Precision measurements of the hyperfine splitting of ${\mathrm{Cs}}^{133}$ or ${\mathrm{Rb}}^{87}$, in a cell with buffer gases and using optical pumping, show a frequency shift when the intensity of the exciting resonance light is varied. Use of high buffer gas pressures will reduce considerably the light intensity shift. Also, in some cases, the magnitude and sign of the light intensity shift can be changed appreciably by varying slightly the frequency of the hyperfine components of the exciting light. Tentative explanations of the light shift are discussed. Frequency shift variations with light intensity, buffer gas pressure, and temperature of the cell, show the existence of an invariant point whose frequency, when reduced to zero pressure and zero field, is very close to the value obtained by the atomic beam resonance method. This invariant point is the basis for a definition of the pressure shift and temperature shift coefficients. Experimental determination of these coefficients is given for ${\mathrm{Cs}}^{133}$ and ${\mathrm{Rb}}^{87}$.

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