turns on and off many times per second. At a certain frequency, the atoms resonate and line up with the second field. When it turns off, the atoms bounce back, aligning with the background. This creates a signal that can be received and converted into an image.

Tissues with a large amount of hydrogen, which is present in the human body in the composition of water, creates a bright image, and with a small amount or absence of it (for example, bones) look dark. The brightness of the MRI is enhanced by a contrast agent such as gadodiamide, which patients take before the procedure. Although these agents can improve image quality, the sensitivity of the procedure remains relatively limited. Techniques are being developed to increase the sensitivity of MRI. The most promising is the use of parahydrogen, a form of hydrogen with unique molecular spin properties that is very sensitive to magnetic fields.

Improvements in the performance of the magnetic fields used in MRI have led to the development of highly sensitive imaging modalities such as diffusion and functional MRI, which are designed to display very specific tissue properties. In addition, a unique form of MRI technology called magnetic resonance angiography is used to image the movement of blood. It allows visualization of arteries and veins without the need for needles, catheters or contrast agents. As with MRI, these techniques have helped revolutionize biomedical research and diagnostics.

Advanced computer technology has allowed radiologists to create three-dimensional holograms from digital sections obtained by MRI scanners, which serve to determine the exact location of lesions. Tomography is especially valuable in examining the brain and spinal cord, as well as pelvic organs such as the bladder, and cancellous bone. The method allows you to quickly and clearly accurately determine the extent of tumor damage and assess the potential damage from a stroke, allowing doctors to prescribe the appropriate treatment in a timely manner. MRI has largely supplanted arthrography, the need to inject a contrast agent into a joint to visualize cartilage or ligament damage, and myelography, the injection of a contrast agent into the spinal canal to visualize disorders of the spinal cord or intervertebral disc.

Application in chemistry

In many laboratories today, nuclear magnetic resonance is used to determine the structures of important chemical and biological compounds. In NMR spectra, different peaks provide information about the specific chemical environment and bonds between atoms. Most widespread the isotopes used to detect magnetic resonance signals are 1 H and 13 C, but many others are suitable, such as 2 H, 3 He, 15 N, 19 F, etc.

Modern NMR spectroscopy has found wide application in biomolecular systems and plays an important role in structural biology. With the development of methodology and tools, NMR has become one of the most powerful and versatile spectroscopic methods for the analysis of biomacromolecules, which makes it possible to characterize them and their complexes up to 100 kDa in size. Together with X-ray crystallography, this is one of the two leading technologies for determining their structure at the atomic level. In addition, NMR provides unique and important information about the functions of a protein, which plays a critical role in drug development. Some of the applications NMR spectroscopy are listed below.

• This is the only method for determining the atomic structure of biomacromolecules in aqueous solutions in close to physiological conditions or membrane-simulating media.
• Molecular dynamics. This is the most powerful method for quantitative determination of the dynamic properties of biomacromolecules.
• Protein folding. NMR spectroscopy is the most powerful tool for determining the residual structures of unfolded proteins and folding mediators.
• The state of ionization. The method is effective in determining the chemical properties of functional groups in biomacromolecules, such as ionization states of ionizable groups of enzyme active sites.
• Nuclear magnetic resonance makes it possible to study weak functional interactions between macrobiomolecules (for example, with dissociation constants in the micromolar and millimolar ranges), which cannot be done using other methods.
• Protein hydration. NMR is a tool for detecting internal water and its interaction with biomacromolecules.
• It's unique direct interaction detection method hydrogen bonds.
• Screening and drug development. In particular, nuclear magnetic resonance is particularly useful in identifying drugs and determining the conformations of compounds associated with enzymes, receptors, and other proteins.
• native membrane protein. Solid state NMR has the potential determination of atomic structures of membrane protein domains in the environment of the native membrane, including those with bound ligands.
• Metabolic analysis.
• Chemical analysis. Chemical identification and conformational analysis of synthetic and natural chemicals.
• Materials Science. A powerful tool in the study of polymer chemistry and physics.

Other uses

Nuclear magnetic resonance and its applications are not limited to medicine and chemistry. The method has proven to be very useful in other areas as well, such as environmental testing, the oil industry, process control, NMR of the Earth's field, and magnetometers. Non-destructive testing saves on expensive biological samples that can be reused if more testing is needed. Nuclear magnetic resonance in geology is used to measure the porosity of rocks and the permeability of underground fluids. Magnetometers are used to measure various magnetic fields.

1. The essence of the phenomenon

   First of all, it should be noted that although the word “nuclear” is present in the name of this phenomenon, NMR has nothing to do with nuclear physics and has nothing to do with radioactivity. If we talk about a strict description, then one cannot do without the laws of quantum mechanics. According to these laws, the interaction energy of a magnetic core with an external magnetic field can take only a few discrete values. If magnetic nuclei are irradiated with an alternating magnetic field, the frequency of which corresponds to the difference between these discrete energy levels, expressed in frequency units, then the magnetic nuclei begin to move from one level to another, while absorbing the energy of the alternating field. This is the phenomenon of magnetic resonance. This explanation is formally correct, but not very clear. There is another explanation, without quantum mechanics. The magnetic core can be thought of as an electrically charged ball rotating around its axis (although, strictly speaking, this is not the case). According to the laws of electrodynamics, the rotation of a charge leads to the appearance of a magnetic field, i.e., the magnetic moment of the nucleus, which is directed along the axis of rotation. If this magnetic moment is placed in a constant external field, then the vector of this moment begins to precess, i.e., rotate around the direction of the external field. In the same way, the spinning wheel axis precesses (rotates) around the vertical, if it is unwound not strictly vertically, but at a certain angle. In this case, the role of the magnetic field is played by the gravitational force.

   The precession frequency is determined both by the properties of the nucleus and by the strength of the magnetic field: the stronger the field, the higher the frequency. Then, if, in addition to a constant external magnetic field, an alternating magnetic field acts on the nucleus, then the nucleus begins to interact with this field - it, as it were, swings the nucleus more strongly, the precession amplitude increases, and the nucleus absorbs the energy of the alternating field. However, this will occur only under the condition of resonance, i.e., the coincidence of the precession frequency and the frequency of the external alternating field. It looks like a classic example from high school physics - soldiers marching across a bridge. If the step frequency coincides with the natural frequency of the bridge, then the bridge sways more and more. Experimentally, this phenomenon manifests itself in the dependence of the absorption of an alternating field on its frequency. At the moment of resonance, the absorption increases sharply, and the simplest magnetic resonance spectrum looks like this:

   

2. Fourier spectroscopy

   The first NMR spectrometers worked exactly as described above - the sample was placed in a constant magnetic field, and RF radiation was continuously applied to it. Then either the frequency of the alternating field or the intensity of the constant magnetic field changed smoothly. The energy absorption of the alternating field was recorded by a radio frequency bridge, the signal from which was output to a recorder or an oscilloscope. But this method of signal registration has not been used for a long time. In modern NMR spectrometers, the spectrum is recorded using pulses. The magnetic moments of the nuclei are excited by a short powerful pulse, after which a signal is recorded, which is induced in the RF coil by freely precessing magnetic moments. This signal gradually decreases to zero as the magnetic moments return to equilibrium (this process is called magnetic relaxation). The NMR spectrum is obtained from this signal using a Fourier transform. This is a standard mathematical procedure that allows you to decompose any signal into frequency harmonics and thus obtain the frequency spectrum of this signal. This method of recording the spectrum allows you to significantly reduce the noise level and conduct experiments much faster.

   

   One excitation pulse to record the spectrum is the simplest NMR experiment. However, there can be many such pulses, of different durations, amplitudes, with different delays between them, etc., in the experiment, depending on what kind of manipulations the researcher needs to perform with the system of nuclear magnetic moments. However, almost all of these pulse sequences end in the same thing - recording a free precession signal followed by a Fourier transform.

3. Magnetic interactions in matter

   In itself, magnetic resonance would remain nothing more than an interesting physical phenomenon, if it were not for the magnetic interactions of nuclei with each other and with the electron shell of the molecule. These interactions affect the resonance parameters, and with their help, NMR can be used to obtain a variety of information about the properties of molecules - their orientation, spatial structure (conformation), intermolecular interactions, chemical exchange, rotational and translational dynamics. Thanks to this, NMR has become a very powerful tool for studying substances at the molecular level, which is widely used not only in physics, but mainly in chemistry and molecular biology. An example of one of these interactions is the so-called chemical shift. Its essence is as follows: the electron shell of the molecule responds to an external magnetic field and tries to shield it - partial shielding of the magnetic field occurs in all diamagnetic substances. This means that the magnetic field in the molecule will differ from the external magnetic field by a very small amount, which is called the chemical shift. However, the properties of the electron shell in different parts of the molecule are different, and the chemical shift is also different. Accordingly, the resonance conditions for nuclei in different parts of the molecule will also differ. This makes it possible to distinguish chemically nonequivalent nuclei in the spectrum. For example, if we take the spectrum of hydrogen nuclei (protons) of pure water, then there will be only one line in it, since both protons in the H 2 O molecule are exactly the same. But for methyl alcohol CH 3 OH, there will already be two lines in the spectrum (if we neglect other magnetic interactions), since there are two types of protons - protons of the methyl group CH 3 and a proton associated with an oxygen atom. As the molecules become more complex, the number of lines will increase, and if we take such a large and complex molecule as a protein, then in this case the spectrum will look something like this:

   

4. Magnetic cores

   NMR can be observed on different nuclei, but it must be said that not all nuclei have a magnetic moment. It often happens that some isotopes have a magnetic moment, while other isotopes of the same nucleus do not. In total, there are more than a hundred isotopes of various chemical elements that have magnetic nuclei, but no more than 1520 magnetic nuclei are usually used in research, everything else is exotic. Each nucleus has its own characteristic ratio of the magnetic field and the precession frequency, called the gyromagnetic ratio. For all nuclei these ratios are known. Using them, you can choose the frequency at which, for a given magnetic field, a signal from the nuclei that the researcher needs will be observed.

   The most important nuclei for NMR are protons. They are most abundant in nature, and they have a very high sensitivity. For chemistry and biology, the nuclei of carbon, nitrogen and oxygen are very important, but scientists were not very lucky with them: the most common isotopes of carbon and oxygen, 12 C and 16 O, do not have a magnetic moment, the natural nitrogen isotope 14 N has a moment, but it for a number of reasons it is very inconvenient for experiments. There are isotopes 13 C, 15 N and 17 O that are suitable for NMR experiments, but their natural abundance is very low and the sensitivity is very low compared to protons. Therefore, special isotopically enriched samples are often prepared for NMR studies, in which the natural isotope of one or another nucleus is replaced by the one needed for experiments. In most cases, this procedure is very difficult and expensive, but sometimes it is the only way to get the necessary information.

5. Electron paramagnetic and quadrupole resonance

   Speaking of NMR, one cannot fail to mention two other related physical phenomena - electron paramagnetic resonance (EPR) and nuclear quadrupole resonance (NQR). EPR is essentially similar to NMR, the difference lies in the fact that the resonance is observed on the magnetic moments not of atomic nuclei, but of the electron shell of the atom. EPR can be observed only in those molecules or chemical groups whose electron shell contains the so-called unpaired electron, then the shell has a non-zero magnetic moment. Such substances are called paramagnets. EPR, like NMR, is also used to study various structural and dynamic properties of substances at the molecular level, but its scope is much narrower. This is mainly due to the fact that most molecules, especially in living nature, do not contain unpaired electrons. In some cases, it is possible to use a so-called paramagnetic probe, i.e. a chemical group with an unpaired electron that binds to the molecule under study. But this approach has obvious drawbacks that limit the possibilities of this method. In addition, in EPR there is no such high spectral resolution (ie, the ability to distinguish one line from another in the spectrum) as in NMR.

   It is most difficult to explain the nature of NQR "on the fingers". Some nuclei have a so-called electric quadrupole moment. This moment characterizes the deviation of the distribution of the electric charge of the nucleus from spherical symmetry. The interaction of this moment with the gradient of the electric field created by the crystalline structure of the substance leads to the splitting of the energy levels of the nucleus. In this case, resonance can be observed at a frequency corresponding to transitions between these levels. Unlike NMR and EPR, NQR does not require an external magnetic field, since level splitting occurs without it. NQR is also used to study substances, but its scope is even narrower than that of EPR.

6. Advantages and disadvantages of NMR

   

   NMR is the most powerful and informative method for studying molecules. Strictly speaking, this is not one method, but a large number of different types of experiments, i.e., pulse sequences. Although they are all based on the NMR phenomenon, but each of these experiments is designed to obtain some specific specific information. The number of these experiments is measured by many tens, if not hundreds. Theoretically, NMR can, if not everything, then almost everything that all other experimental methods for studying the structure and dynamics of molecules can, although in practice this is, of course, far from always feasible. One of the main advantages of NMR is that, on the one hand, its natural probes, i.e., magnetic nuclei, are distributed throughout the molecule, and, on the other hand, it makes it possible to distinguish these nuclei from each other and obtain spatially selective data on properties of the molecule. Almost all other methods provide information either averaged over the entire molecule, or only about one of its parts.

   There are two main disadvantages of NMR. First, this is a low sensitivity compared to most other experimental methods (optical spectroscopy, fluorescence, EPR, etc.). This leads to the fact that in order to average the noise, the signal must be accumulated for a long time. In some cases, the NMR experiment can be carried out for even several weeks. Secondly, it is its high cost. NMR spectrometers are among the most expensive scientific instruments, costing at least hundreds of thousands of dollars, with the most expensive spectrometers costing several million. Not all laboratories, especially in Russia, can afford to have such scientific equipment.

7. Magnets for NMR spectrometers

   One of the most important and expensive parts of a spectrometer is the magnet, which creates a constant magnetic field. The stronger the field, the higher the sensitivity and spectral resolution, so scientists and engineers are constantly trying to get the highest possible fields. The magnetic field is created by an electric current in the solenoid - the stronger the current, the greater the field. However, it is impossible to increase the current indefinitely; at a very high current, the solenoid wire will simply begin to melt. Therefore, superconducting magnets, i.e., magnets in which the solenoid wire is in the superconducting state, have been used for a very long time for high-field NMR spectrometers. In this case, the electrical resistance of the wire is zero, and no energy is released at any current value. The superconducting state can only be obtained at very low temperatures, just a few degrees Kelvin - this is the temperature of liquid helium. (High-temperature superconductivity is still only a matter of purely fundamental research.) It is with the maintenance of such a low temperature that all the technical difficulties in the design and production of magnets are connected, which cause their high cost. The superconducting magnet is built on the principle of a thermos matryoshka. The solenoid is in the center, in the vacuum chamber. It is surrounded by a shell containing liquid helium. This shell is surrounded by a shell of liquid nitrogen through a vacuum layer. The temperature of liquid nitrogen is minus 196 degrees Celsius, nitrogen is needed so that helium evaporates as slowly as possible. Finally, the nitrogen shell is isolated from room temperature by an outer vacuum layer. Such a system is able to maintain the desired temperature of the superconducting magnet for a very long time, although this requires regular pouring of liquid nitrogen and helium into the magnet. The advantage of such magnets, in addition to the ability to obtain high magnetic fields, is also that they do not consume energy: after the start of the magnet, the current runs through the superconducting wires with virtually no loss for many years.

8. Tomography

   In conventional NMR spectrometers, they try to make the magnetic field as uniform as possible, this is necessary to improve the spectral resolution. But if the magnetic field inside the sample, on the contrary, is made very inhomogeneous, this opens up fundamentally new possibilities for using NMR. The inhomogeneity of the field is created by the so-called gradient coils, which are paired with the main magnet. In this case, the magnitude of the magnetic field in different parts of the sample will be different, which means that the NMR signal can be observed not from the entire sample, as in a conventional spectrometer, but only from its narrow layer, for which resonance conditions are met, i.e., the desired ratio of magnetic field and frequency. By changing the magnitude of the magnetic field (or, which is essentially the same thing, the frequency of observing the signal), you can change the layer that will give the signal. Thus, it is possible to "scan" the sample throughout its volume and "see" its internal three-dimensional structure without destroying the sample in any mechanical way. To date, a large number of techniques have been developed that make it possible to measure various NMR parameters (spectral characteristics, magnetic relaxation times, self-diffusion rate, and some others) with spatial resolution inside a sample. The most interesting and important, from a practical point of view, the use of NMR tomography was found in medicine. In this case, the "sample" being examined is the human body. NMR imaging is one of the most effective and safe (but also expensive) diagnostic tools in various fields of medicine, from oncology to obstetrics. It is curious to note that doctors do not use the word "nuclear" in the name of this method, because some patients associate it with nuclear reactions and the atomic bomb.

9. Discovery history

   

   The year of the discovery of NMR is considered to be 1945, when the Americans Felix Bloch from Stanford and independently Edward Parcell and Robert Pound from Harvard first observed the NMR signal on protons. By that time, much was already known about the nature of nuclear magnetism, the NMR effect itself was theoretically predicted, and several attempts were made to observe it experimentally. It is important to note that a year earlier in the Soviet Union, in Kazan, the EPR phenomenon was discovered by Evgeny Zavoisky. It is now well known that Zavoisky also observed the NMR signal, this was before the war, in 1941. However, he had a poor quality magnet with poor field uniformity at his disposal, the results were poorly reproducible and therefore remained unpublished. In fairness, it should be noted that Zavoisky was not the only one who observed NMR before its "official" discovery. In particular, the American physicist Isidore Rabi (Nobel Prize winner in 1944 for the study of the magnetic properties of nuclei in atomic and molecular beams) also observed NMR in the late 1930s, but considered this to be an instrumental artifact. One way or another, but our country remains a priority in the experimental detection of magnetic resonance. Although Zavoisky himself soon after the war began to deal with other problems, his discovery for the development of science in Kazan played a huge role. Kazan is still one of the world's leading research centers for EPR spectroscopy.

10. Nobel Prizes in Magnetic Resonance

    In the first half of the 20th century, several Nobel Prizes were awarded to scientists without whose work the discovery of NMR could not have taken place. Among them are Peter Szeeman, Otto Stern, Isidor Rabi, Wolfgang Pauli. But there were four Nobel Prizes directly related to NMR. In 1952, Felix Bloch and Edward Purcell received the prize for the discovery of NMR. This is the only "NMR" Nobel Prize in physics. In 1991, the prize in chemistry was awarded to the Swiss Richard Ernst, who worked at the famous ETH Zurich. He was awarded it for the development of multidimensional NMR spectroscopy methods, which made it possible to radically increase the information content of NMR experiments. In 2002, the prize winner, also in chemistry, was Kurt Wüthrich, who worked with Ernst in neighboring buildings at the same Technical School. He received the award for developing methods for determining the three-dimensional structure of proteins in solution. Prior to this, the only method that allowed determining the spatial conformation of large biomacromolecules was only X-ray diffraction analysis. Finally, in 2003, the American Paul Lauterbur and the Englishman Peter Mansfield received the Medical Prize for the invention of NMR imaging. The Soviet discoverer of the EPR E.K. Zavoisky, alas, did not receive the Nobel Prize.

Quantum electromagnetic resonator

Quantum electromagnetic resonator (QER) (Quantum Electromagnetic Resonator) is a closed topological object in three-dimensional space, in the general case, a ‘’cavity’’ of arbitrary shape, which has a certain ‘’surface’’ with a certain ‘’thickness’’. In contrast to the classical case, there are no “electromagnetic waves” and radiation losses in it, but there are “endless” oscillations of the phase-shifted electromagnetic field, which follow from the quantum properties of QER.

Background

It so happened historically that physical reactive quantities such as capacitance and inductance were practically not considered not only in quantum, but even in classical theoretical electrodynamics. The fact is that the latter are not explicitly included in the system of Maxwell equations, as a result of which electromagnetic fields were always obtained, and if sometimes in the solutions obtained there were dimensional coefficients that could be associated with capacitance or inductance, then the relation to them was appropriate. It is no less known that the "field approach" leads to the appearance of "bad infinities", due to the consideration of the motion of a "mathematical point" (with an electric charge) under the influence of force fields. The generally recognized quantum electrodynamics did not escape the "bad infinities", within the framework of which powerful methods of "compensation of bad infinities" were also developed.

On the contrary, in applied physics, the concept of capacitance and inductance has found wide application, first in electrical engineering, and then in radio electronics. The main result of the application of reactive parameters in applied physics is today the widespread use of information technologies, which are based on the generation, reception and transmission of electromagnetic waves at different frequencies. At the same time, the lack of development at the theoretical level of physical concepts for capacitance and inductance today is already becoming, to a certain extent, a limiting factor in the development of information technologies in general and quantum computing in particular. It suffices to recall that the quantum consideration of the classical mechanical oscillator was implemented in the era of the creation of quantum mechanics (as one of the illustrations of its practical application), while the quantum consideration of the contour was theoretically posed only in the early 70s of the 20th century, and a detailed consideration began only in the mid 90s.

For the first time, the need to solve the Schrödinger equation for a quantum circuit was posed in the monograph Louisell (1973) . Since at that time there was still no understanding of what quantum reactive parameters were (and there were no practical examples then), this approach was not widely used. The theoretically correct introduction of quantum capacitance, which was based on the density of states, was first introduced by Luria (1988) when considering the quantum Hall effect (QHE). Unfortunately, quantum inductances, which also followed from the density of states, were not introduced at that time, and therefore a full consideration of the quantum reactive oscillator did not occur even then. A year later, Yakimaha (1989) considered an example of a series-parallel connection of quantum circuits (or rather, their impedances) in explaining QHE (integer and fractional). But in this paper, the physical nature of these quantum reactive parameters was not considered, and the quantum Schrödinger equation for the reactive oscillator was not considered either. For the first time, a simultaneous consideration of all quantum reactive parameters was carried out in the work of Yakimahi (1994), during spectroscopic studies of MIS transistors at low frequencies (sound range). Flat quantum capacitances and inductances here had a thickness equal to the Compton wavelength of an electron, and the characteristic resistance was equal to the wave resistance of vacuum. Three years later, Devoret (1997) presented a complete theory of the quantum reactive oscillator (applied to the Josephson effect). The application of quantum reactive parameters in quantum computing is covered in Devoret (2004) .

Classic electromagnetic resonator

In the general case, the classical electromagnetic resonator (CLER) is cavity in 3D space. Therefore, CLER has an infinite number of resonant frequencies due to the three-dimensionality of space. For example, a rectangular Clair has the following resonant frequencies:

where ; respectively width, thickness and length, dielectric constant, relative permeability, magnetic constant, relative susceptibility. In contrast to the classical LC circuit, in the CER the electric and magnetic fields are located in the same volume of space. These oscillating electromagnetic fields in the classical case form electromagnetic waves, which can be radiated to the outside world outside the resonator. Today, CLARE are widely used in the radio frequency range of waves (centimeters and decimeters). Moreover, CLAE is also used in quantum electronics, which deals with monochrome light waves.

quantum approach

Quantum LC Circuit

In classical physics we have the following correspondence relations between mechanical and electrodynamic physical parameters:

magnetic inductance and mechanical weight:

;

electrical capacity and reverse elasticity:

;

electric charge and coordinate offset:

.

Quantum momentum operator in charge space can be presented in the following form:

where is the reduced Planck constant, is the complex conjugate momentum operator. Hamilton operator in charge space can be presented as:

where is the complex conjugate charge operator, and resonant frequency. Consider the case without energy dissipation (). The only difference between charge space and the traditional 3D coordinate space is its one-dimensionality (1D). The Schrödinger equation for a quantum LC circuit can be defined as:

To solve this equation, it is necessary to introduce the following dimensionless variables:

where massive charge. Then the Schrödinger equation takes the form of the Chebyshev-Hermite differential equation:

The eigenvalues ​​for the Hamilton operator will be:

where for we will have zero oscillations:

In general scale charge can be rewritten in the form:

where is the fine structure constant. It's obvious that scale charge differs from the "metallurgical" charge of the electron. Moreover, its quantization will look like:

.

Resonator as quantum LC circuit

Luria's approach, using the density of energy states (DOS), gives the following definition for quantum capacitance:

and quantum inductance:

where is the surface area of ​​the resonator, and PES in two-dimensional space (2D), electric charge (or flux), and magnetic charge (or flux). It should be noted that these streams will be defined later with additional conditions.

The energy accumulated on the quantum capacitance:

Energy stored on quantum inductance:

Resonator angular frequency:

Law of energy conservation:

This equation can be rewritten as:

from which it can be seen that these "charges" are actually "field flows", and not "metallurgical charges".

Characteristic impedance of the resonator:

where is the quantum of the magnetic flux.

From the equations above, we can find the following values ​​for the electric and magnetic field fluxes:

It is necessary to remind once again that these quantities are not "metallurgical charges", but the maximum amplitude values ​​of the field fluxes, which maintain the energy balance between the energy of the resonator oscillations and the total energy on the capacitance and inductance.

EVOLUTION OF ELECTROMAGNETIC RESONATORS

The resonator can maintain periodic oscillations caused by an external pulse for a long time. The resonator has frequency selectivity with respect to external harmonic effects: the amplitude of its oscillations is maximum at the resonant frequency and decreases with distance from it. Oscillations in electromagnetic resonators represent the mutual transformation of electric and magnetic fields. Resonators are widely used in radio engineering devices, being an integral part of many amplifiers, most generators, receivers, frequency filters and frequency meters.

The simplest electromagnetic resonator is an (oscillating LC circuit. It is easy to establish that the electrical energy is generated in the capacitor, and the magnetic reserve is created in the inductor. The transition of energy from the electric field to the magnetic field is accompanied by the spatial movement of energy from the capacitor to the inductance. The dimensions of the circuit must be small compared with the wavelength.Already in the meter wavelength range, the circuit ceases to work satisfactorily: the inter-thread capacitances of the coils, the inductance of the inputs and the capacitor plates affect.Increase in frequency requires a reduction in the size of the coil and capacitor, which entails a decrease in the allowable oscillatory power.

In the range of decimeter and shorter waves (partially in the meter range), resonators are used in which electromagnetic oscillations occur inside a limited volume; therefore they are called volumetric.

The gradual transformation of the circuit into a cavity resonator is shown in Fig. 11.1. Let the circuit (Fig. 11.1a) be designed for a very high frequency and have only one turn. The inclusion of a few more turns parallel to it (Fig. 11.16) increases the oscillation frequency of this system and reduces harmful radiation into space. Combining all the turns into a continuous surface of revolution (Fig. 11.1 c) leads to a completely shielded toroidal resonator with an even higher oscillation frequency; this resonator belongs to the class of quasi-stationary ones.

Quasi-stationary resonators have clearly defined regions of existence of electric and magnetic fields, which are equivalent to capacitance and inductance; we can assume that such a resonator is a completely shielded oscillatory circuit. The dimensions of a quasi-stationary resonator are small compared to the wavelength of its natural oscillations.

By spreading the plates (of the capacitor), we turn the boundary of the resonator into a convex surface, for example, a spherical one (Fig. 11.1 d). The natural frequency of this will increase even more and the wavelength will become comparable with the dimensions of the resonator. Now the entire volume of the resonator is almost equally filled with electric and magnetic zeros, therefore, it is possible to single out separate regions with the properties of capacitance and inductance. The field in a cavity resonator of such a type can be represented as a sum of partial waves successively reflected from its walls. Resonance occurs if a wave circulating inside the resonator arrives at a certain point always in one and the same phase Such an in-phase addition of the fields significantly increases the amplitude of the oscillations.

Significant changes occurred during the development of the optical range, in which the wavelengths are much smaller than the dimensions of the resonator. At the same time, closed volumes with metal walls had to be abandoned. Open cavity resonators generating optical waves retained only a part of the reflecting wall. In the simplest case, they are a system of two opposing mirrors made of a multilayer dielectric that reflect an electromagnetic wave to each other.

OWN AND FORCED VIBRATIONS

Natural oscillations, as is known from the theory of oscillatory circuits, arise in the resonator under an external pulse action, when a portion of energy enters it. After the process of establishment, they become enharmonic damped and depend on time according to the law:

where (Oc is the natural circular frequency of oscillations, the time constant of the resonator, the intrinsic quality factor of the resonator, the complex natural frequency of oscillations.

The cavity resonator has a number of natural oscillations, each of which corresponds to a certain field structure and certain values. Therefore, an external electromagnetic pulse creates a complex oscillation in the resonator, consisting of a number of frequency components of the form (11.1).

Forced oscillations are caused (by external periodic influences, while energy enters the system every period. If the frequency of these oscillations coincides with one of the resonant frequencies of the oscillatory system, a resonance occurs, (accompanied by a sharp increase in the amplitude of the oscillations. The reserves of electrical and magnetic energy in the resonator resonance in the average for the period are the same, so that the energy is completely transferred from one (state to another. The communication line from (the external source) delivers to the oscillatory system only a relatively small amount of energy necessary to replenish heat losses.

CAVITY PARAMETERS IN THE FORCED OSCILLATION MODE

The resonant frequency or only slightly differs from the natural frequency. For example, at this difference (is less than. The value is determined by the geometric dimensions of the resonator and the structure of the electromagnetic field of the considered oscillation. The study of a certain type of oscillation, independently of others, is possible only in a relatively narrow band near if other types of oscillations have resonant frequencies far enough away from or unrelated to the exciter.

The quality factor can be determined through energy parameters. (In the theory of circuits where is the inductance of the coil, resistance (losses. Multiply the numerator and denominator of this formula (by

Energy stored in the resonator at resonance. It is equal to twice the magnetic energy in the inductance due to the fact that the average power of losses in the resonator over the period.

Therefore, the intrinsic quality factor of the resonator is expressed as

i.e. equal to the multiplied by the ratio of the energy accumulated in the resonator at [resonance, the energy loss (in the resonator for one period. Formula (11.2) for is more universal than the original ratio. It includes energy quantities that are easily determined for any system .

The input resistance at resonance (or conductivity is measured in the line at the entrance to the resonator in front of the communication device (Fig. 11.2). We will call this section of the line the reference plane. In the steady state, power is consumed from the generator equal to the power losses in the resonator. Therefore

Thus, resistance is a measure of the losses in a resonator. Its value depends on the design of the communication device and the place of its inclusion in a given resonator.

Resonance characteristic - frequency dependence of the complex input resistance of the resonator or input conductivity Accordingly, at (parallel resonance