========
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. :term:`DEPT` and :term:`Complex Numbers `.
.. glossary::
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 :term:`Cartesian Product
Operators ` and :term:`Spherical
Tensor Operators `.
Cartesian Product Operator
Products of individual spin operators :math:`I_{kx}`,
:math:`I_{ky}`, :math:`I_{kz}`, where k is the spin label. In
a three-spin system, examples of Cartesian product operators
are :pton:`I1x`, :pton:`I2z`, :pton:`I3y`, :pton:`2I1yI2z`, or
:pton:`4I1zI2zI3x`. (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 :ref:`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 :math:`B_0`.
Coherence Order
An operator :math:`A` has a well-defined :ref:`coherence
order` :math:`p` if a rotation around the z axis by an
arbitrary angle :math:`α` reproduces the operator :math:`A` up
to an additional phase factor :math:`exp(-ipα)`. For a visual
explanation see :ref:`Droplet Symmetry and Coherence Order Rules`.
Complex Number
Recall that a complex number :math:`c = a + ib` can be
expressed in terms of its real and imaginary parts :math:`a`
and :math:`b`, but also in terms of their amplitude :math:`r`
and phase φ, where :math:`r2=a2+b2` and
:math:`tan(φ)=b/a`. The amplitude :math:`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 :math:`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. :ref:`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 :ref:`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 :math:`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 :ref:`Heteronuclear Pulse Sequences` and
:ref:`Solution 12` for more information.
Linearity
The linearity of an operator reflects the number of involved
single-spin operators. For example, :pton:`I1x`,
:pton:`2I1xI2y` and :pton:`4I1xI2zI3y` are linear, bilinear
and trilinear operators, respectively.
LISA
This :ref:`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 :ref:`LISA Basis`.
Pauli Matrices
The `Pauli Matrices `_ are
- :math:`\sigma_1 = \sigma_x = \left( \begin{matrix}0 & 1\\ 1 & 0\end{matrix} \right)`
- :math:`\sigma_2 = \sigma_y = \left( \begin{matrix}0 & -i\\ i & 0\end{matrix} \right)`
- :math:`\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 :term:`Basis Operator` can be drawn
from the :term:`Tensor Operator` set, or the NMR-scaled
:term:`Pauli Matrices`.
Plain Text Operator Notation
PTON
The :ref:`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 :code:`I1xI2y` corresponds to
the familiar operator :pton:`I1xI2y`.
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 :pton:`I1m` by a
complex phase, giving :pton:`e^{-i \phi} I1m`.
Spherical Tensor Operator
Tensor Operator
An irreducible spherical tensor :math:`T_j` with rank
:math:`j` has :math:`2j+1` components :math:`T_{jm}` with
order :math:`m ∈ \{-j, … , j \}`. The operators :math:`T_{jm}`
form a basis of a space which stays invariant under
rotations. In the Condon-Shortley phase convention, only the
operators :math:`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 :math:`Y_{jm}`. See also :ref:`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
:pton:`t=1/(J12)`. 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).