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Assuming that the only thing that changes in the experiment is the magnetic moment µ. We can then measure a for both the first nucleus with known µ_1 and the second nucleus with unknown µ_2. The ratio of a_1 / a_2 will then be equal to µ_1 / µ_2. Since only µ_2 is unknown at this point, its value can easily be extracted from the experimental data.
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