Glossary

Glossary of Terms and Acronyms

This is a summary of terms and acronyms used in SpinDrops and in the accompanying tutorial and help files. For an introduction to basic NMR concepts related to the dynamics of coupled spin systems, we refer to excellent books by Keeler (Understanding Understanding NMR Spectroscopy), Cavanagh, Fairbrother, Palmer, Skelton, Rance (Protein NMR Spectroscopy, Principles and Practice), Levitt (Spin Dynamics: Basics of Nuclear Magnetic Resonance), Ernst, Bodenhausen, Wokaun (Principles of Nuclear Magnetic Resonance in One and Two Dimensions) or Goldman (Quantum Description of High-Resolution NMR in Liquids). For more information concerning concepts related to the DROPS representation, see [GARON2015] and references cited therein. DEPT and Complex Numbers.

Basis Operator

Just as a vector can be expressed as a unique combination of orthogonal basis vectors, any operator can be expressed as a unique combination of orthogonal basis operators. In NMR, the most widely used basis operators are Cartesian Product Operators and Spherical Tensor Operators.

Cartesian Product Operator

Products of individual spin operators \(I_{kx}\), \(I_{ky}\), \(I_{kz}\), where k is the spin label. In a three-spin system, examples of Cartesian product operators are \(I_{1x}\), \(I_{2z}\), \(I_{3y}\), \(2I_{1y}I_{2z}\), or \(4I_{1z}I_{2z}I_{3x}\). (The prefactors 2 and 4 of the bilinear and trilinear Cartesian product operators ensure that all terms in a basis for spin systems of a particular size have the same norm.) See also Cartesian Basis.

Chemical Shift

The portion of a spin’s resonance frequency due to the chemical environment’s effect on the magnetic field it experiences. Often measured in ppm (parts per million) as it scales linearly with the applied magnetic field \(B_0\).

Coherence Order

An operator \(A\) has a well-defined Coherence Order \(p\) if a rotation around the z axis by an arbitrary angle \(α\) reproduces the operator \(A\) up to an additional phase factor \(exp(-ipα)\). For a visual explanation see Droplet Symmetry and Coherence Order Rules.

Complex Number

Recall that a complex number \(c = a + ib\) can be expressed in terms of its real and imaginary parts \(a\) and \(b\), but also in terms of their amplitude \(r\) and phase φ, where \(r2=a2+b2\) and \(tan(φ)=b/a\). The amplitude \(r\) corresponds to the distance of the complex number from the origin of the complex plane. For a given phase φ, the corresponding phase factor is given by the complex number \(cos(φ)+i sin(φ)=exp(iφ)\), which has an amplitude of 1 and is located on the unit circle in the complex plane.

DEPT

(Distortionless Enhancement of Polarization Transfer) is a technique for heteronuclear polarization transfer and spectral editing. Example 5: Spectral Editing describes the sequence in more detail.

Density Operator

This operator describes the state of a spin system. More precisely, it encodes the information about the state of an ensemble of spin systems and allows us to calculate experimentally relevant expectation values of observables, such as the detectable transverse magnetization of spins.

DROPS

stands for Discrete Representation of spin OPeratorS. This is a general approach to visualize abstract quantum mechanical operators of coupled spin systems [GARON2015]. More information can be found in the section Mathematical Background of the DROPS Visualization.

Droplet

In the DROPS representation, operators are mapped to a set of complex functions on a sphere. Each of these function is plotted at a different location. The shape and color of each droplet represent the orientation-dependent amplitude and phase of the complex function, respectively.

E

represents the identity operator.

Hamilton Operator

Hamiltonian

The quantum mechanical operator that corresponds to the energy of a spin system. It includes terms for frequency offsets, couplings and pulses.

Hermitian Operator

Self-Adjoint Operator

Hermitian operators play an important role in quantum mechanics as they have real eigenvalues and expectation values. Observables correspond to Hermitian operators. Cartesian product operators are Hermitian. Any Hermitian operator can be expressed as a linear combination of Cartesian product operators with real coefficients. The multiplication of a Hermitian operator by \(i\) results in a skew-Hermitian operator.

I1

I2

I3

denote the first (I1), second (I2), and third (I3) spins of a spin system.

INEPT

stands for Insensitive Nuclei Enhanced by Polarization Transfer, see Heteronuclear Pulse Sequences and Solution 12 for more information.

Linearity

The linearity of an operator reflects the number of involved single-spin operators. For example, \(I_{1x}\), \(2I_{1x}I_{2y}\) and \(4I_{1x}I_{2z}I_{3y}\) are linear, bilinear and trilinear operators, respectively.

LISA

This LISA Basis is a tensor basis which is defined to adhere to criteria of Linearity, Subsystems and Auxiliary. In addition to the number of involved spins (linearity) and the subset of involved spins (subsystem), permutation symmetry provides a sufficient auxiliary criterion to uniquely define the tensor basis (up to algebraic signs) for systems consisting of up to five spins 1/2. Additional criteria are necessary for more than five spins. For a rigorous definition, see [GARON2015]. and auxiliary criteria, such as permutation symmetry). See also LISA Basis.

Pauli Matrices

The Pauli Matrices are

• \(\sigma_1 = \sigma_x = \left( \begin{matrix}0 & 1\\ 1 & 0\end{matrix} \right)\)

• \(\sigma_2 = \sigma_y = \left( \begin{matrix}0 & -i\\ i & 0\end{matrix} \right)\)

• \(\sigma_3 = \sigma_z = \left( \begin{matrix}1 & 0\\ 0 & -1\end{matrix} \right)\)

Phase

The term phase typically refers to the argument of a periodic function. This term can be somewhat confusing because it appears in different contexts in NMR, where it is associated with different properties.

In the context of pulses, the phase corresponds to the transverse rotation axis in the rotating frame. It can be defined in terms of the angle between the x axis and the rotation axis (in units of degree, e.g. 90° or in units of radians, e.g. π/2) or by the rotation axis itself (e.g. y).

In the context of complex numbers, the phase refers to the angle between the real axis and the line between the origin and the location of a given complex number in the complex plane.

The relation between z rotations and phase factors of operators plays an important role in the definition of coherence order. In the DROPS representation, individual droplets represent complex functions on a sphere, where the orientation-dependent phase of each complex function is represented graphically by colors.

Product Operator

An operator composed of the direct sum of other Operators, used to represent the states available to Spin Systems. The most basic elements of a Basis Operator can be drawn from the Tensor Operator set, or the NMR-scaled Pauli Matrices.

Plain Text Operator Notation

PTON

The Plain Text Operator Notation (PTON) is a canonical textual way of expressing the common product operators in plain text, rather than in the usual mathematical typeset notation. For example, the PTON I1xI2y corresponds to the familiar operator \(I_{1x}I_{2y}\).

Receiver Phase

The NMR receiver is a quadrature detector producing a complex valued signal. The receiver phase can be altered to select the phase of this signal relative to the base-frequency resonator. From the theoretical standpoint, this simply changes the detection operator, which is typically \(I_{1}^{-}\) by a complex phase, giving \(e^{-i \phi} I_{1}^{-}\).

Spherical Tensor Operator

Tensor Operator

An irreducible spherical tensor \(T_j\) with rank \(j\) has \(2j+1\) components \(T_{jm}\) with order \(m ∈ \{-j, … , j \}\). The operators \(T_{jm}\) form a basis of a space which stays invariant under rotations. In the Condon-Shortley phase convention, only the operators \(T_{j0}\) (with order 0) are Hermitian. Tensor operators form an ideal basis for the DROPS representation because of their favorable properties under non-selective rotations and their close relationship with spherical harmonics \(Y_{jm}\). See also Spherical Tensor Basis.

TOCSY

Total Correlation Spectroscopy

is based on the efficient transfer of polarization or coherence between coupled spins under isotropic mixing conditions. Isotropic mixing conditions can be created by TOCSY multiple-pulse sequences. In homonuclear spin systems, the effective coupling constants of the isotropic mixing Hamiltonian are ideally identical to the actual couplings between the spins. For the simple case of two coupled spins 1/2, polarization or coherence can be transferred from one spin to the other spin, resulting in cross peaks in two-dimensional TOCSY experiments. The optimal mixing time is \(t=1/(J_{12})\). For systems consisting of more than two spins 1/2, polarization and coherence is transferred between all spins of a coupling network, resulting in „total correlation“ spectra. (Isotropic mixing conditions can also be created in heteronuclear spin systems, but the effective coupling constants of the isotropic mixing Hamiltonian are scaled down to 1/3 of the actual couplings between the spins).