# Nuclear magnetic resonance

Nuclear magnetic resonance , and nuclear magnetic resonance or nuclear magnetic resonance (abbreviated as NMR to English nuclear magnetic resonance ) is a (nuclear) physical phenomenon in which atomic nuclei of a material sample in a constant magnetic field alternating electromagnetic fields absorb and emit. Nuclear magnetic resonance is the basis of both nuclear magnetic resonance spectroscopy (NMR spectroscopy), one of the standard methods for examining atoms , molecules , liquids and solids , as well as nuclear magnetic resonance tomography (magnetic resonance tomography, MRT) for medical imaging diagnostics.

The nuclear magnetic resonance is based on the Larmor precession of the nuclear spins around the axis of the constant magnetic field. Through the emission or absorption of alternating magnetic fields that are in resonance with the Larmor precession , the nuclei change the orientation of their spins in relation to the magnetic field. If the emitted alternating field is observed by means of an antenna coil , this is also referred to as core induction . The absorption of an irradiated alternating field is observed on the basis of the energy transfer to the nuclear spins.

The resonance frequency is proportional to the strength of the magnetic field at the location of the core and to the ratio of the magnetic dipole moment of the core to its spin ( gyromagnetic ratio ). The amplitude of the measured signal is u. a. proportional to the concentration of the relevant type of nuclei ( nuclide ) in the sample. The amplitude and especially the frequency of the nuclear magnetic resonance can be measured with very high accuracy. This allows detailed conclusions to be drawn about the structure of the nuclei as well as their other interactions with the near and far atomic environment.

A nuclear spin not equal to zero is a prerequisite for nuclear magnetic resonance. The nuclei of the isotopes 1 H and 13 C are most frequently used to observe nuclear magnetic resonance. Other nuclei examined are 2 H, 6 Li, 10 B, 14 N, 15 N, 17 O, 19 F, 23 Na, 29 Si, 31 P, 35 Cl, 113 Cd, 129 Xe, 195 Pt and the like. v. a., each in their basic state. All nuclei with an even number of protons and neutrons are excluded , unless they are in a suitable excited state with a spin other than zero. In some cases, nuclear magnetic resonance has been observed on nuclei in a sufficiently long-lived excited state.

For the analogous observation of electrons see electron spin resonance .

## History and Development

### Before 1940: Zeeman effect and Rabi method

In 1896 it was discovered that optical spectral lines split up in a magnetic field ( Zeeman effect ). Hendrik Antoon Lorentz soon interpreted this to mean that the (circular) frequency of the light wave shifts by the amount of the Larmor frequency because the atom represents a magnetic top that is excited by the magnetic field to a precession movement with the Larmor frequency . ${\ displaystyle \ pm \ omega _ {\ text {L}}}$

According to the light quantum hypothesis ( Einstein 1905), the frequency shift corresponds to a change in energy , which in turn could be explained by the directional quantization of the angular momentum discovered by Arnold Sommerfeld in 1916 . With the angular momentum vector, the magnetic dipole of the atom parallel to it also has only discrete permitted setting angles to the magnetic field and correspondingly different discrete values ​​of the magnetic energy. The magnetic field causes an energy level to split into several so-called Zeeman levels. This picture was directly confirmed in the Stern-Gerlach experiment in 1922 . It was shown there that the smallest possible (non-vanishing) angular momentum (i.e. quantum number ) can only have two possible angles of incidence to an external field. ${\ displaystyle \ pm \ omega _ {\ text {L}}}$${\ displaystyle \ Delta E = \ pm \ hbar \ omega _ {\ text {L}}}$ ${\ displaystyle J = {\ tfrac {1} {2}}}$

At the end of the 1920s it was discovered that atomic nuclei have a magnetic moment that is around 1000 times smaller than that of atoms, which is why the splitting of the energy levels they cause is referred to as hyperfine structure . The transition frequencies between neighboring hyperfine levels are in the range of radio waves (MHz). In 1936, Isidor Rabi succeeded in proving experimentally that the precession motion of atoms flying through a constant magnetic field in an atomic beam is disturbed by the radiation of an alternating magnetic field if its frequency is in resonance with such a transition frequency . As a result, the magnetic moments of numerous cores could be determined with high accuracy, which u. a. enabled the development of more accurate core models .

### 1940s: nuclear magnetic resonance in liquids and solids

Nuclear magnetic resonance in the narrower sense, i.e. the change in the setting angle of the nuclear spins to the static external magnetic field without significant involvement of the atomic shell in the precession movement, was first realized in 1946 in two different ways. Edward Mills Purcell used the energy transfer from the magnetic alternating field to the nuclear spins and further into their atomic environment to demonstrate the resonance. Felix Bloch observed the alternating voltage that is induced by the precessing dipole moment of the nuclei in a coil when this is no longer parallel to the direction of the static field in the case of resonance (method of "nuclear induction"). The prerequisite is that the static magnetic field causes the strongest possible polarization of the nuclear spins, which has oriented the development of devices towards ever stronger magnetic fields (today with superconducting coils up to 24 Tesla). These methods now enabled measurements on liquid and solid matter and a further increase in measurement accuracy to soon 6–8 decimal places. The measured values ​​for the nuclear magnetic moments thus obtained were correspondingly accurate. In reverse of the question, nuclear magnetic resonance has also become a common method for determining the precision of magnetic fields. In addition, various additional influences of the atomic environment on the magnetic field acting at the location of the nuclei could be measured, which are small but allow detailed conclusions to be drawn about the structure and bonding of the molecules and their mutual influence. Therefore, nuclear magnetic resonance spectroscopy is still a standard method in chemical structure research and one of the most important instruments in analytical organic chemistry.

Applications in chemistry were initially considered unlikely. One of the pioneers was Rex Edward Richards in England, who was supported by Linus Pauling in not listening to skeptics. In 1946, in the group of Felix Bloch, Martin Everett Packard first recorded the NMR spectrum of an organic molecule. A breakthrough for the commercial market of using NMR spectrometers in organic chemistry was the A-60 NMR spectrometer from Varian Associates , developed in 1961 by James Shoolery at Varian, which also did an essential job in disseminating knowledge of NMR among chemists performed and in its popularization. Another pioneer of NMR spectroscopy in organic chemistry was John D. Roberts .

### 1950s: high frequency pulses and spin echo

The measurement possibilities of the core induction method expanded in the 1950s, when the direction of the polarization of the nuclei could be manipulated through the use of the 10–20 MHz alternating field in the form of short-term pulses. If the polarization is initially parallel to the constant magnetic field, z. For example, a “90 ° pulse” can rotate the entire dipole moment of the sample in a certain direction perpendicular to the field direction. This enables direct observation of the subsequent free Larmor precession of the dipole moment about the field direction, because it induces (such as the rotating magnet in a generator of electrical engineering) in an antenna coil, an AC voltage ( "free induction decay", FID, for engl. Free induction decay ). The amplitude then decreases over time because the degree of alignment of the nuclear spins decreases along the common direction perpendicular to the field, partly because the polarization parallel to the static magnetic field is restored (longitudinal relaxation ), partly due to field inhomogeneities and fluctuating interference fields (transverse relaxation). Both processes can be observed separately here, especially using the spin-echo method first described by Erwin Hahn .

### 1970 / 80s: NMR tomography and imaging

From the 1970s on, nuclear magnetic resonance was further developed into an imaging method, magnetic resonance tomography , based on the work of Peter Mansfield and Paul C. Lauterbur . When a strongly inhomogeneous static field is applied, the resonance frequency becomes dependent in a controlled manner on the location of the nuclei ( field gradient NMR ), but only in one dimension. A three-dimensional image of the spatial distribution of the nuclei of the same isotope can be obtained from this if the measurements are repeated one after the other with different directions of the inhomogeneous static fields. To create an image as rich in information as possible, e.g. B. for medical diagnoses , then not only the measured values ​​for the concentration of the relevant isotope are used, but also those for the relaxation times. These devices use superconducting magnets and 400 to 800 MHz alternating fields.

### Special developments

Of fundamental physical interest are two more seldom used methods:

• As early as 1954, the FID method was used to demonstrate the Larmor precession of the hydrogen nuclei (protons) in a water sample in the earth's magnetic field (approx. 50 μT). The protons had been polarized by a stronger field perpendicular to the earth's field, which had been quickly switched off at a certain point in time. The immediately onset Larmor precession induces an alternating voltage with a frequency of approx. 2 kHz, which z. B. is used to precisely measure the earth's magnetic field. Absorption from a resonant alternating field is not required here. Therefore, this is the purest case of observing nuclear induction.
• Nuclear magnetic resonance has been successfully demonstrated on nuclei in a sufficiently long-lived excited state (shortest lifetime so far 37 μs), whereby the changed angular distribution of the γ-radiation emitted by the nuclei was used as proof.

## Physical basics

With nuclear magnetic resonance, macroscopic explanations according to classical physics and microscopic explanations according to quantum mechanics can easily be combined with one another ( here is a more detailed explanation ). The decisive factor here is that the Larmor precession of the nuclear spins has a magnitude and direction that is independent of its orientation. The corresponding effect of the static field can therefore be completely transformed away by transitioning into a reference system that rotates around the field direction with the Larmor frequency, regardless of the respective state of the individual nuclei of the sample under consideration and the size and direction of the macroscopic magnetic moment formed by them .

### polarization

A nucleus with the magnetic moment has a potential energy dependent on the angle in a magnetic field . The lowest energy belongs to the parallel position of the moment to the field, the highest energy applies to the antiparallel position. In thermal equilibrium at temperature , the moments are distributed over the various energies according to the Boltzmann factor (  : Boltzmann constant ). With typical nuclear moments and typical thermal energies , the Boltzmann factors differ only by less than 10 −4 , but the statistical preference of the small setting angles over the large ones is expressed by a mean value other than zero . A polarization arises and with it a macroscopic magnetic moment parallel to the external field (therein : number of nuclei). So much for the classic explanation of polarization through (nuclear) paramagnetism . ${\ displaystyle {\ vec {\ mu}}}$${\ displaystyle {\ vec {B}} _ {0}}$${\ displaystyle E _ {\ mathrm {mag}} = - ({\ vec {\ mu}} \ cdot {\ vec {B}} _ {0})}$${\ displaystyle T}$ ${\ displaystyle \ exp {(-E _ {\ mathrm {mag}} / k _ {\ mathrm {B}} T)}}$${\ displaystyle k _ {\ mathrm {B}}}$${\ displaystyle | {\ vec {\ mu}} | \ approx 10 ^ {- 7} \ mathrm {eV} / \ mathrm {T}}$${\ displaystyle k _ {\ mathrm {B}} T \ approx {\ tfrac {1} {40}} \, \ mathrm {eV}}$${\ displaystyle \ langle {\ vec {\ mu}} \ rangle}$${\ displaystyle {\ vec {M}} = N _ {\ text {core}} \ cdot \ langle {\ vec {\ mu}} \ rangle}$${\ displaystyle {\ vec {B}} _ {0}}$${\ displaystyle N _ {\ text {core}}}$

### Zeeman levels

According to quantum mechanics, in states with a certain angular momentum, every vector operator acts parallel to the angular momentum operator , one writes ${\ displaystyle {\ hat {\ vec {I}}}}$

${\ displaystyle {\ hat {\ vec {\ mu}}} = \ gamma \ cdot {\ hat {\ vec {I}}}}$.

The constant is called the gyromagnetic ratio , it has a characteristic value for each nuclide (see also Landé factor ). ${\ displaystyle \ gamma}$

The direction quantification known from angular momentum therefore also applies to the vector , according to which, for a given angular momentum quantum number, the cosine of the setting angle to the field direction in the natural energy states can only assume the values , whereby the magnetic quantum number passes through the values . The largest possible component of along the field, also known as the magnitude of the magnetic moment , is therefore . ${\ displaystyle {\ vec {\ mu}}}$ ${\ displaystyle I}$${\ displaystyle {\ tfrac {m} {\ sqrt {I (I + 1)}}}}$${\ displaystyle m = -I, - (I-1), \ dots, (I-1), \, I}$${\ displaystyle {\ vec {\ mu}}}$${\ displaystyle \ mu = \ gamma \ hbar I}$

The component of the moment parallel to the field therefore has one of the values ${\ displaystyle \ mu _ {z}}$

${\ displaystyle \ mu _ {z} = {\ frac {\ mu} {I}} m = \ gamma \ hbar m}$

and the magnetic energy accordingly: ${\ displaystyle E _ {\ mathrm {mag}}}$

${\ displaystyle E_ {m} = - {\ frac {\ mu} {I}} B_ {0} m = - \ gamma \ hbar mB_ {0}}$

( : Amount of .) This formula gives the energies of the Zeeman levels , which result from the equidistant splitting of the level with nuclear spin . The distance between neighboring Zeeman levels corresponds exactly to the Larmor frequency , i.e. the frequency with which a (classical as well as quantum mechanical) magnetic top precesses in the field : ${\ displaystyle B_ {0}}$${\ displaystyle {\ vec {B}} _ {0}}$${\ displaystyle (2I {\ mathord {+}} 1)}$ ${\ displaystyle I}$ ${\ displaystyle \ omega _ {\ mathrm {L}}}$${\ displaystyle {\ vec {B}}}$

${\ displaystyle \ Delta E = \ hbar \ omega _ {\ mathrm {L}} = \ gamma \ hbar B_ {0}}$.

The occupation numbers of the Zeeman levels decrease in thermal equilibrium from to (in the case of positive , otherwise the other way round), but on the order of not more than 10 −4 relative. ${\ displaystyle m = + I}$${\ displaystyle m = -I}$${\ displaystyle \ gamma}$

### Relaxation

The adjustment of the equilibrium polarization of the nuclear spins parallel to the external field is called longitudinal relaxation . It takes up to several seconds in liquid and solid samples (in gases it can take weeks) if the sample does not contain any paramagnetic admixtures, i.e. atoms with permanent magnetic dipole moments that cause transitions between the Zeeman levels through fluctuating magnetic fields and thus the exchange of energy accelerate with the nuclear spins. The time constant is denoted by. The reduction of a polarization perpendicular to the field down to the equilibrium value zero is called transverse relaxation and (mostly) takes place more quickly (time constant ), because no energy conversion is necessary for this; rather, it is sufficient that the nuclear spins aligned transversely to the magnetic field lose their common alignment due to small fluctuations in their constant Larmor precession around the field direction. In terms of time, the approximation to equilibrium follows a simple decaying exponential function as a good approximation. ${\ displaystyle T_ {1}}$${\ displaystyle T_ {2} ^ {*}}$

### Bloch equations

The Bloch equations summarize the Larmor precession and the longitudinal and transversal relaxation in a single equation of motion for the vector of the magnetic moment (with magnetic field and equilibrium magnetization , both parallel to the -axis): ${\ displaystyle {\ vec {M}} = (M_ {x}, M_ {y}, M_ {z})}$${\ displaystyle {\ vec {B}} _ {0} = (0,0, B_ {0})}$${\ displaystyle {\ vec {M}} _ {0} = (0,0, M_ {0})}$${\ displaystyle z}$

${\ displaystyle {d {\ vec {M}} \ over dt} = \ gamma {\ vec {M}} {\ mathord {\ times}} {\ vec {B}} _ {0} \ - \ {\ begin {pmatrix} {\ tfrac {M_ {x}} {T_ {2} ^ {*}}} \\ [0.5em] {\ tfrac {M_ {y}} {T_ {2} ^ {*}}} \\ [0.5em] {\ tfrac {M_ {z} -M_ {0}} {T_ {1}}} \ end {pmatrix}}}$

The cross product describes the Larmor precession with the angular velocity . In the second term, the relaxation is phenomenologically summarized as a first-order process (i.e. simple exponential decay), with the time constant for the component parallel to the field being different from that for the transverse component . According to quantum mechanics, the Bloch equations also apply to the expected value of the magnetic moment of each individual nucleus${\ displaystyle {\ vec {\ omega}} _ {\ mathrm {L}} = \ gamma {\ vec {B}} _ {0}}$${\ displaystyle {\ vec {M}}}$${\ displaystyle \ langle {\ vec {\ mu}} \ rangle.}$

### Transverse alternating field and absorption of energy

A weak additional alternating field, e.g. B. in -direction, can always be understood as the sum of two circularly polarized alternating fields that z. B. rotate about the axis (i.e. the direction of the strong constant field) in the opposite sense. ${\ displaystyle x}$${\ displaystyle z}$

• From a quantum mechanical perspective, this alternating field induces transitions between the Zeeman levels in one direction or the other in the case of resonance, because its circularly polarized quanta have the right angular momentum ( component ) and then just the right energy. These transitions disturb the thermal equilibrium because they reduce existing differences in the occupation numbers. This means a net energy consumption, because before there were more nuclei in lower energy states than in higher ones, corresponding to the thermal equilibrium. This energy flow from the alternating field into the system of nuclear spins would come to a standstill when the equal population was reached. The thermal contact of the spin system with the environment, which is decisive for producing the original equilibrium magnetization, continuously withdraws energy from the disrupted spin system. A flow equilibrium is established with a slightly reduced magnetization. The relevant parameter for this is the longitudinal relaxation time . The first evidence and applications of nuclear magnetic resonance according to the Purcell method are based on this continuous wave method.${\ displaystyle z}$${\ displaystyle \ pm 1 \ hbar}$${\ displaystyle \ hbar \ omega _ {\ mathrm {L}}}$${\ displaystyle T_ {1}}$
• In a macroscopic view it is easier to overlook which movement of the macroscopic dipole moment results from this: The two components of the alternating field rotating with the Larmor precession represents a constant field perpendicular to the axis in the case of resonance in the rotating reference system . It acts on the dipole with a torque, that imposes a further Larmor precession around the axis of this additional field ( rotating in the xy plane ). Since the setting angle has to change to the much stronger static field , the dipole absorbs or gives off energy from the alternating field. If the dipole was previously parallel to the field direction , it can itself induce an alternating voltage in a receiver coil in the twisted state. If the alternating field is pulsed, the dipole moment z. B. specifically rotated exactly by 90 ° or completely reversed (as far as the relaxation time allows). This results in the numerous different pulse methods with their versatile measurement options (e.g. the spin echo for the separate determination of and ).${\ displaystyle z}$${\ displaystyle {\ vec {B}} _ {0}}$${\ displaystyle {\ vec {B}} _ {0}}$${\ displaystyle T_ {1}}$${\ displaystyle T_ {1}}$${\ displaystyle T_ {2} ^ {*}}$

## Apparatus and methods

Basic structure for the detection of nuclear magnetic resonance

An NMR apparatus typically consists of a magnet to generate a static magnetic field that is as strong and homogeneous as possible, into which the sample can be introduced, and a small magnetic coil each for generating or detecting a high-frequency transverse magnetic field (see Fig.). In their first successful apparatus, Bloch and Purcell used a static field of the order of 1 T, generated by an electromagnet. Small additional coils were installed to improve the spatial constancy of the field and to fine-tune it. At is the resonance frequency of protons . In Bloch's apparatus, the coil for receiving the high-frequency magnetic field was perpendicular to the transmitter coil in order to eliminate direct reception of the alternating field it generated. The alternating voltage emitted by the receiver coil is then only caused by the protons whose magnetic moments rotate around the field direction with the Larmor precession after they have been successfully rotated away from the direction of the static field by the radiated alternating field in the case of resonance. Purcell used only one coil for transmission and reception in his apparatus , whereby the resonance is noticeable in that the alternating voltage induced by the cores in the coil is opposite to the alternating voltage applied, which removes more energy from the transmitter. In the case of a deliberately weak transmitter, this leads to an easily detectable reduction in the oscillation amplitude. In order to find the resonance without having to adjust the frequency of the transmitter, the field strength of the static field was varied using the additional coils. For this reason, the resonance curves according to the Purcell method in the NMR spectra were not plotted against the frequency, but rather against the applied magnetic field . ${\ displaystyle B_ {0} = 1 \ mathrm {T}}$${\ displaystyle f _ {\ mathrm {L}} \ equiv {\ tfrac {\ omega _ {\ mathrm {L}}} {2 \ pi}} = 42 {,} 5 \; \ mathrm {MHz}}$${\ displaystyle B_ {0}}$

In organic chemistry, NMR spectroscopy prevailed with the availability of cheap, easy-to-use equipment, first the A-60 from Varian Associates , developed under the direction of James Shoolery .

Nowadays, NMR pulse spectrometers are used almost exclusively for energy- resolved NMR ( nuclear magnetic resonance spectroscopy ), for time-resolved NMR ( relaxation time measurements ) and for spatially resolved NMR ( field gradient NMR ) . The first commercial and “quartz-controlled” pulse spectrometers were developed and manufactured in the 1960s in Germany by a group led by physicists Bertold Knüttel and Manfred Holz in the Bruker company . The nuclei are excited with high-frequency pulses and the NMR signal is measured as a free induction decay (FID) or spin echo . With the "quartz-controlled" devices, all transmission frequencies and all times in the pulse program (pulse spacing, pulse duration, etc.) in the NMR experiment are derived from a single mother crystal and the result is a crystal-stable but variable phase relationship between the transmitter high frequency and e.g. B. the start of the pulse. This allows the RF phase and thus the RF radiation direction of the individual high-frequency transmission pulses to be set in a complex series of pulses, which is an indispensable requirement for most modern NMR experiments. In the Fourier transform (FT) spectrometers , also developed in the 1960s , the signals recorded in the time domain (e.g. the FID) are then transformed by computers into signals in the frequency domain (spectrum). Almost all nuclear magnetic resonance apparatus today work on this basis .

## literature

• Hermann Haken, Hans Chr. Wolf : Atom- und Quantenphysik, Springer, 1996, ISBN 3-540-61237-8 , chap. 20th
• Manfred Holz, Bertold Knüttel : Pulsed nuclear magnetic resonance, A physical method with a variety of possible applications, Phys. Blätter, 1982, 38 , pp. 368-374
• SW Homans : A Dictionary of Concepts in NMR, Clarendon Press, Oxford, 1989, ISBN 0-19-855274-2
• Malcom H. Lewitt : Spin Dynamics. Wiley & Sons, Chichester 2001, ISBN 0-471-48922-0 .

## Individual evidence

1. ^ EM Purcell, HC Torrey, RV Pound: Resonance absorption by nuclear magnetic moments in a solid . In: Phys Rev . tape 69 , no. 1-2 , 1946, pp. 37-38 , doi : 10.1103 / PhysRev.69.37 .
2. ^ F. Bloch, WW Hansen, M. Packard: Nuclear induction . In: Phys Rev . tape 69 , no. 3–4 , 1946, pp. 127 , doi : 10.1103 / PhysRev.69.127 .
3. ^ F. Bloch: Nuclear induction . In: Phys Rev . tape 70 , no. 7–8 , 1946, pp. 460-474 , doi : 10.1103 / PhysRev.70.460 .
4. Derek Lowe, Das Chemiebuch, Librero 2017, p. 398
5. ^ EL Hahn: Nuclear induction due to free Larmor precession . In: Phys Rev . tape 77 , no. 2 , 1950, p. 297-298 , doi : 10.1103 / PhysRev.77.297.2 .
6. EL Hahn: Spin echoes . In: Phys Rev . tape 80 , no. 4 , 1950, p. 580-594 , doi : 10.1103 / PhysRev.80.580 .
7. P. Mansfield, PK Grannell: NMR 'diffraction' in solids? In: J Phys C . tape 6 , 1973, p. L422-L426 , doi : 10.1088 / 0022-3719 / 6/22/007 .
8. PC Lauterbur: Image formation by induced local interactions: Examples employing nuclear magnetic resonance . In: Nature . tape 242 , 1973, pp. 190-191 , doi : 10.1038 / 242190a0 ( nature.com ).
9. ^ ME Packard, R. Varian, Phys. Rev. A93 (1954) p. 941. Also referred to in Georges Bené et al., Physics Reports Vol. 58, 1980, pp. 213-267
10. N. Bräuer, B. Focke, B. Lehmann, K. Nishiyama, D. Riegel, Zeitschrift für Physik A, 1971, Vol. 244, pp. 375-382
11. A. Geiger, M. Holz: Automation and Control in high power pulsed NMR In: J. Phys. E: Sci.Instrum. 13, 1980, pp. 697-707.
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