XRR and XRD fringes

Spin glass or ferromagnets?

Neel temperature of RTO3

Nature of band gap in conjugate polymers

Power generation of a photo cell as a function of the electronic band gap

Following the post on

http://xiaoshanxu.blogspot.com/2010/07/blog-post.html

we calculated the power generation of a photo cell as a function of band gap, assuming the filling factor is unity.

As one can see that 1.1 eV is the optimal gap in terms of power generation.

Change of supersaturation

Systematic extinction rules

Systematic extinction rules

The structural factor in the crystallographic diffraction is:

$F=\Sigma { f_j exp(j \vec{r} \vec{k})}$,

where $j$ runs through all the atoms in the unit cell.

When the unit cell has more than one primitive cell in it, there might be some cases that $F$ can be zero. This is called systematic extinction (absence).

For example in BCC, for every atom at position $r$, there is another identical one at $\vec{r}+(1/2,1/2,1/2)$. Then the structural factor becomes:

$F=1+exp[i\pi (h+k+l)]$,

where $(h,k,l)$ are the reciprocal indices.

Obviously, $F$ is zero when $h+k+l=2n+1$.

TABLE I, Systematic absence because of lattice type

Lattice type	Symbol	Absence
a-center (100)	A	k+l=2n+1
b-center (010)	B	h+l=2n+1
c-center (001)	C	h+k=2n+1
face center	F	h,k,l not all odd and not all even
body center	I	h+k+l=2n+1
rhomohedral	R	-h+k+l=3n+1
hexagonal	H (triple unit cell)	h-k=3n+1 or h-k=3n+2
primitive	P	no absence

Above is only the systematic absence caused by typical symmetry of lattice types. There are other systematic absences coming from existence of screw axis and glide planes. In the end, for every space group the systematic absence is more complicated than the cases shown in the above table.

To get complete information or to find out the systematic absence for every space group, one can use Bilbao server : http://www.cryst.ehu.es/. Just click on " HKLCONDReflection conditions of Space Groups" and input the space group number.

Long wavelength dispersion relation in a solid

Long wavelength dispersion relation in a solid

In solid state physics, perhaps the most visited two elementary excitations are phonons (vibration) and magnons (spin wave). It is interesting to notice that the long wavelength dispersion relation are different for them, i.e.
phonon:
magnon: ,
where is the frequency and is the wave vector.

One can go back to the derivation of those two excitations and find the mathematical reason. On the other hand, the origin can be resorted to the fundamental difference between the two equation of motions.
phonon:
magnon: ,
where x is the displacement, is the angular momentum, is the spring constant and is the torque.

In solid, the restoring force for an oscillator comes from the imbalance of the interaction between neighbors. For long sinusoidal wave, it is always propotional to .

Hence, the final dispersion relation will be decided by the equation of motion. Since for phonon, it involves second derivative, one gets ; for magnon, the first derivative gives .

Supersaturation

Change of supersaturation

In gas-solid phase transition, the phase boundary as a function of  divides the two phases. In equilibrium, if a  combination falls on the solid side, the matter is in its solid phase. On the other hand, at inequilibrium, for a finite amount of time, a system can be with a  combination on the solid side but still remain in gas phase. This is called supersaturation. As shown in the Fig below for the points  and if they are in gas phase.
The quantitative definition of supersaturation is
,
where is the pressure corresponding to the that on the phase boundary line for the temperature . This is actually the chemical potential difference between state and state  in gas phase.

It is useful to know the dependence of on and . In other words, . Here we will start with the differential form of the relation, i.e. and . The second partial differential can be found from the definition:
,
while /Tneeds some effort to find.

To find , we look at  


From Gibbs-Duham equation:

The first term
.
Or

The second term is the chemical potential change along the phase boundary.
Since on the phase boundary, one has

where is the enthalpy change between solid and gas phase per mole, which does not vary too much with the temperature.
Again using Gibbs-Duham equation,
.

Therefore,

Or
.
Finally:
.
Or

$d\Delta\mu=RTdlnP-hdlnT$

Differential pumping

Differential pumping

Problem:

The governing equation of the problem is the following:
P2=G/S (1)
C(P1-P2)=P2S (2)

Note that P1>>P2, so the equation (2) can be written as
CP1=P2S.

In fact, if we believe these equation are true, the quantity P1 is not affected by the pumping speed Sas long as G is kept constant and P1>>P2. So changing the speed of the pump (e.g. by closing the gate valve upstream of pump to some extend) does not affect the pressure P1, at lest to the first order.

On the other hand, if all the quantities except P2are unknown, there is not way to know the pressure P1.

gas mix