Laue equations

In crystallography, the Laue equations relate the incoming waves to the outgoing waves in the process of diffraction by a crystal lattice. They are named after physicist Max von Laue (1879–1960). They reduce to Bragg's law.

Laue equation

The Laue equations

Let ${\displaystyle \mathbf {a} \,,\mathbf {b} \,,\mathbf {c} }$ be the primitive vectors of the crystal lattice ${\displaystyle L}$, whose atoms are located at the points ${\displaystyle \mathbf {x} =p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} }$ that are integer linear combinations of the primitive vectors.

Let ${\displaystyle \mathbf {k} _{\mathrm {in} }}$ be the wavevector of the incoming (incident) beam, and let ${\displaystyle \mathbf {k} _{\mathrm {out} }}$ be the wavevector of the outgoing (diffracted) beam. Then the vector ${\displaystyle \mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {\Delta k} }$ is called the scattering vector (also called transferred wavevector) and measures the change between the two wavevectors.

The three conditions that the scattering vector ${\displaystyle \mathbf {\Delta k} }$ must satisfy, called the Laue equations, are the following: the numbers ${\displaystyle h,k,l}$ determined by the equations

${\displaystyle \mathbf {a} \cdot \mathbf {\Delta k} =2\pi h}$

${\displaystyle \mathbf {b} \cdot \mathbf {\Delta k} =2\pi k}$

${\displaystyle \mathbf {c} \cdot \mathbf {\Delta k} =2\pi l}$

must be integer numbers. Each choice of the integers ${\displaystyle (h,k,l)}$, called Miller indices, determines a scattering vector ${\displaystyle \mathbf {\Delta k} }$. Hence there are infinitely many scattering vectors that satisfy the Laue equations. They form a lattice ${\displaystyle L^{*}}$, called the reciprocal lattice of the crystal lattice. This condition allows a single incident beam to be diffracted in infinitely many directions. However, the beams that correspond to high Miller indices are very weak and can't be observed. These equations are enough to find a basis of the reciprocal lattice, from which the crystal lattice can be determined. This is the principle of x-ray crystallography.

Mathematical derivation

The incident and diffracted beams are planar wave excitations

${\displaystyle f_{\mathrm {in} }(t,\mathbf {x} )=A_{\mathrm {in} }\cos(\omega \,t-\mathbf {k} _{\mathrm {in} }\cdot \mathbf {x} )}$

${\displaystyle f_{\mathrm {out} }(t,\mathbf {x} )=A_{\mathrm {out} }\cos(\omega \,t-\mathbf {k} _{\mathrm {out} }\cdot \mathbf {x} ).}$

of a field that for simplicity we take as scalar, even though the main case of interest is the electromagnetic field, which is vectorial.

The two waves propagate through space independently, except at the points of the lattice, where they resonate with the oscillators, so their phase must coincide.[1] Hence for each point ${\displaystyle \mathbf {x} }$ of the lattice ${\displaystyle L}$ we have

${\displaystyle \cos(\omega \,t-\mathbf {k} _{\mathrm {in} }\cdot \mathbf {x} )=\cos(\omega \,t-\mathbf {k} _{\mathrm {out} }\cdot \mathbf {x} ),}$

or equivalently, we must have

${\displaystyle \omega \,t-\mathbf {k} _{\mathrm {in} }\cdot \mathbf {x} =\omega \,t-\mathbf {k} _{\mathrm {out} }\cdot \mathbf {x} +2\pi n,}$

for some integer ${\displaystyle n}$, that depends on the point ${\displaystyle \mathbf {x} }$. Simplifying we get

${\displaystyle \mathbf {\Delta k} \cdot \mathbf {x} =(\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} })\cdot \mathbf {x} =2\pi n.}$

Now, it is enough to check that this condition is satisfied at the primitive vectors ${\displaystyle \mathbf {a} ,\mathbf {b} ,\mathbf {c} }$ (which is exactly what the Laue equations say), because then for the other points ${\displaystyle \mathbf {x} =p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} }$ we have

${\displaystyle \mathbf {\Delta k} \cdot \mathbf {x} =\mathbf {\Delta k} \cdot (p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} )=p\,2\pi h+q\,2\pi k+r\,2\pi l=2\pi (hp+kq+lr)=2\pi n,}$

where ${\displaystyle n}$ is the integer ${\displaystyle hp+kq+lr}$.

This ensures that if the Laue equations are satisfied, then the incoming and outgoing wave have the same phase at all points of the crystal lattice, so the oscillation of the atoms, that follows the incoming wave, can at the same time generate the outgoing wave.

Relation to Bragg's Law

If ${\displaystyle \mathbf {G} =h\mathbf {A} +k\mathbf {B} +l\mathbf {C} }$ is the reciprocal lattice vector, we know by definition of the reciprocal lattice basis vectors that ${\displaystyle \mathbf {G} \cdot \mathbf {x} =\mathbf {G} \cdot (p\mathbf {a} +q\mathbf {b} +r\mathbf {c} )=2\pi (hp+kq+lr)=2\pi n}$, where ${\displaystyle n}$ is an integer (we use the definition for a reciprocal lattice vector which gives the factor of ${\displaystyle 2\pi }$). But notice that this is nothing but the Laue equations. Hence we identify ${\displaystyle \mathbf {\Delta k} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {G} }$, this is sometimes called the Laue condition. In a sense, diffraction patterns are a way to experimentally measure the reciprocal lattice.

Rewriting the Laue condition:[2]

${\displaystyle {\begin{aligned}:\mathbf {G} &=\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }\\|\mathbf {k} _{\mathrm {in} }|^{2}&=|\mathbf {k} _{\mathrm {out} }-\mathbf {G} |^{2}\\|\mathbf {k} _{\mathrm {in} }|^{2}&=|\mathbf {k} _{\mathrm {out} }|^{2}-2\mathbf {k} _{\mathrm {out} }\cdot \mathbf {G} +|\mathbf {G} |^{2}\end{aligned}}}$

Applying the elastic scattering condition ${\displaystyle |\mathbf {k} _{\mathrm {out} }|^{2}=|\mathbf {k} _{\mathrm {in} }|^{2}}$ to the above equation, we obtain:

${\displaystyle 2\mathbf {k} _{\mathrm {out} }\cdot \mathbf {G} =|\mathbf {G} |^{2}}$.

Essentially, the Laue condition is the conservation of momentum and is a consequence of the very general statement that the crystal momentum is only conserved up to a reciprocal lattice vector, while the elastic condition is the conservation of energy carried by the X-rays (i.e. the crystal gains no energy from scattering radiation).

The result ${\displaystyle \;2\mathbf {k} _{\mathrm {out} }\cdot \mathbf {G} =|\mathbf {G} |^{2}}$ is an equation for a plane (geometry). The vector ${\displaystyle \mathbf {G} }$ specifies a set of Bragg planes in reciprocal space normal to it. Note that this implies a corresponding set of Bragg planes in real space, i.e. integer solutions for ${\displaystyle p,q,r}$ to the equation ${\displaystyle (hp+kq+lr)=n}$ for integer coefficients ${\displaystyle h,k,l}$ and order ${\displaystyle n}$. The vectors ${\displaystyle \mathbf {k} _{\mathrm {in} }}$, ${\displaystyle \mathbf {k} _{\mathrm {out} }}$, and ${\displaystyle \mathbf {G} }$ form an isosceles triangle. This means that X-rays seemingly "reflect" off these planes at the same angle as their angle of approach ${\displaystyle \theta }$ (with respect to the plane).

Since the angle between ${\displaystyle \mathbf {k} _{\mathrm {out} }}$ and ${\displaystyle \mathbf {G} }$ is ${\displaystyle \pi /2-\theta }$, this implies that ${\displaystyle \mathbf {k} _{\mathrm {out} }\cdot \mathbf {G} =|\mathbf {k} _{\mathrm {out} }||\mathbf {G} |\sin \theta }$. Clearly, ${\displaystyle |\mathbf {k} _{\mathrm {out} }|=2\pi /\lambda }$. If the lattice constant is ${\displaystyle d}$, then ${\displaystyle |\mathbf {G} |=2\pi n/d}$; this is because by definition we require ${\displaystyle \mathbf {G} \cdot \mathbf {x} =2\pi n}$, and moreover we can pick a set of Bragg planes in real space with interplane separation ${\displaystyle |\mathbf {x} |=d}$, and without loss of generality choose ${\displaystyle \mathbf {G} }$ parallel to ${\displaystyle \mathbf {x} }$. With these, we now recover Bragg's law:

${\displaystyle {\begin{aligned}2\mathbf {k} _{\mathrm {out} }\cdot \mathbf {G} =|\mathbf {G} |^{2}\\2|\mathbf {k} _{\mathrm {out} }||\mathbf {G} |\sin \theta =|\mathbf {G} |^{2}\\2(2\pi /\lambda )(2\pi n/d)\sin \theta =(2\pi n/d)^{2}\\2d\sin \theta =n\lambda .\end{aligned}}}$

References

• Kittel, C. (1976). Introduction to Solid State Physics, New York: John Wiley & Sons. ISBN 0-471-49024-5

Notes

1. More realistically, the oscillators of the lattice should lag behind the incoming wave, and the outcoming wave should lag behind the oscillator. But since the lag is the same at all point of the lattice, the only effect of this correction would be global shift of phase of the outcoming wave, which we are not taking into consideration.
2. Chaikin, P. M.; Lubensky, T. C. Principles of condensed matter physics. p. 47. ISBN 0521794501.