Geometric Factor

7/28/2019 Geometric Factor

1/20

INTRODUCTION

The use of X-ray diffraction as a technique for crystal structure analysis dates from von Laue'sdiscovery of the X-ray effect for single crystal samples in 1912. Laue predicted that the atoms ofa single crystal specimen would diffract a parallel, monochromatic X-ray beam, giving series ofdiffracted beams whose directions and intensities would be dependent upon the lattice structureand chemical composition of the crystal. The predictions were soon verified by the experimentalwork of Friedrich and Knipping. The location of the diffraction maxima was explaind byW.L.Bragg on basis of a very simple model in which it is assumed that the X-rays are reflectedspeculary from successive planes of various (hkl) families in the crystal. His famous equation is:

Where is the glancing angle between the atomic plane and the incident beam, d is the spacingbetween the atomic planes of a given (hkl) family and is the wave length of the beam.

Experimental methods of X-ray diffraction

There are essentially three methods which may be employed. If one uses monochromatic X-rays,the equation above cannot be satisfied for an arbitrary value of theta. This has led to the rotating-crystal method, whereby reflection occurs for a discrete set of theta values. This method can be


2/20

applied only if single crystals of reasonable size are available. If this is not the case, one canemploy monochromatic X-rays when the sample is in powder form and held in a fixed position.The reason that a diffraction pattern is observed is that there are always enough crystallites of theright orientation available to satisfy the Bragg relation. By a proper analysis it is possible toidentify the indices (hkl) of a particular reflection, and this enables one to calculate the

interatomic parameters when the wavelength of the employed radiation is known. Finally, thereis the Von Laue method, in which the sample (a single crystall) is held stationary in a beam ofwhite X-rays. Each set of planes then "chooses" its own wavelength to satisfy the Bragg relation.

The Geometrical Structure Factor

If the crystal structure is that of a monoatomic lattice with an n-atom basis, then the contents ofeach primitive cell can be analyzed into a set of identical scatterers at positions d1,..,dn within thecell. The intensity of radiation in a given Bragg peak will depend on the extent to which the raysscattered fron these basis sites interfere with one another, being greatest when there is completeconstructive interference and vanishing altogether should there happen to be complete

destructive interference. The net ray scattered by the entire primitive cell is the sum of theindividual rays, and will therefore have an amplitude containing the factor:

K=(hb1,kb2,lb3) ; (b1,b2, b3) is the unit vector of a reciprocal lattice.

The quantity SK, known as the geometrical structure factor, expresses the extent to wichinterference of the waves scattered from identical ions within the basis can diminish the intensityof the Bragg peak associated with the reciprocal lattice vector K. The intensity in the Bragg peak,being proportional to the square of the absolute value of the amplitude, will contain a factor |SK|2.It is important to note that this is not the only source of K dependence in the intensity. Furtherdependence on the change in wave vector comes both from the ordinary angular dependence ofany electromagnetic scattering, together with the influence on the scattering of the detailedinternal structure of each individual ion in the basis. Therefore the structure factor alone cannot

be used to predict the absolute intensity in a Bragg peak. It can however, lead to a characteristicdependence on K that is easily discerned even though other less distinctive K dependences havebeen superimposed upon it. The one case in which the structure factor can be used withassurance is when it vanishes. This occurs when the elements of the basis are so arranged thatthere is complete destructive interference for the K in question. For each of the possible spacegroups there are characteristic absences of reflections, and from these the space group isdetermined.


3/20

---> (h2+k2+l2)Powder patterns for differet cubic crystals, illustratingcharacteristic reflections and absences for each type. [1]

Atomic Form Factor

If the ions in the basis are not identical, the structure factor assumes the form:

where fj, known as the atomic form factor, is entirely determined by the internal structure of theion that occupies position dj in the basis. This form factor is the ratio of the amplitude of the


4/20

radiation scattered by the atom to the amplitude of the radiation which a single electron wouldscatter under the same conditions according to classical theory.In the nonrelativistic approximation the form factor is given by:

is the total wave function of the atom and subscripts i and f refer to the initial and final states,* is the charge distribution. The vector s bisects the angle 180-2 between the incident andscattered wave vectors. The magnitude of s is |s|=4-1sin . The vector s/2 represents thechange in momentum of the X-ray. If the charge density of the atom is spherically symmetric theform factor will be:

Note that 4r2(r)dr is equal to the total number of electrons Z in the atom. Hence the atomicform factor is equal to Z only for =0, and less then Z for all other angles of scattering.


5/20

Atomic form factor for Zn as a function of sin()/.[2]

The charge distributions on which such curves are based may be obtained from a Hartreeapproximation or for atoms with a large number of electrons (beyond rubidium) from a statisticalatomic model developed by Thomas and Fermi.

These normalized scattering curves have been fitted to a 9- parameter equation by Don Cromerand Mann[3]. Knowing the 9coefficients, a(i), b(i) and c and the wavelength, we can calculatethe form factor of each atom at any given scattering angle.

This analytic expression gives an excellent fit to the form factor curves in the range sin()/ 2and is a convenient form for entering the information into a computer.
http://tx.technion.ac.il/~katrin/f0_CromerMann.txthttp://tx.technion.ac.il/~katrin/f0_CromerMann.txthttp://tx.technion.ac.il/~katrin/f0_CromerMann.txt


6/20

[1] Structure of Metals, McGraw-Hill 2d ed, 1952, p. 136.[2] Graph plotted in Matlab from the analytic expression.[3] INTERNATIONAL TABLES FOR X-RAY CRYSTALLOGRAPHY, MacGillavry, KluwerAcademic Pub.


7/20

In this lecture the following are introduced for ideal gases:

Internal energy change

Internal energy of a monatomic gas

Molar specific heat of a monatomic gas at constant volume Molar specific heat at constant pressure

Equipartition of energy

Temperature dependence of molar specific heat

Adiabatic changes

The change in Internal Energy of an Ideal Gas

By adding heat to a fixed volume of gas, the pressure and temperature increase but no

work is done by the system (no expansion).

For any ideal gas at constant volume,

the 1st Law of Thermodynamics gives

Notice that the change depends on the difference in temperature. This implies that the

change in internal energy does notdepend on the typeof process, i.e. whether the

change is at constant volume or constant pressure or both vary.


8/20

The graph shows three processes to gefinal temperature. In each case:

The Internal Energy of an Ideal Monatomic Gas

Since the atoms in an ideal monatomic gas are all mathematical points, with no

interactions between them (except collisions), the Internal Energy of the gas is simply

the sum of the translational kinetic energies of all the atoms. Starting rom the Kinetic

Theory Equation of the previous lecture):

Writing the Internal Energy as U, for an ideal monatomic gas:


9/20

In diatomic and more complex molecules, the energy can be distributed in other ways

than simple translation, because the atoms can vibrate and/or rotate with respect to

each other. In these cases the simple theory above has to be modified to include these

modes.

Example

A cylinder contains 0.03 m3 of Argon gas at a temperature of 250 C and a pressure of

1.2 MPa. Find the internal energy of the gas.

Since Argon is a noble gas, it consists only of atoms and the energy is purely

translational kinetic energy. Therefore:

The Molar Specific Heat for an Ideal MonatomicGas at Constant Volume

From above, the change in internal energy is:

The heat added to the system is:

From the 1st Law of Thermodynamics:

Now R = 8.314 J.(g mol)-1.K-1, so the Molar Specific heat for a monatomic gas atconstant volume is 12.5 J.(g mol)-1.K-1

Example

The specific heat at constant volume of a particular mass of Argon gas is 313.5 J.kg-1.

Find the mass of an Argon atom.


10/20

The Molar Specific Heat for an Ideal Gas at Constant Pressure

As an ideal gas expands its pressure will tend to drop

along the green line shown in the diagram.

To keep the pressure constant, an amount of heat

(Q) has to be added to the system, as indicated by

the temperature rise in the diagram.

For any ideal gas at constant pressure, the 1st Law of Thermodynamics gives


11/20

Example

1.5 mole of an ideal gas at 300K is heated at constant Atmospheric pressure till the

temperature is 320 K. Find the change in volume.

The Equipartition of Energy

As given above, for all practical purposes, the atoms in an ideal monatomic gas are

mathematical points. Thus the only energy that they can store is translational kinetic

energy along the 3 perpendicular axes of normal space. In contrast to this, diatomic

and polyatomic molecules, have definite shapes - as shown below.

The atoms in such a molecule can store energy in vibrations and rotations as well as

translations. Each way energy can be stored in the molecule is called a degree of

freedom.


12/20

The Oxygen molecule can store energy by rotation about the two perpendicular axes

shown in green, but not along the third perpendicular axis because the mass is too

close to that axis.

The Oxygen molecule has 2 degrees of rotational freedom as well as its 3 translational

ones.

The Methane molecule is a tetrahedron and can store energy in rotations about three

perpendicular axes.

The Methane molecule has 3 degrees of rotational freedom in addition to its 3

translational ones.

As well as rotations, the molecules can vibrate in a number of ways. For example, a

diatomic molecule can have its atoms vibrating in line with each other or parallel to

each other:

James Clerk Maxwell proposed the idea of equipartition of energy, which states that:

Each molecule in a gas is given an energy, , for each degree of freedom.

Since the Helium atom has only 3 translational degrees of freedom, Helium gas will

have an internal energy given by: per molecule. With for a

mole of an ideal monatomic gas.

Since the Oxygen molecule has 3 translational and 2 rotational degrees of freedom,

Oxygen gas will have an internal energy: per molecule.

With for a mole of an ideal diatomic gas.


13/20

Since the Methane molecule has 3 translational and 3 rotational degrees of freedom,

Methane gas will have an internal energy: per molecule.

With for a mole of an ideal polyatomic gas.

Including the Specific Heat at constant Pressure (with R added as above), the following

table can be constructed.

Molecule C v Cp

Monatomic

Diatomic

Polyatomic 3R 4R

Example

20 g of Oxygen is heated at constant Atmospheric pressure from 200C to 1200C. Find

(a) the heat transferred to the Oxygen, and

(b) what fraction of the heat raised the internal energy.


14/20

Temperature dependence of Molar Specific Heat at constant volume

The theory of Maxwell above doesn't work quite as simply as stated, because the

vibrational modes haven't been included! The reason is that there is a thresholdeffect.

The different modes are only switched on above certain temperatures. The graph below

shows the temperature variation of CV/R for Hydrogen.

At low temperatures, Hydrogen gas has only translational degrees of freedom available,

i.e. it is as if its molecules don't rotate.

At higher temperatures rotation cuts in, and at the highest temperatures vibrations

become possible.

At room temperatures Maxwell's equipartition of energy usually holds for normal gases.

The Adiabatic Expansion of an Ideal Gas

In an adiabatic expansion, no heat is transferred between the system and the

environment, i.e. Q goes to zero in the 1st Law of Thermodynamics. This happens

when the change occurs very quickly (as with sound waves) or slowly with a system

completely insulated from its environment.


15/20

On a PV graph, an adiabatic process has a steeper curve than isothermal processes.

The drop in temperature that occurs in an adiabatic expansion is due to small but non-

zero attractive forces between molecules (despite the kinetic theory assumptions).

Kinetic Energy is used up in overcoming the attractive forces.

To find an expression that characterises adiabatic processes, the starting point is the

ideal gas equation. This describes the state of the gas. When the state changes, there

are changes in pressure (p), volume (V) and temperature (T).

Introducing differentials into the state

equation gives:

From the 1st Law of Thermodynamics, for

an adiabatic change

Substituting the Adiabatic Temperature change int

differential 1st Law:


16/20

Here the parameter is the ratio of the specific heats,

Expanding and simplifying the differential 1st Law equation gives:

Dividing this bypVgives:

Integrating the result gives:

Finally, an equation for adiabatic change is:

Using the ideal gas equation, this can also be re-written

as:

Example

A sample of gas with =1.4 is at atmospheric pressure and 330 K. It is compressed

adiabatically to one third of its volume. Find its final pressure and temperature.

Example

A heat engine carries 12.03 moles of ideal monatomic gas around the cycle shown.


17/20

(a) Find the change in internal energy, the

the system and the work done by the syste

process, and the whole cycle.

(b) Find the pressure and volume for states

Answer (a)

Process U kJ

1->2 +120 +

2->3 +195 0

3->1 -315 -

Cycle 0 -

Answer (b)


18/20

State 1 2 3

Pressure kPa 200 200 1250

Volume m3 0.2 0.6 0.2

Summarising:

Internal energy change:

Internal energy of a monatomic gas:

Molar specific heat of a monatomic gas at constant volume:

Molar specific heat at constant pressure:

The Equipartition of Energy: Each molecule in a gas has energy, , for each

degree of freedom.

Adiabatic changes: and


19/20

Kinetic Theory of Matter

This theory explains the physical properties of matter in terms of motion of its molecules. According to this

theory, every substance (solid, liquid or gas) consists of a large number of minute particlescalled molecules. A molecule may be defined as the smallest particle of a substance that can exist in

free state and has all the characteristics of the present substance.

The molecules are in continuous random motion. They possess all possible velocities in all possible

directions. When a body is cooled, there is a decrease in the molecular motion. When a body is heated,

there is an increase in molecular motion.

The energy possessed by molecules is of two forms - kinetic energy and potential energy. The kinetic

energy manifests itself in molecular motion. The potential energy manifests either in the expansion of a

substance or in change of its state. The heat supplied to a body partly increases the kinetic energy andpartly increases the potential energy of the molecules.

To sum it up, the kinetic theory of matter is based upon the following points:

Matter consists of molecules. These are the smallest particles, which are capable of free

existence and retain all the chemical properties of the parent substance.

The molecules are always in a state of random continuous motion.

The molecules exert forces on one another. These forces depend upon intermolecular distance.

The intermolecular distances are greater in gases than in solids or liquids. As a result, the intermolecular

forces of attraction are very weak. So, the molecules of a gas are free to move about in the entire space

available to them. That is why a gas neither has a fixed volume nor a fixed shape.

When a gas is heated, the random motion of the molecules increases. This increases the rate of collision

and as a consequence, increases the pressure exerted by the gas on the walls of the container.


20/20

The kinetic theory of gases attempts to develop a model of the molecular behavior, which should result in

the observed behavior of an ideal gas.

Documents

Geometric Factor