The EFG can be described by a real, symmetric, traceless 3x3 tensor. Such a tensor can always be made diagonal by choosing an appropriate set of coordinate axes known as principal axes (Vxx,Vyy,Vzz) [1]. It is common to define:

**eq = V _{zz}** and

where η is the asymmetry parameter which indicates the degree of the deviation of the EFG from its axisymmetric shape. Usually the principal axes are chosen as follows:

**|V _{xx}| ≤ |V_{yy}| ≤ |V_{zz}|**

One of the goals of an NQR measurement is to determine the quadrupole coupling constant e^{2}qQ and the asymmetry parameter η, which contain information about the environment surrounding the nucleus [1]. Based on this parameters conclusions with respect to changes of the chrystallographic structures can be made. In case of axial symmetry, η=0, 2*I+1 quadrupole energy levels are observable. For ease of representation we define as follows:

**A = e ^{2}qQ / ( 4I ( 2I - 1 ) )**

The quadrupole energy levels can be calculated as follows:

**E _{Q}(m) = A * ( 3m^{2} - I ( I + 1 ) )**

where m is the magnetic quantum number (m = -I, -I+1, ..., I-1, I). The quadrupole frequencies f_{Q}(m-m') are corresponding to the difference between two energy levels (E_{m} - E_{m'}). For the more general case of η ≠ 0, a correction function f_{η} hast to be multiplied with f_{Q}(m-m'). If η=0, f_{η}=1 for every kind of nucleus. For quadrupole nuclei with a higher spin quantum number I than 3/2 an exact solution for f_{η} is not known [1]. One option to get accurate results is to use tabulated iterative calculation results. The Landolt-Börnstein tables provide an easy and precise method to determine values for quadrupole frequencies and correction functions for nuclei with spin quantum number I > 3/2. Another possibility is just to diagonalize the quadrupolar Hamiltonean numerically and calculate the respective eigenvalues for arbitraty η .

**Spin 3/2 nuclei:**

Spin 3/2 nuclei have two quadrupolar energy levels and therefore only one transition frequency [1]. Fig. 1 shows the energy spectrum in case of η=0 in the absence (E_{Q}) and in the presence (E_{QZ}) of an external magnetic field B_{0}. We here assume that the quadrupolar coupling is much stronger than the Zeeman interaction. The corresponding frequency spectrum is shown in fig. 2. In the presence of a weak B_{0} the initial transition frequency (f_{Q}) is split into two distinct frequencies (f_{Q} - f_{0}, f_{Q} + f_{0}). The shift frequency f_{0} depends on f_{Larmor} and on the angle α between the direction of V_{zz} and the direction of B_{0}.

**f _{0} = f_{Larmor} * cos (α)**

Two discrete frequency peaks are just observable if the compounds have perfect mono-crystalline structures. In case of powders or substances with many crystalline defects the spectrum shows a broad frequency distribution (pake-doublet). The energy levels under the presence of an external magnetic field B_{0} can be calculated as followed:

**E _{QZ}(m) = E_{Q}(m) + E_{Z}(m) * cos (α)**

The animation shows how the energy levels and the frequency spectrum of fig. 1 and 2 build up. It is clearly recognizable that the pure quadrupole frequency f_{Q} disappears under the presence of an external magnetic field B_{0}.

Animation 1

**Spin > 3/2 nuclei**

Fig. 3 shows the energy levels in the absence (E_{Q}) and in the presence (E_{QZ}) of a weak external magnetic field **B _{0} **for quadrupole nuclei with a spin quantum number of I=9/2 (η=0). The frequency spectrum corresponding to the energy spectrum is shown in fig 4. In the presence of

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