I know that in nano size the periodicity of lattice is only in one direction that is why in nano size can h,k and l one of them has a value and others equal to zero for example h have the value k and l equal to zero
k have the value l and h equal to zero
l have the value k and l h equal to zero
But most papers showed that nano size but in XRD present h,k and l have the value
Yes, but you can also get a (h00) peak for crystalline non-nano objects.
Miller indices are just a way to describe the respective lattice plane. Thus miller indices are a theoretical concept that holds true regardless of the dimension of the particle. It does not depend on the size of the object:
If you think back to the time when you learned about miller indices, you will probably remember that a unit cell (unit cube) of unit length is enough to describe (or draw) all the possible miller indices (or equivalent planes). For the determination of miller indices, in principle, the crystalline system is always assumed to be infinite, which is obviously never the case in reality, no matter if on the nano or micro or meter scale.
Nevertheless the result of the scattering changes, when you go to nano object, i.e., when only a limited number of lattice planes contribute to the scattering. You will observe a broadening of the peak (you may understand this if you consider the scatting in the description involving Fourier transformationation: infintite periodicity gives you a single delta peak, limited periodicity gives you a wider peak). This broadening is basically described by the Scherrer-Equation. If there is no periodicity at all of the respective plane, you will not get a signal.
And as a side remark, I do not think your assumption that periodicity in nano objects is only in one direction is generally true. Although there might be examples like a graphene nanoribbon,
Maybe you can paraphrase you question a bit. E.g. if you mean that for certain nano objects only (h00) peaks were allowed...
http://en.wikipedia.org/wiki/Scherrer_equation
Yes, but you can also get a (h00) peak for crystalline non-nano objects.
Miller indices are just a way to describe the respective lattice plane. Thus miller indices are a theoretical concept that holds true regardless of the dimension of the particle. It does not depend on the size of the object:
If you think back to the time when you learned about miller indices, you will probably remember that a unit cell (unit cube) of unit length is enough to describe (or draw) all the possible miller indices (or equivalent planes). For the determination of miller indices, in principle, the crystalline system is always assumed to be infinite, which is obviously never the case in reality, no matter if on the nano or micro or meter scale.
Nevertheless the result of the scattering changes, when you go to nano object, i.e., when only a limited number of lattice planes contribute to the scattering. You will observe a broadening of the peak (you may understand this if you consider the scatting in the description involving Fourier transformationation: infintite periodicity gives you a single delta peak, limited periodicity gives you a wider peak). This broadening is basically described by the Scherrer-Equation. If there is no periodicity at all of the respective plane, you will not get a signal.
And as a side remark, I do not think your assumption that periodicity in nano objects is only in one direction is generally true. Although there might be examples like a graphene nanoribbon,
Maybe you can paraphrase you question a bit. E.g. if you mean that for certain nano objects only (h00) peaks were allowed...
http://en.wikipedia.org/wiki/Scherrer_equation
You may encounter a situation when ordering is limited to two dimensions. Typical example is what one call turbostratic structures observed in petroleum coke. In powder X-ray diffraction patterns you may observe only hk type reflections. On heat treatment, three dimensional or hkl reflections evolve. Note that turbostratic structure in carbon means graphite sheets that displaced through rotation or translationional displacements.
Yes, you can have h00, 0k0, etc. The best was to know the hkl for a specific plane/s in your material is to run XRD. Then you can use the XRD pattern to identify the hkl for each crystal plane in your material.
Nano-crystalline materials can have three dimensional periodicity. There are special cases in which the periodicity is limited to two or even one dimension, but it is not the general case. So your assumption is incorrect, and the papers you are reading reflect that. You may be generalizing from a particular type of nano-material which you are familiar with to all nano-materials.
The scattering of nano-particles is not limited to crystalline scattering. Amorphous materials can produce maximums in the wide angle x-ray scattering region. The scattering from amorphous to crystalline is usually referred to as paracrystalline scattering. This scattering can be described as a precrystalline state that may not have three dimensional order and do not have the atoms in their final crystaloine atomic positions.. The paracrystalline phase will not have the same crystallographic axis lengths as the same crystallographic material and therefore (hkl or (hk-) or (h--) may or may not coincide with the crystalline (hkl)'s. In nano-materials, the formation of crystals that have sharp well defined crystalline peaks are rare( based on 20 years of diffraction work on nanosystems). As mentioned previously by Arras, the crystallite size of a material leads to peak broadening based on the limited number of repeat units of a particular plane defined by (hkl). The broader the peak the harder it is to make the distinction between amorphous and crystalline material. In conclusion, the peaks in the wide angle diffraction scan need to have their position(s) determined to compare it to crystalline values, their peak shape(s) analyzed for crystallite size, and use this data to determine if the nanomaterial is amorphous, paracrystalline and/or crystalline. One note, polymeric systems can have all three types of scattering entities present.
I agree with Lawrence. In nanosize (? I understand that as a nanomaterial), the materials can have periodicity in 3D. If you have a (h00) diffraction pattern in XRD this doesn't mean that you have a 1D or if you have a (hk0) pattern, it doesn't mean that you have a 2D periodicity. You can also have (h00) pattern on a non nano material in 3D. You can also have a nanosized material without periodicity (amorphous). I don't think that there is an obvious relation between nanosize and Miller index in XRD.
I ask myself, if we have only 2D periodicity, can we get the diffraction patterns in XRD? Because as far as I remerber diffraction=scattering+interference. The 3rd dimension permits to have the interference (path-length difference).
A 2D periodic material can give a diffraction pattern. If the 3-rd periodicity is missing the "diffraction" pattern involving the 3-rd dimension is continuous. Typically that happens in layered materials (like clay minerals) when layers are randomly stacked along the 3-rd direction.
There is quite a mess in the literature concerning the finite size of a real lattice and its periodicity (and quite often you see some absurd/unphysical interpretation of the corresponding diffraction data). To understand what's going on, you should also not forget the difference between single crystal and powder .
Let's take just ONE atomistic crystalline object with a 3D periodic lattice. In the nano world what you do is to limit the size of the real space (the crystalline lattice), as each object is made of a finite number of atoms. This finite real lattice, however, has a corresponding infinite reciprocal lattice.and therefore if you do a single crystal experiment you see all reflections permitted by the space group. The shape of the reciprocal lattice points is the Fourier transform of the real shape
If you take a powder of those objects, you just integrate on spheres in reciprocal space and you see again all reflections permitted by the space group.
If you take an oriented powder, then you don't integrate anymore on spheres but on sectors and therefore some of the reflections might be absent (typical in platelet or acicular shaped systems)
When the lattice is no longer 3D periodic, things become more complex. If it is just 2D periodic you can still apply what I said above (the transform a of a finite-width plane is just a finite-width rod) and you can use planar groups in place of spacegroups. Of course you will see the typical "2D reflections" coming from the reciprocal space rods when you do the corresponding integral on spheres. Expect to see some oddidties due to the extension of the rods in places where there should be no Bragg intensity. If the lattice has a periodicity in between, then welcome to my world: the reciprocal lattice becomes non periodic and intensity spreads a lot in between those that in the 3D periodic case were the Bragg spots. Here you can definitely see 3D, 2D and forbidden reflections in positions that seems disobeying Bragg's law. Talking about Miller indices is not trivial there.
Just as a final remark to the note of Matthias Arras:. the width of the peak is not given by Scherrer equation. Under very restrictive hypotheses (that people tend to forget), the equation can give you a number that correspond to the size of the coherently scattering domains. In general, the peak shape (and thus also the width that corresponds only to two points of it) is related to the Fourier transform of the shape function integrated over the whole size distribution and convoluted with all other broadening sources (instrument, defects).. it is easy to generate a diffraction pattern that looks well behaving, but that can completely fool Scherrer equation
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