Page 40 out of 81 total pages , Page 13 out of 15 pages in this chapter

3.19 Literature Comparisons

3.19.1 P.P. Man's "Quadrupolar Interactions"

The following figure lists some of the equations found in Pascal P. Man's article1 along with the corresponding GAMMA equations. Aside from difference in scaling factors, GAMMA is in full agreement with these equations.

Comparison of GAMMA & P.P. Man's Quadrupolar Equations

Note that Man puts an hbar in front of his Hamiltonians to indicate that they are expressed in angular frequency units (I've left them out). In GAMMA, the Hamiltonian functions associated with class IntQuad will typically be in units of Hz. To be precise, they will have the unit that are used during construction of the interaction (e.g. if DCC is set in Hz, the returned Hamiltonian(s) will be in Hz as well).

The following table indicates the variables in Man's equations and the conversion factor required to switch between his nomenclature and GAMMA's.

G & P.P. Man's Quadrupolar Equation Variables
Mathematical Construct G Man G = X * Man
Spin Tensor Components

Spatial Tensor Components

(Interaction) Constant

Other

Quadrupolar Hamiltonian

11

1
Man says his Hamiltonians are in angular frequency units, but the Hamiltonian units in GAMMA are relative. So there may indeed be a conversion factor but that depends on what on chooses to express is "eQ" factor in.

Of the equations tested, GAMMA is in perfect agreement with Man's equations. The correspondence benefits from use of the same PAS definition

Users who wish to make direct computational comparisons between the two treatments should see the literature comparison programs IntQu_LC0.cc and IntQu_LC2.cc which are found at the end of this chapter.

3.19.2 Alexander Vega's "Quadrupolar Nuclei in Solids"

The following figure lists some of the equations found in Alexander Vega's article2 along with the corresponding GAMMA equations. Please take careful not of the differences between this article and what GAMMA's IntQuad class contains.

Comparison of GAMMA & A.J. Vega's Quadrupolar Equations 3

Note that Vega puts an hbar in front of his Hamiltonians to indicate that they are expressed in angular frequency units (I've left them out). In GAMMA, the Hamiltonian functions associated with class IntQuad will typically be in units of Hz. To be precise, they will have the unit that are used during construction of the interaction (e.g. if DCC is set in Hz, the returned Hamiltonian(s) will be in Hz as well).

The following table indicates the variables in Vega's equations and the conversion factor required to switch between his nomenclature and GAMMA's.

G & A.J. Vega's Quadrupolar Equation Variables
Mathematical Construct G Vega G = X * Vega
Spin Tensor Components

Spatial Tensor Components

1

(Interaction) Constant

Quadrupolar Hamiltonian

12

1
Because Vega uses a different convention for defining his PAS than GAMMA, a sign change is requited to relate
 2
Vega says his Hamiltonians are in angular frequency units, but the Hamiltonian units in GAMMA are relative. So there may indeed be a conversion factor but that depends on what on chooses to express is "eQ" factor in.

Of the equations tested, GAMMA is NOT in very good agreement with Vega's equations. A small part of the problem is that Vega has used a different convention for his PAS definition that does GAMMA.

GAMMA:       VEGA:

This simply causes all of his equations to have the sign changed on . Of a more serious nature is the (misprint?) definition of the m=0 spin tensor component. The one in the article is NOT a rank 2 irreducible tensor component. Although the scaling factor on all components can be (simultaneously) changed, the relationship between the components must be strictly adhered to or else there will be trouble when the quadrupolar Hamiltonian is rotated in space.

In addition, it seems that Vega's Hamiltonian for an I=1 case, his Eq. (32) is incorrect and does NOT correspond to (28), although the I=3/2 case in Eq. (33) appears correct. Users who wish to make direct computational comparisons between the two treatments should see the literature comparison programs IntQu_LC0.cc - IntQu_LC3.cc which are found at the end of this chapter.



Page 40 out of 81 total pages , Page 13 out of 15 pages in this chapter

1
"Quadrupolar Interactions", P.P. Man, Encyclopedia of Nuclear Magnetic Resonance, Editors-in-Chief D.M.Grant and R.K. Harris, Vol. 6, Ped-Rel, pgs. 3838-3948.
 2
"Quadrupolar Nuclei in Solids", Alexander J. Vega, Encyclopedia of Nuclear Magnetic Resonance, Editors-in-Chief D.M.Grant and R.K. Harris, Vol. 6, Ped-Rel, pgs 3869-3889.
 3
I've taken the liberty to correct Vega's equation herein which could not (should not) be correct as printed. Note that Vega's PAS definition does NOT coincide with GAMMA's, thus the sign on must be changed on all equations containing it if one is to do comparisons.
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