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We propose a new ion-trap geometry to carry out accurate measurements of the quadrupole shifts in the 171Yb ion. This trap will minimize the quadrupole shift due to the harmonic component of the confining potential by an order of magnitude. This will be useful to reduce the uncertainties in the clock frequency measurements of the 6s 2S1/2 → 4f136s22F7/2 and 6s 2S1/2 → 5d 2D3/2 transitions, from which we can deduce the precise values of the quadrupole moments (Θs) of the 4f136s22F7/2 and 5d 2D3/2 states. Moreover, it may be able to affirm the validity of the measured Θ value of the 4f136s22F7/2 state, for which three independent theoretical studies defer almost by one order of magnitude from the measurement. We also calculate Θs using the relativistic coupled-cluster (RCC) method. We use these Θ values to estimate the quadrupole shift that can be measured in our proposed ion trap experiment. [ABSTRACT FROM AUTHOR]