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3.17 Description

3.17.1 Overview

There is an orientationally energy dependence that occurs between two point charges (two nuclei) in an externally applied magnetic field. This dipolar interaction is of rank 2 and symmetric about the internuclear axis. It produces relaxation effects in liquid NMR and orientationally dependent shifts in solids.

3.17.2 Coordinate Systems

We will shortly concern ourselves with the mathematical representation of dipole-dipole interactions, in particular their description in terms of spatial and spin tensors. The spatial tensors will be cast in both Cartesian and spherical coordinates and we will switch between the two when convenient. The figure below relates the orientation angles theta and phi to the standard right handed coordinate system in all GAMMA treatments.

3.17.3 Internal Structure

The internal structure of class IntDip contains the quantities listed in the following table (names shown are also internal).

The values of I and S are the spin quantum number of the two spins involved in the dipole-dipole interaction and both will have values which are positive non-zero integer multiples of 1/2. These dictate how many energy levels (and transitions) are associated with the dipolar interaction. These are intrinsically tied into the values and dimensions of the matrices in the vector Tsph.

The value DELZZ is used to specify the dipolar interaction strength. In GAMMA the value of DELZZ is factored out of the spatial tensor such that all rank two interactions (such as the dipolar interaction) have the same spatial tensor scaling.

The two angles THETA and PHI indicate how the diolar interaction (internuclear vector) is aligned relative to the interaction principal axes (PAS). These are one in the same as the angles shown in Figure 3-1 when the Cartesian axes are those of the PAS with the origin vaguely being the center of the nucleus I. These are intrinsically tied into the values in the array Asph.

The asymmetry ETA indicates how the diolar interaction varies with the angle phi. Normally this is zero because the interaction is symmetric about the internuclear vector. However, users may set this to a nonzero value in cases where the dipolar vector is modulated (due to exchange for example) and it may be advantageous to use a single averaged diipole.

There are five values in the complex vector Asph and these are irreducible spherical components of the dipolar spatial tensor oriented at angle THETA down from the PAS z-axis and over angle PHI from the PAS x-axis. Note that these 5 values are not only orientation dependent, they are also ETA dependent. If either of the three the interaction values {ETA, THETA, PHI} are altered these components will all be reconstructed. The values in Asph will be scaled such that they are consistent with other rank 2 spatial tensors in GAMMA which are independent of the interaction type.

The vector of matrices relates to the sperical spin tensor components according to:

Tsph:	[0]	[1]	[2]	[3]	[4]
:

and the vector of complex numbers relate to the spherical spatial tensor components via

Asph:	[0]	[1]	[2]	[3]	[4]
:

3.17.4 Classical Dipole-Dipole Treatment

The classical interaction energy between two dipoles, and , separated by a distance r is¹

where is the magnetic moment, and spin indices, the energy, and the vector connecting the two spins. The superscript is used to denote a dipolar interaction.

3.17.5 Quantum Mechanical Formulation

The associated Hamiltonian is obtained from substitution of for (here ).

Using normalized unit vectors pointing in the direction of , , the equation becomes

where is the matrix formed from the dyadic product of the two unit vectors. A dipolar tensor between the two spins can be defined as

(1)

and thus the dipolar Hamiltonian for a spin pair & , , given by

where is the spin angular momentum operator of spin and the dipolar tensor between the two spins. In expanded matrix form this equation looks like

(2)

An equivalent equation explicitly showing the matrix multiplication is (with )

(3)

.

3.17.6 Cartesian Tensor Formulation

Equation (3) can be rearranged to produce an equation involving two rank 2 Cartesian tensors by taking the dyadic product of the vectors and .

The dyadic product to produce is explicitly done via

.

and from Equation (1) the matrix in the principle axis system ( ) given by

(4)

.

Letting , the Hamiltonian is expressed as a scalar product of two rank 2 Cartesian tensors.

(4-1) or equivalently

3.17.7 Spherical Tensor Formulation

Equation (4-1) can be rewritten in terms of irreducible spherical components rather than the current Cartesian components² using the substitution

(5)

The result is

(6)

3.17.8 Dipole-Dipole Spherical Tensor Spin Components

We can thus obtain the 9 irreducible spherical components of the dipolar spin tensor (rank 2), , directly from the Cartesian components, , as indicated in GAMMA Class Documentation on Spin Tensors. The nomenclature used here for a tensor component is

,

where the subscript l spans the rank (in this case 2) as l = [0, 2], and the subscript m spans +/- l, m = [-l, l]. The nine formulas for these quantities a listed in the following figure.

The matrix representation of these nine tensor components will depend upon the matrix representations of the individual spin operators from which they are constructed³. These in turn depend upon the spin quantum numbers of the two spins involved. For a treatment of two spin 1/2 particles the dipolar tensor components are expressed in their matrix form (spanning the composite Hilbert space of the two spins) in the default product basis of GAMMA as follows⁴ (spin indices implicit).

   

3.17.9 Dipole-Dipole Spherical Spatial Tensor Components

The 9 irreducible spherical components of a rank two spatial tensor, , are related to its Cartesian components by the following formulas⁵.

(7)

Again the subscript l spans the rank as l = [0, 2], and the subscript m spans +/- l, m = [-l, l]. In this dipolar treatment we then have components as indicated in equation (6). Thus, the irreducible spherical tensor components can be obtained by substituting the Cartesian elements of the dipolar spatial tensor, from equation (2), into equations (7).

(8)

However, it is more convenient to rewrite the general rank two Cartesian tensor in terms of a sum over tensors of ranks 0 through 2 as follows,

(9)

where

(10)

The rank 0 part is isotropic (scalar), the rank 1 part is antisymmetric and traceless, and the rank 2 part traceless and symmetric. We shall apply this same nomeclature to our dipolar spatial tensor in order to produce

(11)

.

where

(12)

Note that is for most spatial tensors is NOT equivalent to the unscaled spatial tensor xy-component ( here) unless there is no isotropic component. That turns out to be the case for the dipolar interacation as seen from Equation (4). As with any rank 2 spatial tensor, the dipolar spatial tensor can be specified in its principal axis system(PAS), the set of axes in which the irreducible rank 2 component is diagonal⁶,⁷. The spatial tensor values are experimentally determined in the tensor principal axes. Employing (9) when the irreducible rank 2 component is diagonal,

Indeed the anti-symmetric components are also all zero since . So we can rewrite out dipolar spatial tensor in the principle axis system as

(13)

.

where and .

3.17.10 Unscaled Spherical Spatial Tensor PAS Components

Rank 2 spatial tensors are also commonly specified in their principal axis system by the three components; the isotropic value , the anisotropy , and the asymmetry . These are generally given by

,                            

A set of Euler angles is normally also given to relate the spatial tensor principle axes to another coordinate system. For the dipolar spatial tensor we have

(14)

We have already seen that both the isotropic and anti-symmetric terms are zero for the dipolar interacation. Again, we note that is for most spatial tensors is NOT equivalent to the unscaled spatial tensor z-component ( here) and that is usually NOT equivalent to unscaled ratio (here ) unless there is no isotropic component. But we have also seen that there is no isotropic component in the dipolar spatial tensor.

,                            

We can also figure out the asymmetry value from the PAS representation of the tensor. Again from The irreducible spherical elements of the dipolar tensor, , in the principal axis system are, by placement of (14) into (7),

and these values should be equivalent to those given in (8) on page 3-103. Fortunately, there is no isotropic component of the dipolar tensor, nor is there any asymmetry. If we then use these fact we obtain a much simpler result

(15)

All but one of the spherical components is zero because the dipolar spatial tensor is symmetric and traceless.

3.17.11 Scaled Dipolar Spherical Spatial Tensor PAS Components

Throughout GAMMA we desire all spatial components to be scaled so that they are independent of the particular interaction. To do so, we adjust them to be as similar to normalized spherical harmonics as possible. Thus, we here scale the dipolar spatial tensor such that the component (the only non-zero one in this instance) will have a magnitude of the rank two spherical harmonic when the two spherical angles are set to zero. Our "normalization" factor "X" is obtained by

(16)

Using we define the GAMMA dipolar spatial tensor such that

(17)

and the components are given in the next figure.

The scaling factor which was multiplied into the "D" components will be compensated for in the dipolar interaction constant. The dipole-dipole Hamiltonian given in equation (6) becomes

(18)

3.17.12 Dipolar Interaction Constant

In GAMMA, since we have defined our generic spatial and spin tensors to be scaled independent of the type of interaction, we use an interaction constant as a scaling factor when formulating Hamiltonians. The dipolar Hamiltonian may be produced from

(19)

so evidently

(20)

Such interaction constants are not very common in the literature (except with regards to some papers treating relaxation in liquid NMR) and thus not intuitive to many GAMMA users. So, one simply needs to be aware of the relationships between the interaction constant and any commonly used dipolar tensor definitions. Many treatments retain the dipolar tensor in Cartesian components, whereas in GAMMA we (internally) work with the spherical components consistently across the magnetic resonance interaction types. Perhaps the only quantity worthy of mention is the , the dipolar anisotropy . This is readily related to the typical D tensor Cartesian components.

3.17.13 Spatial Tensor Rotations

We can express these spatial tensor components relative to any arbitrary axis system (AAS) by rotating the tensor from the principal axes to the new axes via the formula

(21)

where are the rank Wigner rotation matrix elements and the set of three Euler angles which relate the principal axes to the arbitrary axes. As is evident from equations (18-15) - (21), regardless of the coordinate system, only the rank two components will contribute to the dipolar Hamiltonian. This is now demonstrated by combining the last two equations.

(22)

Using now the Wigner rotation matrix element relationship

(23)

we have

(24)

3.17.14 Dipolar Hamiltonian

Because only the rank 2 dipolar spatial tensors will be non-zero, it is sufficient to utilize only the irreducible spherical rank 2 components of both the spatial and spin dipolar tensors in construction of the dipolar Hamiltonian. Equation (18-15) can be rewritten as

(25)

where

(26)

               

When working with an entire spin system one must sum over all spin pairs with the spatial tensors being placed in the same coordinate system, usually the laboratory system. The dipolar Hamiltonian for a spin system becomes the following.

(27)

Here the angles and are the polar angles of the dipolar vector between spins i and j when written relative to the coordinate system in which is expressed. are the normalized rank two spherical harmonics, the superscript indicating the dipolar interaction is unnecessary but used for consistency with other Hamiltonian definitions.

3.17.15 Dipolar Orientation Angles

A set of Euler angles is normally given to relate the spatial tensor principle axes to another coordinate system. In the dipolar Hamiltonian derivation we have instead used for the Euler angle designations because, due to the dipolar tensor symmetry, we ultimately utilize only the angles which are equivalent to the common angle designations in spherical coordinates. In turn the spherical harmonics are written in these coordinates.

For the dipolar spatial tensor we have

                                  

This is perhaps more easily seen visually by examination of the matrix breakdown into these components (superscript has been added so there is an association with the dipolar spatial tensor).

The following figure contains a grouping of the applicable dipolar Hamiltonian equations.

3.17.16 Summary

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