Magnetism and Superconductivity in Iron-based Superconductors as Probed by Nuclear Magnetic Resonance
Nuclear Magnetic Resonance (NMR) has been a fundamental player in the studies of superconducting materials for many decades. This local probe technique allows for the study of the static electronic properties as well as of the low energy excitations of th
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le anisotropies, it is convenient to use [36, 52, 55]: 1 Kiso = (KX + KY + KZ ) 3 1 Kaniso = (KY − KX ) 2 1 (A.2) Kaxial = (2KZ − KX − KY ) , 6 such that in the end the Knight shift Kz along the direction z of the applied magnetic field can be expressed as [36, 52, 55]: Kz (θ, φ) = Kiso + Kaxial (3 cos2 θ − 1) + Kaniso sin2 θ cos 2φ .
(A.3)
The isotropic part Kiso shifts the resonance line without affecting its line width, while the axial and the anisotropic contributions to the Knight shift, Kaxial and Kaniso , entail a broadening of the line.
F. Hammerath, Magnetism and Superconductivity in Ironbased Superconductors as Probed by Nuclear Magnetic Resonance, DOI 10.1007/978-3-8348-2423-3, © Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden 2012
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A Appendix
The first order quadrupole interactions [see Eq. (2.33)] lead to the appearance of broadened quadrupolar satellites symmetrically distributed around the central transition [32, 36] (see Fig. A.1). In second order, the quadrupole interaction [see Eq. (2.34)]: νq2 1 3 Δ(2) νm (θ, φ, η) = − I(I + 1) − A(φ, η) cos4 θ + B(φ, η) cos2 θ + C(φ, η) (A.4) νL 6 4 with the prefactors [54–56]: 27 9 3 − η cos 2φ − η 2 cos2 2φ 8 4 8 3 30 1 2 − η + 2η cos 2φ + η 2 cos2 2φ B(φ, η) = 8 2 4 3 1 3 1 C(φ, η) = − + η 2 + η cos 2φ − η 2 cos2 2φ 8 3 4 8 A(φ, η) = −
(A.5)
gives rise to an asymmetric shape of the central line, where, depending on the value of η, five (η < 1/3) or six (η > 1/3) characteristic features can be observed [52, 55]. Fig. A.2 shows the line shape of the central transition for the case η < 1/3. Two singularities, two shoulders and a step appear as characteristic features in the line shape. For the prominent case of η = 0, the shoulders merge with their neighbouring singularities and the step appears at the Larmor frequency ν = νL (neglecting magnetic hyperfine corrections, which will shift νL by the Knight shift K). If the second order quadrupole interaction is sufficiently large, it may also affect the satellites [52, 55]. For details of this effect, the reader is referred to [52, 53].
Figure A.1: Theoretical powder pattern for I = 3/2 in case of first order quadrupole effects for η = 0. For clarity the intensity of the satellites is artificially enhanced. Magnetic hyperfine contributions are neglected in this sketch (K = 0).
A.1 NMR Powder Spectra
137
4h² -16(1+h) -16(1-h)
0
(3-h)² (3+h)² n-nL
Figure A.2: Theoretical powder pattern for the central transition (m = − 1/2 ↔ m = +1/2) in case of second order quadrupole effects for η < 1/3 (reproduced from [55]). Frequencies are given in units of (νq2 /144νL )[I(I + 1) − 3/4]. Magnetic hyperfine interactions are neglected (K = 0).
Combining second order quadrupole and first order magnetic hyperfine corrections corresponding to Equations (A.3), (A.4) and (A.5), and assuming that the principal axes of the EFG and the Knight shift tensor coincide with each other, the powder pattern of the central resonance line resembles the one depicted in Fig. A.2. The characteristic features correspondin
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