Bragg equation

The Bragg equation , also called the Bragg condition , was developed by William Lawrence Bragg in 1912 . It describes when there is constructive interference of waves when scattered on a three-dimensional grid. She explains the patterns that arise during the diffraction of X-ray or neutron radiation on crystalline solids from the periodicity of lattice planes. The analogue of the Bragg condition in reciprocal space is the Laue condition .

principle

Diffraction image of a crystal, recorded with monochromatic X-rays (beam points perpendicular to the center of the detector)

If X-rays strike a crystal , a large part of the radiation penetrates it unhindered. However, it is also observed that a small part is deflected by the crystal - a phenomenon known as X-ray diffraction . If you mount a suitable detector behind the crystal, for example a photo plate , in order to make the deflected radiation components visible, characteristic patterns are created on it.

The cause of the diffraction is the scattering of the X-rays on the individual atoms of the lattice. This can also be seen as a weak reflection of the X-ray radiation on the individual lattice planes of the crystal, the radiation being reflected only in those directions in which the individual reflections constructively overlap. This condition describes the Bragg equation:

{\ displaystyle n \ lambda = 2d \, \ sin (\ theta)}

The Bragg equation links:

the distance d between parallel lattice planes,
the wavelength of the X-rays as well ${\ displaystyle \ lambda}$
the angle between the X-ray beam and the lattice plane, the so-called glancing or Bragg angle , ${\ displaystyle \ theta}$

n is a natural number that indicates the diffraction order .

Each family of parallel lattice planes has a characteristic lattice plane spacing d and thus, according to the Bragg equation, also a characteristic Bragg angle . For different orientations in which radiation hits the crystal, one almost always receives different images on the detector behind the crystal, because there are always different sets of parallel lattice planes (with different Bragg angles and with different orientations in the crystal) in the reflection position to the incident beam . ${\ displaystyle \ theta}$

The lattice levels are usually marked with Laue indices hkl , so that, for example, in the cubic system, this results in:

{\ displaystyle d_ {hkl} = {\ frac {a_ {0}} {\ sqrt {h ^ {2} + k ^ {2} + l ^ {2}}}}}

with the lattice constant . The Bragg condition in the cubic system is: ${\ displaystyle a_ {0}}$

{\ displaystyle {\ sin (\ theta)} ^ {2} = {\ frac {\ lambda ^ {2}} {4 \, {a_ {0}} ^ {2}}} (h ^ {2} + k ^ {2} + l ^ {2})}

For n = 1 , the Laue indices are identical to the Miller indices . For higher orders n , hkl stands for the Laue indices , the Miller indices of the grating plane multiplied by the diffraction order n .

Physical background

Schematic representation of the diffraction phenomenon

In fact, it is a diffraction phenomenon . In the electromagnetic field of the incident radiation, the electrons of the atoms are excited to forced oscillations and begin to emit radiation themselves in the form of spherical waves. Since the waves of the individual electrons add up as a first approximation to waves of the associated atoms, and furthermore the distances in the crystal lattice and the wavelength of the X-rays are of a similar order of magnitude, interference phenomena occur.

If the Bragg equation is fulfilled at a given wavelength for a family of parallel lattice planes, i.e. if the X-rays hit the crystal at the correct angle, there will be constructive interference of the spherical waves that are produced during diffraction at the electron shells. Macroscopically, there is the impression of a reflection of the radiation on the crystal. ${\ displaystyle \ lambda}$

Derivation

Diffraction geometry scheme

Diffraction geometry with constructive interference (left) and destructive interference (right)

The blue lines in the graphic opposite (scheme for diffraction geometry) correspond to rays that hit parallel lattice planes and enclose the angle with the perpendicular . The complementary angle is called the Bragg angle or gloss angle . d is the lattice plane spacing, the black dots are atoms on the lattice planes. ${\ displaystyle \ alpha}$ ${\ displaystyle \ theta = 90 ^ {\ circ} - \ alpha}$

Due to the large number of atoms in a crystal, in the event of not exclusively constructive interference, there is always a second statistically for each atom, which cancels out the diffracted wave of the first so that no reflection can be observed. This is also the situation in non-crystalline material , regardless of the direction of irradiation.

The phase relationship remains unchanged and constructive interference occurs when the path difference between the upper and lower wave train (the path difference ), marked in dark blue in the upper diagram, corresponds to an integral multiple of the wave length : ${\ displaystyle 2 \ delta}$ ${\ displaystyle \ lambda}$

{\ displaystyle 2 \ delta = n \ cdot \ lambda}

where n is also referred to as the diffraction order.

In the illustration above, red, green and purple lines together form a right triangle with the hypotenuse d . The definition of the sine gives the following expression:

{\ displaystyle \ delta = d \ cdot \ sin (\ theta)}

It is important to understand that the red lines are not extensions of the upper light blue lines, but the perpendicular to them.

If you insert the second expression into the first equation, you immediately get the Bragg equation (also Bragg interference ):

{\ displaystyle n \ lambda = 2d \, \ sin (\ theta)}

Carrying out the experiment

Emission spectrum of a copper anode with X-rays of different wavelengths (or here plotted as an angle).

The rotating crystal arrangement is a possible implementation of the experiment. Since earlier X-ray machines were very heavy and therefore not rotatable, the X-ray beam was directed onto a rotating crystal. By rotating the crystal and the receiver, the crystal could then be examined from different angles. A second possibility is the Debye-Scherrer process , in which the crystal is pulverized so that every "direction of rotation" is statistically distributed at the same time.

meaning

X-ray diffraction experiments on crystals offer the opportunity to gain insights into the internal structure of crystals ( see : Crystal structure analysis ).

The Bragg reflection is also important for neutron diffraction . Moderated neutrons have wavelengths comparable to X-rays, so the same phenomenon occurs on the crystal lattice.

Diffraction images of electron beams can be produced and observed in electron microscopes . These are also fundamentally described by the Bragg law.

Bragg reflections occur with so-called white light holograms . There they are responsible for ensuring that the image of the hologram changes color when it is tilted.

Bragg reflectors are used for wavelength selection in lasers or X-rays (see , inter alia, monochromator , DBR laser , DFB laser ).

Acousto-optical modulators are based on the principle of Bragg diffraction; Light rays are diffracted on a moving optical grid, caused by longitudinal sound waves in crystals.

literature

Dieter Meschede : Gerthsen Physics. 22nd edition, Springer-Verlag, Berlin / Heidelberg 2004, ISBN 3-540-02622-3 .
Rudolf Allmann : X-ray powder diffractometry. Verlag Sven von Loga, Cologne 1994, ISBN 3-87361-029-9 .
Werner Massa: Crystal structure determination. 5th edition, Teubner-Verlag, Stuttgart / Leipzig / Wiesbaden 2007, ISBN 3-8351-0113-7 .
Anthony R. West: Fundamentals of Solid State Chemistry. Wiley-VCH, Weinheim 2000, ISBN 3-527-28103-7 .

Web links

Commons : Bragg equation - collection of images, videos and audio files

Individual evidence

↑ Borchardt-Ott, Kristallographie, Springer 2009, p. 284 f.

[1] Borchardt-Ott, Kristallographie, Springer 2009, p. 284 f.