Laue equations

Let ${\displaystyle \mathbf {a} \,,\mathbf {b} \,,\mathbf {c} }$ be primitive translation vectors (shortly called primitive vectors) of a crystal lattice ${\displaystyle L}$, where atoms are located at lattice points described by ${\displaystyle \mathbf {x} =p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} }$ with ${\displaystyle p}$, ${\displaystyle q}$, and ${\displaystyle r}$ as any integers. (So ${\displaystyle \mathbf {x} }$ indicating each lattice point is an integer linear combination of the primitive vectors.)

Let ${\displaystyle \mathbf {k} _{\mathrm {in} }}$ be the wave vector of an incoming (incident) beam or wave toward the crystal lattice ${\displaystyle L}$, and let ${\displaystyle \mathbf {k} _{\mathrm {out} }}$ be the wave vector of an outgoing (diffracted) beam or wave from ${\displaystyle L}$. Then the vector ${\displaystyle \mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {\Delta k} }$, called the scattering vector or transferred wave vector, measures the difference between the incoming and outgoing wave vectors.

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

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

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

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

where numbers ${\displaystyle h,k,l}$ are integer numbers. Each choice of 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 as there are infinitely many choices of Miller indices ${\displaystyle (h,k,l)}$. Allowed scattering vectors ${\displaystyle \mathbf {\Delta k} }$ form a lattice ${\displaystyle L^{*}}$, called the reciprocal lattice of the crystal lattice ${\displaystyle L}$, as each ${\displaystyle \mathbf {\Delta k} }$ indicates a point of ${\displaystyle L^{*}}$. (This is the meaning of the Laue equations as shown below.) This condition allows a single incident beam to be diffracted in infinitely many directions. However, the beams corresponding to high Miller indices are very weak and can't be observed. These equations are enough to find a basis of the reciprocal lattice (since each observed ${\displaystyle \mathbf {\Delta k} }$ indicates a point of the reciprocal lattice of the crystal under the measurement), from which the crystal lattice can be determined. This is the principle of x-ray crystallography.

For an incident plane wave at a single frequency ${\displaystyle \displaystyle f}$ (and the angular frequency ${\displaystyle \displaystyle \omega =2\pi f}$) on a crystal, the diffracted waves from the crystal can be thought as the sum of outgoing plane waves from the crystal. (In fact, any wave can be represented as the sum of plane waves, see Fourier Optics.) The incident wave and one of plane waves of the diffracted wave are represented as

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

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

where ${\displaystyle \displaystyle \mathbf {k} _{\mathrm {in} }}$ and ${\displaystyle \displaystyle \mathbf {k} _{\mathrm {out} }}$ are wave vectors for the incident and outgoing plane waves, ${\displaystyle \displaystyle \mathbf {x} }$ is the position vector, and ${\displaystyle \displaystyle t}$ is a scalar representing time, and ${\displaystyle \varphi _{\mathrm {in} }}$ and ${\displaystyle \varphi _{\mathrm {out} }}$ are initial phases for the waves. For simplicity we take waves as scalars here, even though the main case of interest is an electromagnetic field, which is a vector. We can think these scalar waves as components of vector waves along a certain axis (x, y, or z axis) of the Cartesian coordinate system.

The incident and diffracted waves propagate through space independently, except at points of the lattice ${\displaystyle L}$ of the crystal, where they resonate with the oscillators, so the phases of these waves must coincide.^[1] At each point ${\displaystyle \mathbf {x} =p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} }$ of the lattice ${\displaystyle L}$, we have

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

or equivalently, we must have

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

for some integer ${\displaystyle n}$, that depends on the point ${\displaystyle \mathbf {x} }$. Since this equation holds at ${\displaystyle \mathbf {x} =0}$, ${\displaystyle \varphi _{\mathrm {in} }=\varphi _{\mathrm {out} }+2\pi n'}$ at some integer ${\displaystyle n'}$. So

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

(We still use ${\displaystyle n}$ instead of ${\displaystyle (n-n')}$ since both the notations essentially indicate some integer.) By rearranging terms, 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, at any lattice point ${\displaystyle \mathbf {x} =p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} }$, we have

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

where ${\displaystyle n}$ is the integer ${\displaystyle ph+qk+rl}$. The claim that each parenthesis, e.g. ${\displaystyle (\mathbf {\Delta k} \cdot \mathbf {a} )}$, is to be a multiple of ${\displaystyle 2\pi }$ (that is each Laue equation) is justified since otherwise ${\displaystyle p(\mathbf {\Delta k} \cdot \mathbf {a} )+q(\mathbf {\Delta k} \cdot \mathbf {b} )+r(\mathbf {\Delta k} \cdot \mathbf {c} )=2\pi n}$ does not hold for any arbitrary integers ${\displaystyle p,q,r}$.

This ensures that if the Laue equations are satisfied, then the incoming and outgoing (diffracted) wave have the same phase at each point of the crystal lattice, so the oscillations of atoms of the crystal, that follows the incoming wave, can at the same time generate the outgoing wave at the same phase of the incoming wave.

If ${\displaystyle \mathbf {G} =h\mathbf {A} +k\mathbf {B} +l\mathbf {C} }$ with ${\displaystyle h}$, ${\displaystyle k}$, ${\displaystyle l}$ as integers represents the reciprocal lattice for a crystal lattice ${\displaystyle L}$ (defined by ${\displaystyle \mathbf {x} =p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} }$) in real space, we know that ${\displaystyle \mathbf {G} \cdot \mathbf {x} =\mathbf {G} \cdot (p\mathbf {a} +q\mathbf {b} +r\mathbf {c} )=2\pi (hp+kq+lr)=2\pi n}$ with an integer ${\displaystyle n}$ due to the known orthogonality between primitive vectors for the reciprocal lattice and those for the crystal lattice. (We use the physical, not crystallographer's, definition for reciprocal lattice vectors which gives the factor of ${\displaystyle 2\pi }$.) But notice that this is nothing but the Laue equations. Hence we identify ${\displaystyle \mathbf {\Delta k} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {G} }$, means that allowed scattering vectors ${\displaystyle \mathbf {\Delta k} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }}$ are those equal to reciprocal lattice vectors ${\displaystyle \mathbf {G} }$ for a crystal in diffraction, and this is the meaning of the Laue equations. This fact is sometimes called the Laue condition. In this sense, diffraction patterns are a way to experimentally measure the reciprocal lattice for a crystal lattice.

The Laue condition can be rewritten as the following.^[2]

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

Applying the elastic scattering condition ${\displaystyle |\mathbf {k} _{\mathrm {out} }|^{2}=|\mathbf {k} _{\mathrm {in} }|^{2}}$ (In other words, the incoming and diffracted waves are at the same (temporal) frequency. We can also say that the energy per photon does not change.)

To the above equation, we obtain

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

${\displaystyle 2{{\mathbf {k} }_{\text{in}}}\cdot (-\mathbf {G} )=|\mathbf {G} {{|}^{2}}.}$

The second equation is obtained from the first equation by using ${\displaystyle \mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {G} }$.

The result ${\displaystyle 2\mathbf {k} _{\mathrm {out} }\cdot \mathbf {G} =|\mathbf {G} |^{2}}$ (also ${\displaystyle 2{{\mathbf {k} }_{\text{in}}}\cdot (-\mathbf {G} )=|\mathbf {G} {{|}^{2}}}$) is an equation for a plane (as the set of all points indicated by ${\displaystyle \mathbf {k} _{\mathrm {out} }}$ satisfying this equation) as its equivalent equation ${\displaystyle \mathbf {G} \cdot (2{{\mathbf {k} }_{\text{out}}}-\mathbf {G} )=0}$ is a plane equation in geometry. Another equivalent equation, that may be easier to understand, is ${\displaystyle {{\mathbf {k} }_{\text{out}}}\cdot {\widehat {\mathbf {G} }}={\frac {1}{2}}\left|\mathbf {G} \right|}$ (also ${\displaystyle (-{{\mathbf {k} }_{\text{in}}})\cdot {\widehat {\mathbf {G} }}={\frac {1}{2}}\left|\mathbf {G} \right|}$). This indicates the plane that is perpendicular to the straight line between the reciprocal lattice origin ${\displaystyle \mathbf {G} =0}$ and ${\displaystyle \mathbf {G} }$ and located at the middle of the line. Such a plane is called Bragg plane.^[3] This plane can be understood since ${\displaystyle \mathbf {G} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }}$ for scattering to occur. (It is the Laue condition, equivalent to the Laue equations.) And, the elastic scattering ${\displaystyle |\mathbf {k} _{\mathrm {out} }|^{2}=|\mathbf {k} _{\mathrm {in} }|^{2}}$ has been assumed so ${\displaystyle \mathbf {G} }$, ${\displaystyle \mathbf {k} _{\mathrm {out} }}$, and ${\displaystyle -\mathbf {k} _{\mathrm {in} }}$ form a rhombus. Each ${\displaystyle \mathbf {G} }$ is by definition the wavevector of a plane wave in the Fourier series of a spatial function which periodicity follows the crystal lattice (e.g., the function representing the electronic density of the crystal), wavefronts of each plane wave in the Fourier series is perpendicular to the plane wave's wavevector ${\displaystyle \mathbf {G} }$, and these wavefronts are coincident with parallel crystal lattice planes. This means that X-rays are seemingly "reflected" off parallel crystal lattice planes perpendicular ${\displaystyle \mathbf {G} }$ at the same angle as their angle of approach to the crystal ${\displaystyle \theta }$ with respect to the lattice planes; in the elastic light (typically X-ray)-crystal scattering, parallel crystal lattice planes perpendicular to a reciprocal lattice vector ${\displaystyle \mathbf {G} }$ for the crystal lattice play as parallel mirrors for light which, together with ${\displaystyle \mathbf {G} }$, incoming (to the crystal) and outgoing (from the crystal by scattering) wavevectors forms a rhombus.

Since the angle between ${\displaystyle \mathbf {k} _{\mathrm {out} }}$ and ${\displaystyle \mathbf {G} }$ is ${\displaystyle \pi /2-\theta }$, (Due to the mirror-like scattering, the angle between ${\displaystyle \mathbf {k} _{\mathrm {in} }}$ and ${\displaystyle \mathbf {G} }$ is also ${\displaystyle \pi /2-\theta }$.) ${\displaystyle \mathbf {k} _{\mathrm {out} }\cdot \mathbf {G} =|\mathbf {k} _{\mathrm {out} }||\mathbf {G} |\sin \theta }$. Recall, ${\displaystyle |\mathbf {k} _{\mathrm {out} }|=2\pi /\lambda }$ with ${\displaystyle \lambda }$ as the light (typically X-ray) wavelength, and ${\displaystyle |\mathbf {G} |={\frac {2\pi }{d}}n}$ with ${\displaystyle d}$ as the distance between adjacent parallel crystal lattice planes and ${\displaystyle n}$ as an integer. With these, we now derive Bragg's law that is equivalent to the Laue equations (also called the Laue condition):

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