In Co(1-x)Cd(x) ferrite , I calculated lattice parameter it showed a constant behaviour with variation of x why???? although there is a difference in atomic radius between Co and Cd
Do you observe shift in XRd pattern? Maybe you did some mistake during calculations.
Probably this ferrite crystallizes in fcc unit cell, so you can calculate lattice constant using following formula:
It is rewritten Bragg equation
(h^2+k^2+l^2)^(1/2)=(2sin(theta))/lambda* a
where h,k,l are Miller indices, theta-Bragg degree, lambda - wavelenght of XRd radiation and a is lattice cobstant. If you have several reflexes you can calculate a using linear regression.
I understand you calculated It but you can chęci your results.
Good luck
Greetings
So do you observe shift of peaks toward higher/lower angles in Xrd partern?
Can we interpret your question of, "a constant behaviour with variation of x" to mean "a monotonic change in the lattice parameter as a function of x"?
If so, then your second sentence answered your question. For a given (alloyed) crystal structure, the lattice parameter will vary with ionic radii of the constituents. Google or Wiki "Vegard's Rule".
Do the words "constant behaviour with variation of x" the independence of a from x?
Since the ferrite spinels are compounds that are not alloys, but complex oxides with different substitutions at the positions with different coordination, sometimes having cation- or anion-deficient structures, the Vegard's rule (which itself is not strong law) does not always work well.
I suppose, You could or to carry out the calculation, based on data from X-ray diffraction, or he is referring to the calculation for some models: a) based on the lattice parameters of the initial simple ferrites (e.g., by the Vegard's rule), b) on the basis of ionic radii. In case (b) simple models consider only the linear terms of ionic radii, and we have relation, which corresponds to the Vegard’s rule with a direct dependence on x. But in the General case model includes quadratic forms of the type Ra^2, Rb^2 and Ra*Rb, where Ra and Rb are the ionic radii in tetra- and octahedral positions of crystal. At the same time the cations of the same class have different values of ion radius in positions with different coordination (a and b). Therefore, the results of the calculations may be different.
In Your case, since Cd2+ occupies a-sites (normal spinel) and Co2+ occupies b-sites (inverted spinel), it is formed a mixed spinel. The substitutions occur in different sublattices of the crystal structure, so the Vegard’s rule can enforce poorly. By the way, ionic radii of Cd2+ in a-sites and Co2+ in the b-sites are almost equal (78 pm and 74,5 pm respectively). But at the same time, it take place the transition of the Fe3+ cations from a-sites (where their ionic radius is 49 pm) into b-sites (with ionic radius of 64.5 pm) and the dependence of lattice parameter on the ionic radius and x can be really nonlinear. In this case, the resulting change should be small for moderate values of x. Thus, the peaks remain in place. But with a higher degree of substitution, the difference will be noticeable.
So, approximations lattice parameter of CoGd ferrite by linear function of cation radius and in the form of a second-order polynom give the same values (0.838 and 0.869 nm at x = 0 and 1.0, respectively), while account of nonlinear terms leads to the difference of values (0.844 and 0.843 nm for x = 0.2).
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