1. Surface crytallography and diffraction
Surface symmetry
Any crytslalline solid materials composed of repeated units in 3 dimension structure called “crystals”. These crystals can possess restricted number of types which repeated in a so-called “translational symmetry”. The science which study this is called “crytallography”.
** Many properties of solids are intimately related to the special symmetry properties of these materials.
** The surface has two dimension structure, i.e. a thin layer from the crystals on the surface make like sheet in two dimensional structure. This layer strongly affects the properties of the surface because it has influences on the electronic properties of the surface. Also, this layer plays dominant role in allowing the electron, X-ray and atom diffraction techniques to provide information on the structure of the surface.
** Therefore, “surface structure” phrase means the structure of the solid in the vicinity of the surface “selvedge” .
** We have to understand the crytallography of the surface because it is very important in the analytical techniques. The signal which emerges in the solid is likely to contain large contribution from the top atom layer “surface”, a weaker contribution from the next layer, and so on. Therefore, if there is a layer from adsorbate atoms it will affect the characterization.

2. Surface crystallography
There is a scientific way to characterize the surface crystallography, it is called space groups. The number of the units in the space group depends on the dimensions, for example in two dimension space there are 17 patterns, while 219 patterns in three dimensional space
Space groups in two dimensional space
These are the basic shapes

3. Generally in 2 or 3 dimension the space group consists of 4 letters
1. The first one is p or c p primitive
C non - primitive
In two dimensions crytallography, only the centered rectangular cells are the non-primitive.
2. The second is a digit denotes the number of the highest order of rotational symmetry: 1-fold (none), 2-fold, 3-fold, 4-fold, or 6-fold.
1- fold means can not rotate
2-fold means can rotate with angle 180o fold means can rotate with angle 120o
4-fold means can rotate with angle 90o fold means can rotate with angle 60o
Rotation can be done at one or several centers in the cells

4. The next two symbols indicate symmetries relative to one translation axis of the pattern, referred to as the "main" one; if there is a mirror perpendicular to a translation axis we choose that axis as the main one (or if there are two, one of them). The symbols are either
m for mirror
g glide reflection
none.
P211 (p2) Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections.
p4g (p4gm): Primitive cell, 4-fold rotation,
glide reflection perpendicular
to main axis, mirror axis at 90°.

5. cmm (c2mm): Centered cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis.
p31m (p31m): Primitive cell, 3-fold rotation, mirror axis at 60°.

6. The two dimension space group is not enough to characterize the bulk structure, so the more accurate is the three dimensional space groups.
For example, if we have a material with FCC (faced centered cube), the {100} surface (i.e. the plane par

7. Miller indices Definition
Miller indices are a notation system in crystallography for planes and directions in crystal (Bravais) lattices.
Presentation
In particular, a family of lattice planes is determined by three integers ℓ, m, and n, the Miller indices. They are written (hkl), and each index denotes a plane orthogonal to a direction (x, y, z) in the basis of the reciprocal lattice vectors. Miller index (100) represents a plane orthogonal to direction x; index 010 represents a plane orthogonal to direction y, and index 001 represents a plane orthogonal to z.
General Notes
Sometimes we specify the origin and the axes locations in the reciprocal lattice so we may have negative integers, they are written with a bar, as in 3 for −3. The integers are usually written in lowest terms, i.e. their greatest common divisor should be 1.
The notation {ℓmn} denotes the set of all planes that are equivalent to (ℓmn) by the symmetry of the lattice.

8. Miller indices for directions
In the context of crystal directions (not planes), the corresponding notations are:
[ℓmn], with square instead of round brackets, denotes a direction in the basis of the direct lattice vectors instead of the reciprocal lattice; and
similarly, the notation 〈hkl〉 denotes the set of all directions that are equivalent to [hkl] by symmetry.

9. Spectroscopy Definition
Spectroscopy is the study of the interaction between matter and radiated energy.
Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, e.g., by a prism. Later the concept was expanded greatly to comprise any interaction with radiative energy as a function of its wavelength or frequency. Spectroscopic data is often represented by a spectrum, a plot of the response of interest as a function of wavelength or frequency.

10. Fields
We can find application fields of spectroscopy in our life in many fields.
Neon lamps: Neon lighting is a direct application of atomic spectroscopy. Neon and other noble gases have characteristic emission colors, and neon lamps use electricity to excite these emissions.
Ink, dyes and paints which reflects the complementary colors when a light falls on it.
Astronomy: Most research telescopes have spectrographs. The measured spectra are used to determine the chemical composition and physical properties of astronomical objects (such as their temperature and velocity).
Remote sensing.

11. Type of radiative energy
Types of spectroscopy are distinguished by the type of radiative energy involved in the interaction. In many applications, the spectrum is determined by measuring changes in the intensity or frequency of this energy. The types of radiative energy studied include:
Electromagnetic radiation was the first source of energy used for spectroscopic studies. Techniques that employ electromagnetic radiation are typically classified by the wavelength region of the spectrum and include microwave, terahertz, infrared, near infrared, visible and ultraviolet, x-ray and gamma spectroscopy.

12. 2. Particles, due to their de Broglie wavelength, can also be a source of radiative energy and both electrons and neutrons are commonly used. For a particle, its kinetic energy determines its wavelength.
3. Acoustic spectroscopy involves radiated pressure waves
4. Mechanical methods can be employed to impart radiating energy, similar to acoustic waves, to solid materials. An acoustic wave is an oscillation of pressure that travels through a solid, liquid, or gas in a wave pattern. It transmits sound by vibrating organs in the ear that produce the sensation of hearing. Acoustic waves, or sound waves, are defined by three characteristics: wavelength, frequency, and amplitude.

13. Nature of the interaction
 Types of spectroscopy can also be distinguished by the nature of the interaction between the energy and the material. These interactions include:
 Absorption occurs when energy from the radiative source is absorbed by the material. Absorption is often determined by measuring the fraction of energy transmitted through the material; absorption will decrease the transmitted portion.
Emission indicates that radiative energy is released by the material. A material's blackbody spectrum is a spontaneous emission spectrum determined by its temperature. Emission can also be induced by other sources of energy such as a flames or sparks or electromagnetic radiation in the case of fluorescence.
Elastic scattering and reflection spectroscopy determine how incident radiation is reflected or scattered by a material. Crystallography employs the scattering of high energy radiation, such as x-rays and electrons, to examine the arrangement of atoms in proteins and solid crystals.

14. 5. Impedance spectroscopy studies the ability of a medium to impede or slow the transmittance of energy. For optical applications, this is characterized by the index of refraction.
6. Inelastic scattering phenomena involve an exchange of energy between the radiation and the matter that shifts the wavelength of the scattered radiation. These include Raman and Compton scattering.
7. Coherent or resonance spectroscopy are techniques where the radiative energy couples two quantum states of the material in a coherent interaction that is sustained by the radiating field. The coherence can be disrupted by other interactions, such as particle collisions and energy transfer, and so often require high intensity radiation to be sustained. Nuclear magnetic resonance (NMR) spectroscopy is a widely used resonance method and ultrafast laser methods are also now possible in the infrared and visible spectral regions.

15. Examples for the spectroscopy techniques
As aforementioned, there are many techniques of the spectroscopy based on the natures of the radiant energy and the interaction.
Usually, the particles radiant energy (electrons , neutrons or ions) is used in the surface science due to the good results obtained, however these particles can not be used to investigate the bulk of the material due to the size so the electromagnetic radiation is more preferable. Below is two main techniques for both categories:
1. Low Energy Electron Diffraction (LEED)
Low-energy electron diffraction (LEED) is a technique for the determination of the surface structure of crystalline materials by bombardment with a collimated beam of low energy electrons (20-200eV) and observation of diffracted electrons as spots on a fluorescent screen.

16. LEED may be used in one of two ways:
Qualitatively, where the diffraction pattern is recorded and analysis of the spot positions gives information on the symmetry of the surface structure. In the presence of an adsorbate the qualitative analysis may reveal information about the size and rotational alignment of the adsorbate unit cell with respect to the substrate unit cell.
Quantitatively, where the intensities of diffracted beams are recorded as a function of incident electron beam energy to generate the so-called I-V curves. By comparison with theoretical curves, these may provide accurate information on atomic positions on the surface at hand.
Experimental Setup
In order to keep the sample clean and free from unwanted adsorbates,
LEED experiments are performed in an ultra-high-vacuum environment
(10−9mbar).
The most important elements in an LEED experiment are
A sample holder with the prepared sample
An electron gun
A display system, usually a hemispherical fluorescent screen
on which the diffraction pattern can be observed directly
A sputtering gun for cleaning the surface
5. An Auger-Electron Spectroscopy system in order to determine
the purity of the surface.
LEED pattern of a Si(100) reconstructed surface.

17. X-Ray analysis
Because it is relatively easy to use electrons or neutrons having wavelengths smaller than a nanometre, electrons and neutrons may be used to study crystal structure in a manner very similar to X-ray diffraction. Electrons do not penetrate as deeply into matter as X-rays, hence electron diffraction reveals structure near the surface; neutrons do penetrate easily and have an advantage that they possess an intrinsic magnetic moment that causes them to interact differently with atoms having different alignments of their magnetic moments.
X-ray scattering techniques are a family of non-destructive analytical techniques which reveal information about the crystallographic structure, chemical composition, and physical properties of materials and thin films. These techniques are based on observing the scattered intensity of an X-ray beam hitting a sample as a function of incident and scattered angle, polarization, and wavelength or energy.

18. X-ray diffraction techniques
X-ray diffraction yields the atomic structure of materials and is based on the elastic scattering of X-rays from the electron clouds of the individual atoms in the system.
Single-crystal X-ray diffraction is a technique used to solve the complete structure of crystalline materials, ranging from simple inorganic solids to complex macromolecules, such as proteins.
Thin film diffraction and grazing incidence X-ray diffraction may be used to characterize the crystallographic structure and preferred orientation of substrate-anchored thin films.
High-resolution X-ray diffraction is used to characterize thickness, crystallographic structure, and strain in thin epitaxial films. It employs parallel-beam optics.
X-ray pole figure analysis enables one to analyze and determine the distribution of crystalline orientations within a crystalline thin-film sample.
X-ray rocking curve analysis is used to quantify grain size and mosaic spread in crystalline materials.
3. Powder diffraction (XRD) is a technique used to characterize the crystallographic structure, crystallite size (grain size), and preferred orientation in polycrystalline or powdered solid samples. Powder diffraction is commonly used to identify unknown substances, by comparing diffraction data against a database maintained by the International Centre for Diffraction Data. It may also be used to characterize heterogeneous solid mixtures to determine relative abundance of crystalline compounds and, when coupled with lattice refinement techniques.