Nuclear quadrupole resonance (NQR) spectroscopy is a method to characterize chemical compounds containing quadrupolar nuclei (QN). Similar as in nuclear magnetic resonance (NMR) the sample under investigation is irradiated with strong radiofrequency (RF) pulses to induce and detect transitions between sublevels of nuclear ground states [1] [2]. NMR refers to the situation where the sublevel energy splitting is predominantly due to nuclear interaction with an applied static magnetic field, while NQR referes to the case where the predominant splitting is due to an interaction with electric field gradients (EFG) within the material [1]. Quadrupole splitting is possible if the charge distribution of the nucleus has an electric quadrupole moment. This is the case if the spin quantum-number I is greater than 1/2. The electric quadrupole moment of the nucleus interacts with the EFG which is caused by a non-spherical charge distribution of the environment of the nucleus.

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 .

The presence of an external magnetic field B_{0} results in Zeeman splitting. The eigenstates E_{Z}(m) of a pure Zeeman splitting are defined as follows:

**E _{Z}(m) = m * f_{Larmor} * h = m * γ * B_{0} * h**

where f_{Larmor} is the Larmor frequency at the presence of a static magnetic field B_{0}. γ is the specific gyromagnetic constant of the nucleus.

**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

The correction function f_{η}(m-m') depends on the asymmetry parameter η. The correct transition frequency can be calculated as follows:

**f _{Q}(m-m') = A/h * 3(m^{2} - m'^{2}) * f_{η}(m-m')**

In systems with small values of η only transitions of ∆m = ±1 are allowed [3]. Figures 3-5 show f_{η}(m-m') depending on η for transitions of ∆m = ±1.

[1] B. H. Suits: Nuclear Quadrupole Resonance Spectroscopy. In: Vij, D.R. (Ed.) Handbook of Applied Solid State Spectroscopy. Berlin, Springer Verlag (2006)

[2] H. Scharfetter, A. Petrovic, H. Eggenhofer, R. Stollberger: A no-tune no-match wide- band probe for nuclear quadrupole resonance spectroscopy in the VHF frequency range. Meas. Sci. Technol. 25: 125501 (2014)

[3] T. P. Das, E. L. Hahn: Nuclear quadrupole resonance spectroscopy. In: F. Setz, D. Turn- bull (Ed.) Solid State Physics. New York - London, Academic Press (1958)