Does the lattice constant of a unit cell vary over isotopes? e.g Si 28 and Si 29
No. Isotope number only affects the size of the nucleus, which is negligible compared to the electron cloud around each nucleus. It is the interaction between electron clouds that determines lattice constants.
Some elements do exhibit isotope nuclear spin effects, which show up in the rotational spectra of diatomic molecules containing two of the same atom, but I doubt if this effect would not show up in solid lattices.
Then why does the new silicon kilogram prototype need to be made out of pure isotope silicon-28?
That is a good question, so I had to check. It has to do with wanting to define the kg in terms of a precise Avogadro number of a particular atom, without specifying a "natural" isotope ratio. See the Wiki article on "kilogram". It contains this quote:
<<All silicon-based approaches would fix the Avogadro constant but vary in the details of the definition of the kilogram. One approach would use silicon with all three of its natural isotopes present. About 7.77% of silicon comprises the two heavier isotopes: silicon-29 and silicon-30. As described in Carbon–12 above, this method would define the magnitude of the kilogram in terms of a certain number of carbon-12 atoms by fixing the Avogadro constant; the silicon sphere would be the practical realization. This approach could accurately delineate the magnitude of the kilogram because the mass of the three silicon isotopes relative to carbon-12 are known with great precision. An alternative method for creating a silicon sphere-based kilogram proposes to use isotopic separation techniques to enrich the silicon until it is nearly pure silicon-28, which has an atomic mass of 27.9769271(7) g/mol. With this approach, the Avogadro constant would not only be fixed, but so too would the atomic mass of silicon-28. As such, the definition of the kilogram would be decoupled from carbon-12 and the kilogram would instead be defined as 1000/27.9769271 × 6.02214179 × 1023 atoms of silicon-28 (≅35.743739699 fixed moles of silicon-28 atoms). Physicists could elect to define the kilogram in terms of silicon-28 even when kilogram prototypes are made of natural silicon (all three isotopes present). Even with a kilogram definition based on silicon-28, a silicon-sphere prototype made of nearly pure silicon-28 would necessarily deviate slightly from the defined number of moles of silicon in order to compensate for various chemical and isotopic impurities as well as the effect of surface oxides.[24]>>
The suggested choice of isotopically pure silicaon as a kg standard has nothing to do with the lattice constant. I stand by my previous answer.
The lattice constant of a unit cell is determined by the arrangement of atoms within the crystal structure. Isotopes, which are atoms with the same number of protons but different numbers of neutrons, have slightly different atomic masses but the same chemical properties. Therefore, the difference in the atomic mass of isotopes should not significantly affect the lattice constant of a crystal.
To confirm this, you can check experimental data or consult reliable sources such as scientific journals or textbooks. Experimental techniques like X-ray diffraction or neutron scattering can measure the lattice constant of crystals accurately. By comparing the lattice constants of different isotopes of the same element, you'll be able to see if there are any significant differences. Keep in mind that for most elements, the isotopic composition is dominated by one or a few isotopes, so the overall effect on the lattice constant may be negligible.