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I have a simple question (but I precise that I am very a beginner in solid states physics).

I know that the reciprocal vectors have this form : $ \overrightarrow{G}=n_1\overrightarrow{b_1}+n_2\overrightarrow{b_2}+n_3\overrightarrow{b_3} $ with n1, n2, n3 relative integers.

But at the same time, I have found a course where when we try to find bloch waves, it is said that k has this form : $ \overrightarrow{k}=\frac{n_1}{N_1}\overrightarrow{b_1}+\frac{n_2}{N_2}\overrightarrow{b_2}+\frac{n_3}{N_3}\overrightarrow{b_3} $ (N1,N2,N3 are the limit dimension of the lattice, n1,n2,n3 are relative integers). And it is said that these k are in the reciprocal lattice.

BUT it can't be in the reciprocal lattice because $ \frac{n_1}{N_1} $ is not an integer ?!

This is what I don't understand.

Could you help me ?

Thank you !

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  • $\begingroup$ If I remember correctly the $\mathbf{k}$ values are generally not reciprocal lattice vectors themselves, but they live in the same reciprocal space and since $\frac{n_i}{N_i} < 1$ the $\mathbf{k}$s are all in the first Brillouin zone (defined by the basis vectors $\mathbf{b_i}$). $\endgroup$ – Kyle Arean-Raines Jan 27 '16 at 21:36
  • $\begingroup$ So if I am correct, thoose waves vector are NOT in the reciprocal lattice, but they live in the same space than the reciprocal lattice vectors. And more specially, they live in the first Brillouin zone ? $\endgroup$ – StarBucK Jan 28 '16 at 15:27
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deriving the desired expression for $\boldsymbol k$ and figuring out what $\boldsymbol k$ really is we will begin with the definition of bloch's theorem. $$\psi_{\boldsymbol k}(\boldsymbol r + \boldsymbol t_n) = e^{i \boldsymbol k \cdot \boldsymbol t_n} \psi_{\boldsymbol k} (\boldsymbol r)$$

here $\boldsymbol k$ is a wave vector defined in the primitive cell of the reciprocal lattice for which solutions to the schrodinger equation for a lattice periodic hamiltonian can be characterized.

the definition of $\boldsymbol k$ in bloch's theorem is used to express it in a more condensed form. it arises from the solution to the expanded form of the equation for the eigenvalues of the translation operator $O(t)$ denoted by $t_a$.$$t_a(\boldsymbol t_n) = \prod^3_{i=1}\big[t_a(\boldsymbol a_i)\big]^{n_i}$$ under born-von karman boundary conditions it is implied that $\big [ t_a(\boldsymbol a_i) \big ]^{N_i} = 1$ for $i = 1, 2, 3$; which has the general solution of $t_a(\boldsymbol a_i) = e^{-2\pi ip_i/N_i}$ where $p_i$ is an integer. substituting that into the equation for $t_a(\boldsymbol t_n)$ we find that $$t_a(\boldsymbol t_n) = e^{-2 \pi i(n_1p_1/N_1 + n_2p_3/N_2 + n_3p_3/N_3)}$$ we can now define the wave vector $\boldsymbol k$ as $$\boldsymbol k = \frac{p_1}{N_1}\boldsymbol b_1 + \frac{p_2}{N_2}\boldsymbol b_2 + \frac{p_3}{N_3}\boldsymbol b_3$$ where $\boldsymbol b_i$ $i = 1, 2, 3$ are the primitive vectors of the reciprocal lattice.

therefore the wave vector $\boldsymbol k$ is defined to be within the primitive cell of the reciprocal lattice.

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  • $\begingroup$ Thank you. So am I right if I say that the k vector is not a reciprocal lattice vector but it lives in the same space. More precisely it lives in first brillouin zone. In fact what I misunderstood with my course is that they say that k is a reciprocal lattice vector, but if its coeffcient are not integers it can't be a reciprocal lattice vector by definition. But of course it lives in the same space than reciprocal lattice vectors. It is probably just a question of vocabulary ? $\endgroup$ – StarBucK Jan 28 '16 at 15:34
  • $\begingroup$ @user3183950 well $\boldsymbol k$ can be associated with a wave vector in the brillouin zone. though $\boldsymbol k$ itself is within the primitive cell of the reciprocal lattice. $\endgroup$ – user105307 Jan 28 '16 at 18:23
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$b_1, b_2$ and $b_3$ are reciprocal primitive vectors. $G$ is the set of all vectors that are in the reciprocal lattice and, as you said, is given by the linear combination of the reciprocal primitive vectors. The set of points, $G$, just define the lattice vectors or the locations of the origin of each Brillouin zone.

Now we need to look within each Brillouin zone. There are numerous $k$ points within each Brillouin zone, the number of which is determined by the size of the real space lattice. So within each Brillouin zone whose origin is a member of $G$, are numerous $k$ points given by:

$$k = \frac{n_1}{N_1}b_1 + \frac{n_2}{N_2}b_2+ \frac{n_2}{N_2}b_1$$

Now, $n_1$ is an integer and the resulting coefficient $n_1$/$N_1$ is a fraction. The Brillouin zone is divided up further. Values of $n_1$ greater than $N_1$ simply correspond to $k$ points in the next Brillouin zone along.

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  • $\begingroup$ Thank you. So, is the first Brillouin an important notion because it is used in bloch theory (because the k vectors of the bloch waves live in the first brillouin zone ?). $\endgroup$ – StarBucK Jan 28 '16 at 15:29

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