Page 67 out of 81 total pages , Page 10 out of 12 pages in this chapter
Usage:
Description:
The function delzz is used to either obtain or set the interaction G coupling constant. With no arguments the function returns the coupling in Hz. If an argument, dz, is specified then the coupling constant for the interaction is set. It is assumed that the input value of dz is in units of Hz. The function is overloaded with the name delz for convenience. Note that setting of delzz will alter the (equivalent) value of the G coupling GCC/NGCC as well as the G frequency.Return Value:
Either void or a floating point number, double precision.Example(s):
See Also: GCC, NGCC, wG
Usage:
Description:
The function delzz is used to either obtain or set the interaction G coupling constant. With no arguments the function returns the coupling in Hz. If an argument, dz, is specified then the coupling constant for the interaction is set. It is assumed that the input value of dz is in units of Hz. The function is overloaded with the name delz for convenience. Note that setting of delzz will alter the (equivalent) value of the G coupling GCC/NGCC as well as the G frequency.Return Value:
Either void or a floating point number, double precision.Example(s):
See Also: GCC, NGCC, wG
Usage:
Description:
The function delzz is used to either obtain or set the interaction G coupling constant. With no arguments the function returns the coupling in Hz. If an argument, dz, is specified then the coupling constant for the interaction is set. It is assumed that the input value of dz is in units of Hz. The function is overloaded with the name delz for convenience. Note that setting of delzz will alter the (equivalent) value of the G coupling GCC/NGCC as well as the G frequency.Return Value:
Either void or a floating point number, double precision.Example(s):
See Also: GCC, NGCC, wG
Usage:
Description:
The function print is used to write the interaction G coupling constant to an output stream ostr. An additional flag fflag is set to allow some control over how much information is output. The default (fflag !=0) prints all information concerning the interaction. If fflag is set to zero only the basis parameters are printed.Return Value:
The ostream is returned.Example:
See Also: <<
Usage:
Description:
The operator << defines standard output for the interaction G coupling constant.Return Value:
The ostream is returned.Example:
See Also: print
Usage:
Description:
The function print is used to write the interaction G coupling constant to an output stream ostr. An additional flag fflag is set to allow some control over how much information is output. The default (fflag !=0) prints all information concerning the interaction. If fflag is set to zero only the basis parameters are printed.Return Value:
The ostream is returned.Example:
See Also: <<
Usage:
Description:
The function print is used to write the interaction G coupling constant to an output stream ostr. An additional flag fflag is set to allow some control over how much information is output. The default (fflag !=0) prints all information concerning the interaction. If fflag is set to zero only the basis parameters are printed.Return Value:
The ostream is returned.Example:
See Also: <<
A G interaction is the observed effect from the electron cloud surrounding a nucleus responding to an applied magnetic field. The spin itself experiences not only the applied field but also a field from the perturbed electron cloud, the latter field generally opposing the applied field or "shielding" the nucleus. Not only can the shielding contribution be quite large, it is usually orientationally dependent because the surrounding electron cloud is no spherical (due to chemical bonds). In the following discussion we will not be concerned with the isotropic and anti-symmetric parts of the shielding. The former produces measureable chemical shifts whereas the latter is rarely seen. Rather the focus will be on the symmetric rank 2 contribution, that which produces relaxation effects in liquid NMR and orientationally dependent shifts in solids.
We will shortly concern ourselves with the mathematical representation of G 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.
The internal structure of class IntG contains the quantities listed in the following table (names shown are also internal).
Note that since the spin angular momentum of an electron is I=1/2, the spin tensor components will reside in a spin Hilbert space of dimension 2.
The three values AISO, DELZZ, and ETA are all that is required to specify the G interaction strength and may be used to represent the G spatial tensor. However, in GAMMA the values of AISO and DELZZ are factored out of the spatial tensor such that all rank two interactions (such as the G interaction) have the same spatial tensor scaling.
The two angles THETA and PHI indicate how the G interaction is aligned relative to the interaction principal axes (PAS). These are one in the same as the angles shown in Figure 19-24 when the Cartesian axes are those of the PAS with the origin vaguely being the center of the nucleus. These are intrinsically tied into the values in the array Asph.
There are five values in the complex vector Asph and these are irreducible spherical components of the G 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:
and the vector of complex numbers relate to the GAMMA normalized spherical spatial tensor components via
A chemical shift is the observed effect from the electron cloud surrounding a nucleus responding to an applied magnetic field. The spin itself experiences not only the applied field but also a field from the perturbed electron cloud, the latter field generally opposing the applied field or "shielding" the nucleus. We can write this latter "induced" field in terms of the applied field, 
, as
where 
is the chemical g tensor, a 3x3 array in Cartesian space, and the 
vectors in Cartesian space. In matrix form this is simply1
,
the induced field depends on the applied field strength, the applied field orientation, and the surrounding electron cloud. Note that 
will not necessarily be co-linear with the applied field. Of course, every nuclear spin will have its own associated chemical g tensor. The classical interaction energy between this induced field and a nuclear spin is
where 
is the electron magnetic moment, 
the energy, and superscript 
used to denote an electron G interaction.
The associated G interaction Hamiltonian is obtained from substitution of 
for 
.
;In matrix form this equation looks like
.Taking the magnitude of the applied field out, equation (39-1) is simply
with 
and 
a normalized magnetic field vector in the direction of the applied field.
Equation (39-2) can also be rearranged to produce an equation involving two rank 2 tensors by taking the dyadic product of the vectors 
and 
.
The dyadic product to produce 
is explicitly done via
.
The G interaction Hamiltonian can thus be formulated as a scalar product of two rank 2 tensors. Letting 
, we have
The previous equation, , can also be rewritten in term of irreducible spherical components rather than in terms of the Cartesian components using the substitution
where 
are spherical components of the tensor 
. The result is
and we can expand the summation over the different ranks.
.There is good reason to separate these terms. The rank 0 component of the G Hamiltonian is rotationally invariant and called the isotropic G Hamiltonian. In liquid EPR it will dictate where the electron resonance occurs. The rank 2 part is call the chemical G Anisotropy Hamiltonian. In liquid systems this Hamiltonian averages to zero and thus not affect observed g values. It will contribute to relaxation of the system. On the other hand, in solid systems this component does not average away and will partially determine peak shapes in powder averages. The rank 1 component is the antisymmetric part of the G Hamiltonian. Since the antisymmetric part of the G tensor is difficult to measure, this part of the G Hamiltonian is usually assumed small and neglected.
The isotropic component (
) of the G Hamiltonian is thus written
,
the antisymmetric component (
) of the G Hamiltonian is
,
and the anisotropic component (
) of the G Hamiltonian is
We can obtain the 9 irreducible spherical components of the G rank 2 "spin" tensor2 directly from the Cartesian components, 
, as indicated in GAMMA Class Documentation on Spin Tensors. These are
,
where 
signifies the electron G interaction. The tensor index 
spans the rank: 
while the tensor index 
spans 
: 
The nine formulas for these quantities a listed in the following figure where the field components are those of the normalized field vector 
.3
For 
, the matrix form of these tensor components are shown in the following figure in the single electron spin Hilbert space. The spin index has been omitted, the field components are those of the normalized vector 
.
The matrix representation of these nine tensor components will depend upon the matrix representations of the individual spin operators from which they are constructed4. These in turn depend upon the fact that electrons are spin 1/2 particles. Their G tensor components are, in the previous figure, expressed in matrix form in the default product basis of GAMMA. In this case the spin index is implicit.
The raising an lowering components of the field vector are defined in the standard fashion, namely 
. The simplest situation occurs when magnetic field points along the positive z-axis, 
, i.e. these spin-space tensors are written in the laboratory frame. Then, the (normalized) field vector simplifies, 
and 
. The applicable equations for the shielding space-spin tensors are then as follows.
For 
along the positive z-axis, the matrix form of these tensor components are shown in the following figure5 (in the single spin Hilbert space).
We must very careful in using these single spin rank 2 G tensors of this type because they contain both spatial and spin components. If we desire to express the G Hamiltonian relative to a particular set of axes we must insure that both the spatial tensor and the "spin" tensor are expressed in the proper coordinates. The spatial tensor alone cannot be rotated as it rotates only part of the spatial components6. It is improper to rotate this tensor in spin space because it also rotates spatial variables. Furthermore, note that these rank 2 components are not the same as the rank 1 tensor components.
The 9 irreducible spherical components of a rank 2 spatial tensor, 
, are related to its Cartesian components by the following formulas7.
Again the subscript l spans the rank as l = [0, 2], and the subscript m spans +/- l, m = [-l, l].
In this G interaction treatment, we then have the components 
as indicated in equation (39-5). Thus, the irreducible spherical tensor components can be obtained by substituting the Cartesian elements of the G tensor, 
, into equations (39-10).
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,
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 G spatial tensor to produce
.
As with any rank 2 spatial tensor, the G spatial tensor can be specified in its principal axis system, the set of axes in which the irreducible rank 2 component is diagonal8. The G tensor values are experimentally determined in the tensor principal axes. Employing (39-12) in the case where the irreducible rank 2 component is diagonal,
where (39-15) still applies.
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 g-tensor we have
Note that 
is NOT equivalent to 
and that 
is NOT equivalent to 
. The irreducible spherical elements of the G tensor, 
, in the principal axis system are, by placement of (39-16) into (39-10),
and these values should be equivalent to those given in (39-11) on page 5-334.
Throughout GAMMA, we desire all irreducible spherical rank 2 spatial components to be scaled so as 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 G irreducible rank 2 spatial tensor so that the 
component will have the same magnitude as the 
rank two spherical harmonic when the two spherical angles are set to zero. Our "normalization" factor "X" is obtained by
Using 
we thus define the GAMMA G anisotropy spatial tensor to be scaled such that its normalized spherical components are given by
and the irreducible rank 2 components are given in the next figure.
The scaling factor 
which was multiplied into the spherical G tensor components will subsequently be compensated for in the G interaction by use of a G interaction constant. The Anisotropic G Hamiltonian given in equation (23) becomes
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 G anisotropic Hamiltonian may be produced from
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 G tensor definitions. Most EPR literature retain the G 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 G anisotropy . This is readily related to the typical G tensor Cartesian components.
We can express the spatial tensor components 
relative to any arbitrary axis system (AAS) by a rotation from the principal axes to the new axes via the formula
where 
are the rank 
Wigner rotation matrix elements and 
the set of three Euler angles which relate the principal axes of the spatial tensor to the arbitrary axes9.
The G Hamiltonian can now be expressed with respect to any arbitrary axes through use of its spherical tensor components and the previous equation. Our Hamiltonian in spherical tensor form is
Negletcting the antisymmetric component and recalling that the isotropic component is rotationally invarient we obtain, for an arbitrary axis system
which becomes, if the system is related to the laboratory frame in which the static external field is pointed along +z,
At this point it is evident that the Hamiltonian has units which are dictated by the factor
This factor occurs in both isotropic and anisotropic terms. The G tensor is taken to be unitless and the units of angular momentum from the spin term are considered included in this factor. The value of the Bohr magneton 
is
and H is typicall specified in units of Gauss. Thus 
as shown will have energy units (ergs). We can readily convert to frequency units using h.
For a free electron where 
, the resonance frequency (the transition between 
) in a 3000 G field will be given by
Typical isotropic g factors are larger than that of a free electron so that a higher frequency will be required at any set field. However, most ESR spectrometers operate in CW mode where the frequency is set and the field is swept. As a result it is better to think that at a specified frequency most electrons resonance at a lower field than does a free electon.
The G tensor orientation will affect the observed electron resonance frequency. Unlike isotropic chemical shifts in NMR, the isotropic (rank 0) contribution to G is normally NOT included with the Zeeman Hamiltonian. Furthermore, the anti-symmetric (rank 1) contribution to G is rarely treated. The symmetric rank 2 contribution to the G interaction, that which we are primarily concerned with in class IntG, produces the following amisotropic Hamiltonian10.
The reader should note normally the spin tensors, 
, are specified in the laboratory frame where the applied magnetic field is along the +z axis. When that is true the 
terms are zero and the summation need only be taken over 
.
Furthermore, if we orient the spatial tensor principal axis system (PAS) to coincide with the laboratory axes, the anisotropic contribution to the G Hamiltonian is given by a relatively simple formula because both the 
terms are zero as well.
However, when the G interaction principal axes are not oriented to coincide with the laboratory axes the anisotropic Hamiltonian equation becomes much more complicated than the one above.
Remember, the orientation angles, 
and 
, are spherical angles relative to the laboratory coordinate system. We have thus left off the "LAB" label on all terms. At this point we will substitute in the spin operatiors (assuming H is along +z)
We can use the identities 
to obtain
Upon substitution of the oriented spatial components we obtain
which will condense down into the previous result, equation (39-22) on page 339, when the two angles are set to zero. Often it can be assumed that work is being done in a "high field limit" where the contributions to the anisotropy from the 
and 
terms is negligible. When such is the case the previous equation becomes (hfl => high field limit)
By combining the isotropic and anisotropic parts of the G Hamiltonian we obtain the full Hamiltonian. We are still excluding the anti-symmetric (rank 1) component.
We will define an isotropic resonance condition as 
so that the Hamiltonian can be expressed relative to some base frequency (or field) as
In the high-field limit, we have simply
and this explicitly indicates the dominant way in whcih the G Hamiltonian is modulated by the interaciton orientation.
Having determined what the G Hamiltonian looks like at any orientation we are now in the position to determine the electron transition frequency. Since the electron is only a spin 1/2 particle, there is only one transition and that is between the 
and 
states. We shall examime the energy levels of these states using 
, knowing that the transition frequency will be the difference between the two energies. Our working Hamitonian form is
and we can immediately calculate the isotropic contribution to the transition frequency.
The anisotropic contribution at high field is equally trivial. In fact, we can just read it off of equation (39-27) on page 341.
The third term, due to the x & y spin operator components are a bit more tenacious. We have
We can use the ladder operators define d earlier to determine the
). The isotropic component 
in the treatment of liquid samples will normally be placed into an overall isotropic Hamiltonian, 
because it does not disappear upon rotational averaging. The asymmetric component, 
, is usually zero, the G tensor taken as essentially symmetric. The The Electron G Anisotropy Hamiltonian
When the G interaction has alignment along its principal axes system virtually all of the G spatial tensor equations simplify. However, because the magnetic field components will then be oriented, the space-spin tensor components become complicated. Only when the PAS is aligned with the laboratory z-axis do both space and space-spin simplify. The following figure collects these equations for convenience.
= 0).
Included are the general relationships between the (GAMMA scaled) Cartesian tensor components to the irreducible spherical components. They are valid when is defined accordingly! If is defined by the other common convention (|Azz| 
|Axx| |Ayy|) then the sign on the 
will change as will the sign on the Hamiltonian terms multiplied by .
When the G interaction has a arbitrary alignment (relative to the laboratory frame, where the static field sets the z-axis) the G equations become sligltly more complicated. The figure below depicts them for convenience.
= 0).
This section describes how an ASCII file may be constructed that is self readable by a G interaction. The file can be created with any editor and is read with the G interaction member function "read". An example of one such file is given in its entirety at the end of this section. Keep in mind that parameter ordering in the file is arbitrary. Other parameters are allowed in the file which do not relate to G interactions.
AG: WG, WGkHz, WGKHz, WGHz, WGMHz
The G frequency can be specified. This can be accomplished with parameters using any of the names above or these names with a (#) added as a suffix. The default units for WG are KHz other names can be used to set the value in particular units. Note that this parameter is related to the G coupling constant which is specified with "(N)GCC" parameters. If both GCC and WG are set in the same file, the G frequency will be used to set up the G interaction.
| Parameter | Assumed Units | Examples Parameter (Type=1) : Value - Statement |
|---|---|---|
| WG | KHz | |
| WGMHz | MHz | |
| WGHz | Hz |
G Frequency: WG, WGkHz, WGKHz, WGHz, WGMHz
The G frequency can be specified. This can be accomplished with parameters using any of the names above or these names with a (#) added as a suffix. The default units for WG are KHz other names can be used to set the value in particular units. Note that this parameter is related to the G coupling constant which is specified with "(N)GCC" parameters. If both GCC and WG are set in the same file, the G frequency will be used to set up the G interaction.
| Parameter | Assumed Units | Examples Parameter (Type=1) : Value - Statement |
|---|---|---|
| WG | KHz | |
| WGMHz | MHz | |
| WGHz | Hz |
G Coupling Constant: GCC, GCCkHz, GCCKHz, GCCHz, GCCMHz
The G coupling constant can be specified. This can be accomplished with parameters using any of the names above, these same names with an "N" as a prefix, and/or these names with a (#) added as a suffix. The default units for GCC are KHz other names can be used to set the value in particular units. Note that this parameter is related to the G frequency which is specified with "WG" parameters. If both GCC and WG are set in the same file, the G frequency will be used to set up the G interaction.
| Parameter | Assumed Units | Examples Parameter (Type=1) : Value - Statement |
|---|---|---|
| GCC | KHz | |
| NGCCMHz | MHz | |
| GCCHz | Hz |
G Asymmetry
The asymmetry parameter must be within the range of [0, 1]. This parameter does not need to be set for a G interaction definition, it will be assumed 0 if unspecified.
| Parameter | Assumed Units | Examples Parameter (Type=1) : Value - Statement |
|---|---|---|
| Geta | none |
1
Parameter type 1 indicates an integer parameter. |
G Theta Orientation
The angle theta which relates the G interactions orientation down from the z-axis of its PAS may be set. This is not essential and will be taken as zero in left unspecified.
| Parameter | Assumed Units | Examples Parameter (Type=1) : Value - Statement |
|---|---|---|
| Gtheta | degrees |
1
Parameter type 1 indicates an integer parameter. |
G Phi Orientation
The angle phi which relates the G interactions orientation over from the x-axis of its PAS may be set. This is not essential and will be taken as zero in left unspecified.
| Parameter | Assumed Units | Examples Parameter (Type=1) : Value - Statement |
|---|---|---|
| Gphi | degrees |
1
Parameter type 1 indicates an integer parameter. |
Note that the effect of the G tensor is to alter the overall external field which the electron experiences. This is clearly seen from the product2which produces an effective field vector for the electron.
Due to the nature of the G interaction, the rank 2 tensor treatment produces a "spin" tensor3which contains spatial components, namely the magnetic field vector. As a result, care must be used when performing spatial rotations on G tensors. Any spatial rotations must involve rotations of both
and
For these formulae, it is important to note that it is the second component in the composite spin/space tensor which is set to the normalized magnetic field vector4, although we might just as well have used the first vector instead. The difference is that the
equations would then appear of opposite sign from those given here. Our field vector has be set to point along the positive z-axis in the laboratory frame.
Note that the spin tensors are invariably constructed in the laboratory coordinate system. Here the z-axis corresponds to the direction of the spectrometer static magnetic field and the coordinate system is right-handed.5
The GAMMA program which produced these matrix representations can be found at the end of this Chapter, sosix Rank2SS_SpinT.cc.6
See the discussion in Mehring7
See the GAMMA Class Documentaion on Rank 2 Interactions.8
The principal axis system is set such that |dzz|9
In this instance, i.e. the treatment of an electron G interaction, we must be careful to express the elements10in the same axis system as
. When
is rotated in space, so must be
. Essentially, the field vector changes relative to any new coordinate system when constructing
. In other words, when
is represented in its PAS (normally thought of as
) it does NOT necessarily see the externally applied field point along +z since the latter is defined in the laboratory frame whereas the former is set in an internal (electron cloud fixed) frame.
Keep in mind that this Hamiltonian is for a single electron. In a multi-spin system one will have to sum such Hamiltonians for all electron spins.11
The scaling on both {A2m} and T2m} are arbitrary, GAMMA uses an (uncommon) scaling which independent of the interaction type. What is NOT arbitrary is the scaling within either of the two sets of components. In addition, the combined scaling of the two sets is critical to the proper formation of G Hamiltonians. For that, GAMMA uses an interaction constant.12
The G interaction constant, as well as the relative scalings on the sets of spatial and spin tensors, can be adjusted as desired. However all components of the space or spin tensor must be adjusted by the same scaling. The GAMMA scaling is oriented to liquids where so that all spatial components are related to the spherical harmonics in the spatial tensor PAS.13
The scaling on both {A2m} and T2m} are arbitrary, GAMMA uses a scaling which independent of the interaction type. What is NOT arbitrary is the scaling within either of the two sets of components. In addition, the combined scaling of the two sets is also crucial. For that, GAMMA uses an interaction constant.
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GAMMA Support Provided by the National High Magnetic Field Laboratory
© 1996 Scott A. Smith, The NHMFL, and The Florida State University. All Rights Reserved. |
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