How do the three Miller indices (hkl) denote planes orthogonal to the reciprocal lattice vector?

Question

The Wikipedia page for Miller indices defines Miller indices as follows:

There are two equivalent ways to define the meaning of the Miller indices: via a point in the reciprocal lattice, or as the inverse intercepts along the lattice vectors. Both definitions are given below. In either case, one needs to choose the three lattice vectors $\mathbf{a}_1$, $\mathbf{a}_2$, and $\mathbf{a}_3$ that define the unit cell (note that the conventional unit cell may be larger than the primitive cell of the Bravais lattice, as the examples below illustrate). Given these, the three primitive reciprocal lattice vectors are also determined (denoted $\mathbf{b}_1$, $\mathbf{b}_2$, and $\mathbf{b}_3$).

Then, given the three Miller indices $h, k, \ell, (hk\ell)$ denotes planes orthogonal to the reciprocal lattice vector:

$$\mathbf{g}_{hk\ell} = h \mathbf{b}_1 + k \mathbf{b}_2 + \ell \mathbf{b}_3.$$

That is, ($hk\ell$) simply indicates a normal to the planes in the basis of the primitive reciprocal lattice vectors. Because the coordinates are integers, this normal is itself always a reciprocal lattice vector. The requirement of lowest terms means that it is the shortest reciprocal lattice vector in the given direction.

This idea of the three Miller indices ($hk\ell$) denoting planes orthogonal to the reciprocal lattice vector is something that I'm not confident that I'm understanding. My understanding is that the real (real space) lattice vector is described by the Miller indices as $h{\mathbf {a_{1}}}+k{\mathbf {a_{2}}}+\ell {\mathbf {a_{3}}}$, where the $\mathbf{a}_i$ are linearly independent basis vectors. However, if I am understanding this correctly, the Miller indices are actually not (directly) defined in terms of the real lattice vector, but are actually defined directly in terms of the reciprocal lattice vector $h{\mathbf {b_{1}}}+k{\mathbf {b_{2}}}+\ell {\mathbf {b_{3}}}$, where ${\mathbf {b_{i}}}$ are the basis of the reciprocal lattice vectors; specifically, the Miller indices are defined to denote the family of planes orthogonal to $h{\mathbf {b_{1}}}+k{\mathbf {b_{2}}}+\ell {\mathbf {b_{3}}}$. This is evidenced by the fact that said orthogonal plane is not always orthogonal to the linear combination of real lattice vectors $h{\mathbf {a_{1}}}+k{\mathbf {a_{2}}}+\ell {\mathbf {a_{3}}}$ (see image below), since the reciprocal lattice vectors need not be mutually orthogonal (that is, although the $\mathbf{b}_i$ are basis vectors, they are not necessarily linearly independent, which I think is mathematically correct (?)).

(By Felix King, and posted to https://en.wikipedia.org/wiki/Miller_index)

I would appreciate it if people would please take the time to review everything that I've written so far. Is my understanding correct? Am I misunderstanding anything?

The last point that I'm having a bit of trouble with is how a plane can be defined in such a way that it is orthogonal to three basis vectors? I suspect that this is possible if and only if the basis vectors are not all linearly independent, as I alluded to in my explanation above, but I'm not sure. I think this is more of a mathematical question, so I will instead post it to math.stackexchange for a more mathematics-focused answer, but I would also appreciate it if people could please review my understanding of this and provide clarification.

Answer 1

There are two equivalent ways to define the meaning of the Miller indices: via a point in the reciprocal lattice, or as the inverse intercepts along the lattice vectors.

The reflecting plane are parallel to a plane that includes the following three points:

$$\frac{\mathbf{a}_1}{h}; \frac{\mathbf{a}_2}{k}; \frac{\mathbf{a}_3}{l}$$

To show that the plane defined by these three points is perpendicular to $h \mathbf{b}_1 + k \mathbf{b}_2 + l\mathbf{b}_3$, we can calculate the dot product between vectors in the plane and the supposedly perpendicular reciprocal lattice vector. To get vectors in the plane, we simply subtract pairs of points in the plane.

$$\frac{\mathbf{a}_1}{h} - \frac{\mathbf{a}_2}{k}$$ $$\frac{\mathbf{a}_1}{h} - \frac{\mathbf{a}_3}{l}$$

Now we multiply it with the reciprocal lattice vector:

$$(\frac{\mathbf{a}_1}{h} - \frac{\mathbf{a}_2}{k}) (h \mathbf{b}_1 + k \mathbf{b}_2 + l\mathbf{b}_3) = \mathbf{a}_1 \mathbf{b}_1 - \mathbf{a}_2 \mathbf{b}_2 = 1 - 1 = 0$$ $$(\frac{\mathbf{a}_1}{h} - \frac{\mathbf{a}_3}{l}) (h \mathbf{b}_1 + k \mathbf{b}_2 + l\mathbf{b}_3) = \mathbf{a}_1 \mathbf{b}_1 - \mathbf{a}_3 \mathbf{b}_3 = 1 - 1 = 0$$

The mixed terms vanish because they are orthogonal. Because the dot products are equal to zero, we have shown that the vector is perpendicular to the reflecting plane.

[OP] My understanding is that the real (real space) lattice vector is described by the Miller indices as $h{\mathbf {a_{1}}}+k{\mathbf {a_{2}}}+\ell {\mathbf {a_{3}}}$, where the $\mathbf{a}_i$ are linearly independent basis vectors.

No, the significance of $h, k, l$ in defining reflecting planes in terms of the unit cell vectors is as described in the textbook (inverse intercepts along the lattice vectors).

[OP] The last point that I'm having a bit of trouble with is how a plane can be defined in such a way that it is orthogonal to three basis vectors

That is not how it is defined. It is defined by the intersections of a plane with the three axes, i.e. through three points that are in the plane.