Pulsed nuclear magnetic resonance of $^{175}\mathrm{Lu}$ (I=(7/2) was used to study the electronic phase transition of samples ${\mathrm{Lu}}_{5}$${\mathrm{Ir}}_{4}$${\mathrm{Si}}_{10}$ and ${\mathrm{Lu}}_{5}$${\mathrm{Rh}}_{4}$${\mathrm{Si}}_{10}$ under atmospheric pressure from 49 to 310 K. A calculation of the powder line shape in the quadrupole regime for I=(7/2 shows that the critical frequencies, particularly that of the singularity, are sensitive to the change of the electric-field-gradient (EFG) asymmetry and the Larmor frequency. From the singularity near \ensuremath{\nu}=${\ensuremath{\nu}}_{0}$, the EFG asymmetry of the lutetium nucleus is estimated to be \ensuremath{\eta}\ensuremath{\simeq}0.4. The temperature coefficient of the lutetium EFG asymmetry as estimated from the temperature-dependent shift is close to 3\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}5}$ ${\mathrm{K}}^{\mathrm{\ensuremath{-}}1}$. This small variation of the EFG asymmetry suggests that the crystal structure remains unchanged in the temperature range studied. The possible formation of charge-density-wave states at 83 and 145 K for the two compounds, respectively, is indicated by anomalous variations of the temperature-dependent Knight shift. A loss of the electronic density of states at the Fermi level, as inferred from the jump discontinuity of the Knight shift, is consistent with previous measurements of resistivity and the magnetic susceptibility on these samples. From the correlation of the Knight shift with the magnetic susceptibility, the hyperfine field is determined to be 0.5 and 0.7 MOe, a value consistent with the previous measurements of rare-earth metals. The NMR of $^{45}\mathrm{Sc}$ (I=(7/2) in ${\mathrm{Sc}}_{5}$${\mathrm{Ir}}_{4}$${\mathrm{Si}}_{10}$, isostructural with the above two compounds, was used as a baseline measurement to illustrate the anomalous behavior observed for $^{175}\mathrm{Lu}$ in ${\mathrm{Lu}}_{5}$${\mathrm{M}}_{4}$${\mathrm{Si}}_{10}$ (M=Ir,Rh).
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