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By measuring the hyperfine splitting of both particles under an external magnetic field, will allow us to take the hyperfine field considered in the equations to be dominated by this field and thus the same for both isotopes. This allows us to substitute the hyperfine field in the equation for the hyperfine coupling constant of the second isotope with the equation of the hyperfine coupling constant of the first isotope for which we know both the constant and magnetic dipole moment. Since we know the constant of the second isotope we can effectively calculate the magnetic dipole moment of second isotope.
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