Atoms, electrons and Solids

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1 Atoms, electrons and Solids Shell model of an atom negative electron orbiting a positive nucleus QM tells that to minimize total energy the electrons fill up shells. Each orbit in a shell has a specific set of Q numbers Outermost electrons known as valance electrons When a shell is filled (it can only take a specific number of electrons) the atom becomes stable and inert. To fill shells atoms will share electrons minimizing the energy of the system We tend to imagine the electron orbiting at a radius, but can be misleading at times. Fig 1.1

2 L s h e l l w i t h t w o s u b s h e l l s N u c l e u s L K 1 s 2 s 2 p 1 s 2 2 s 2 2 p 2 o r [ H e ] 2 s 2 2 p 2 The shell model of the atom in which the electrons are confined to live within certain shells and in subshells within shells. Fig 1.1

3

4 The planetary model of the hydrogen atom in which the negatively charged electron orbits the positively charged nucleus. Fig 1.2

5 r o r = M o l e c u l e + S e p a r a t e d a t o m s + Force Attraction 0 Repulsion r o F A = A t t r a c t i v e f o r c e F N = N e t f o r c e I n t e r a t o m i c s e p a r a t i o n, r F R = R e p u l s i v e f o r c e Potential Energy, E(r) Attraction 0 Repulsion E o r o E R = R e p u l s i v e P E E = N e t P E E A = A t t r a c t i v e P E r ( a ) F o r c e v s r ( b ) P o t e n t i a l e n e r g y v s r (a) Force vs interatomic separation and (b) Potential energy vs interatomic separation. Fig 1.3

6 Atomic bonding and molecules Two or more atoms will bond to minimize their total energy. Due to a positive nucleus and negative electron cloud atoms both attract and repulse. These forces produce an energy minimum at characteristic distance apart. The nature of the atoms determines the type of bond (although all are basically energy minimization) Covalent bonds the electrons are shared Metallic bonding the electrons become completely free Ionic bonding An electron is exchanged Fig 1.3

7 Net Force in Bonding Between Atoms F N = F A + F R = 0 F N = net force F A = attractive force F R = repulsive force

8 (a) Force vs r (b) Potential energy vs r (a) Force vs interatomic separation and (b) Potential energy vs interatomic separation. Fig 1.3

9 Covalent Bonding Both atoms have a missing electron from the outermost valence shell. The minimal energy configuration is to share an electron and complete both bonds. This forms a molecule a stable entity with a complete set of electrons The electron becomes concentrated between the two atoms and creates a negative charge region that attracts the two atoms together. Very strong bond, high melting point, insoluble. Highly structured arrangements (crystals) No free electrons

10 E l e c t r o n s h e l l H - a t o m H - a t o m 1 s 1 s C o v a l e n t b o n d H - H M o l e c u l e Formation of a covalent bond between two H atoms leads to the H 2 molecule. Electrons spend majority of their time between the two nuclei which results in a net attraction between the electrons and the two nuclei which is the origin of the covalent bond. Fig 1.4

11 C o v a l e n t b o n d H H H H H H C L s h e l l K s h e l l C C c o v a l e n t b o n d s H H H H H H ( a ) ( c ) ( b ) ( a ) C o v a l e n t b o n d i n g i n m e t h a n e, C H 4, i n v o l v e s f o u r h y d r o g e n a t o m s s h a r i n g e l e c t r o n s w i t h o n e c a r b o n a t o m. E a c h c o v a l e n t b o n d h a s t w o s h a r e d e l e c t r o n s. T h e f o u r b o n d s a r e i d e n t i c a l a n d r e p e l e a c h o t h e r. ( b ) S c h e m a t i c s k e t c h o f C H 4 o n p a p e r. ( c ) I n t h r e e d i m e n s i o n s, d u e t o s y m m e t r y, t h e b o n d s a r e d i r e c t e d t o w a r d s t h e c o r n e r s o f a t e t r a h e d r o n. Fig 1.5

12 Metallic bonds Formed by materials with a mainly empty valence shell The valence electrons are given up completely to form and electron gas. Thus the electrons become de-localized and are shared amongst all of the atoms present. Produce densely packed, crystals (often FCC, BCC) Less ridged then Covalent, ductile. Lots of free electrons electrical, thermal conduction Fig 1.5

13 The diamond crystal is a covalently bonded network of carbon atoms. Each carbon atom is bonded covalently to four neighbors forming a regular three dimensional pattern of atoms which constitutes the diamond crystal. Fig 1.6

14 P o s i t i v e m e t a l i o n c o r e s F r e e v a l e n c e e l e c t r o n s f o r m i n g a n e l e c t r o n g a s In metallic bonding the valence electrons from the metal atoms form a "cloud of electrons" which fills the space between the metal ions and "glues" the ions together through the coulombic attraction between the electron gas and positive metal ions. Fig 1.7

15 Ionic Bonding The complete transfer of an electron from one atom to another (of different type). Known as salt comprised of a Cation (+ive) and Anion (-ive) The ions attract due to opposite charges Repulsed due to inner electron cloud Balance causes an energy minimum and a molecule will form. Periodic packing can build crystals Fig 1.7

16 Na Cl 3 s 3 s 3 p C l o s e d K a n d L s h e l l s ( a ) C l o s e d K a n d L s h e l l s Na + Cl - Na + Cl- F A F A 3 p 3 s r ( b ) r o ( c ) The formation of an ionic bond between Na and Cl atoms in NaCl. The attraction is due to coulombic forces. Fig 1.8

17 N a + C l N a + C l N a + C l C l N a + C l N a + C l N a + N a + C l N a + C l N a + C l C l N a + C l N a + C l N a + N a + C l N a + C l N a + C l C l N a + C l N a + C l N a + ( a ) ( b ) (a) A schematic illustration of a cross section from solid NaCl. NaCl solid is made of Cl and Na+ ions arranged alternatingly so that the oppositely charged ions are closest to each other and attract each other. There are also repulsive forces between the like-ions. In equilibrium the net force acting on any ion is zero. (b) Solid NaCl. Fig 1.9

18 P o t e n t i a l e n e r g y E ( r ), e V / ( i o n - p a i r ) n m Cohesive energy C l 1. 5 e V r = N a + r = C l N a S e p a r a t i o n, r C l N a + r o = n m Sketch of the potential energy per ion-pair in solid NaCl. Zero energy corresponds to neutral Na and Cl atoms infinitely separated. Fig 1.10

19 Secondary, Mixed bonding A weak attraction of the electron cloud of one atom (molecule) and the nucleus of another causes bonding Van der Walls forces. Strongest between polar molecules. Present for all atoms/molecules due fluctuations. Low melting point solids, friction, sticking. All bond types represent a model of energy mimization of the electron/atom configurations. In reality mixed bonds are present with characteristics of a number of types. Fig 1.10

20 H C l A B A B ( a ) ( b ) (a) A permanently polarized molecule is called a an electric dipole moment. (b) Dipoles can attract or repel each other depending on their relative orientations. (c) Suitably oriented dipoles attract each other to form van der Waals bonds. ( c ) Fig 1.11

21 H O H (a) (b) The origin of van der Waals bonding between water molecules. (a) The H2O molecule is polar and has a net permanent dipole moment. (b) Attractions between the various dipole moments in water gives rise to van der Waals bonding. Fig 1.12

22 T i m e a v e r a g e d e l e c t r o n ( n e g a t i v e c h a r g e ) d i s t r i b u t i o n C l o s e d L S h e l l N e I o n i c c o r e ( N u c l e u s + K - s h e l l ) I n s t a n t a n e o u s e l e c t r o n ( n e g a t i v e c h a r g e ) d i s t r i b u t i o n f l u c t u a t e s a b o u t t h e n u c l e u s. A v a n d e r W a a l s f o r c e B S y n c h r o n i z e d f l u c t u a t i o n s o f t h e e l e c t r o n s Induced dipole-induced dipole interaction and the resulting van der Waals force. Fig 1.13

23

24 Physical Properties The physical properties of a solid mechanical, thermal, optical, electrical -- are determined by the bonding and electron sharing As an example the mechanical property of elasticity is determined by how much energy it takes to pull the atoms apart. This requires pulling them out of the energy hole (minimum) in which they sit. The macroscopic quantity relating the force needed to stretch the material is the Young s Modulus and it can be related to the shape of the energy minimum of the material.

25 Definition of Elastic Modulus σ = Yε σ = applied stress (force per unit area), Y = elastic modulus, ε = elastic strain (fractional increase in the length of the solid) Elastic Modulus and Bonding Y 1 df 1 d 2 N r o dr 2 r= r ro dr E r= o r o Y = elastic modulus, r o = interatomic equilibrium separation, F N = net force, r = interatomic separation, E = bonding energy

26 S o l i d F L o + δ L A F 0 Attractive F N Y d F N / d r δ F N r δ F N δ F N Repulsive r o δ r ( a ) ( b ) (a) Applied forces F strech the solid elastically from Lo to δ L. The force is divided amongst chains of atoms that make the solid. Each chain carriers a force δ F N. (b) In equilibrium, the applied force is balanced by the net force δ F N between the atoms as a result of their increased separation. Fig 1.14

27 Elastic Modulus and Bond Energy Y f E bond r o 3 Y = elastic modulus f = numerical factor (constant) that depends on the type of the crystal and the type of the bond E bond = bonding energy r o = interatomic equilibrium separation

28 Heat, Energy and Distributions A lot of the properties of a material are determined by the movement of the atoms and electrons due their kinetic energy. The bulk properties of a material are related to the average velocity/energy of the atoms and electrons. We will start with an idealized gas: Random motion, simple collisions, no forces present, elastic collisions, Newton equations. Wish to relate temperature to average KE Will use for electrons, phonons, just about everything!

29 Kinetic Molecular Theory for Gases PV = 1 3 Nmv2 2 v P = gas pressure = mean square velocity N = number of gas molecules m = mass of the gas molecules

30 Ideal Gas Equation PV = (N/N A )RT N = number of molecules, R = gas constant, T = temperature, P = gas pressure, V = volume, N A = Avogadro s number Change in Momentum of a Molecule p = 2mv x p = change in momentum, m = mass of the molecule, v x = velocity in the x direction

31 S q u a r e C o n t a i n e r F a c e B A r e a A a v y F a c e A v x G a s a t o m s a a The gas molecules in the container are in random motion Fig 1.15

32 Rate of Change of Momentum F = p t = 2mv x (2a/ v x ) = mv 2 x a F = force exerted by the molecule, p = change in momentum, t = change in time, m = mass of the molecule, v x = velocity in the x direction, a = side length of cubic container Total Pressure Exerted by N Molecules P = mnv 2 x V P = total pressure, m = mass of the molecule, = mean square velocity along x, V = volume of the cubic container 2 v x

33 Mean Square Velocity v x 2 = v y 2 = v z 2 Mean square velocities in the x, y, and z directions are the same Mean Velocity for a Molecule v 2 = v x 2 + v y 2 + v z 2 = 3v x 2

34 Gas Pressure in the Kinetic Theory P 2 Nmv 1 = = ρ v 3V 3 2 P = gas pressure, N = number of molecules, m = mass of the gas molecule, v = velocity, V = volume, ρ = density. Mean Kinetic Energy per Atom KE = 1 2 mv2 = 3 2 kt k = Boltzmann constant, T = temperature

35 Internal Energy per Mole for a Monatomic Gas 1 U = N A 2 mv2 = 3 2 N kt A U = total internal energy per mole, N A = Avogadro s number, m = mass of the gas molecule, k = Boltzmann constant, T = temperature Molar Heat Capacity at Constant Volume C m = du dt = 3 2 N Ak = 3 2 R C m = specific heat per mole at constant volume (J K -1 mole -1 ), U = total internal energy per mole, R = gas constant

36 Partition of Energy We attribute 1/2m*v*v of energy to each degree of freedom -- x, y, z If other degrees of freedom need to be added: ie rotational, vibrational. In a solid there is potential energy term due to attraction. Think of a solid as balls connected by springs with spring constant K Simple harmonic oscillator. (KE = PE) This gives 6 degrees of freedom. Due to the relationship 1/2mv^2 = 1/2kT we attribute 1/2kT to each degree of freedom. Useful, but quite limited. This is the definition of temperature!

37 T R A N S L A T I O N A L M O T I O N R O T A T I O N A L M O T I O N I x = 0 v z v x z x I z y I y v y y a x i s o u t o f p a p e r Possible translational and rotational motions of a diatomic molecule. Vibrational motions are neglected. Fig 1.16

38 ( a ) y x z (a) The ball-and-spring model of solids in which the springs represent the interatomic bonds. Each ball (atom) is linked to its neighbors by springs. Atomic vibrations in a solid involve 3 dimensions. (b) An atom vibrating about its equilibrium position stretches and compresses its springs to the neighbors and has both kinetic and potential energy. ( b ) Fig 1.17

39 Internal Energy per Mole U = N A kt = 3RT U = total internal energy per mole, N A = Avogadro s number, R = gas constant, k = Boltzmann constant, T = temperature Dulong-Petit Rule C m = du dt = 3R = 25 J K-1 mol -1 C m = Heat capacity per mole at constant volume (J K -1 mole -1 )

40 Thermal Expansion As materials heat up they expand why? Temperature is KE higher T more KE. The atoms shake more as the energy minimum is asymmetrical they shake more away than towards. The average position moves outwards increasing the separation of the atoms and increasing the size of the solid. Can relate the shape of the energy minimum to the thermal expansion coefficient.

41 E n e r g y U ( r ) = P E 0 r o I n t e r a t o m i c s e p a r a t i o n, r U o B A r a v C T 2 B A C T 1 K E U m i n = U o U m i n The potential energy PE curve has a minimum when the atoms in the solid attain the interatomic separation at r = ro. Due to thermal energy, the atoms will be vibrating and will have vibrational kinetic energy. At T = T 1, the atoms will be vibrating in such a way that the bond will be stretched and compressed by an amount corresponding to the KE of the atoms. A pair of atoms will be vibrating between B and C. Their average separation will be at A and greater than ro. Fig 1.18

42 S t a t e A S t a t e B, K E = 0, E = U B S t a t e A S t a t e C, K E = 0, E = U C Vibrations of atoms in the solid. We consider, for simplicity a pair of atoms. Total energy is E = PE + KE and this is constant for a pair of vibrating atoms executing simple harmonic motion. At B and C, KE is zero (atoms are stationary and about to reverse direction of oscillation) and PE is maximum. Fig 1.19

43 Definition of Thermal Expansion Coefficient λ = 1 L o δl δt λ = thermal coefficient of linear expansion or thermal expansion coefficient, L o = original length, L = length at temperature T Thermal Expansion L = L o [1+ λ(t T o )] L = length at temperature T, L o = length at temperature T o

44 Distributions v and T A number of important properties/effects are due to the distribution of atoms/electrons in velocity or energy. A distribution describes the number of particles at certain v or E For example dn = nv(t,v) dv -- where nv is the number of particles at the velocity v for the temperature T in incremental dv. It can be shown or measured that in thermal equilibrium a Maxwell-Boltzman distribution is found for simple classical particles. We also have the equation for the energy distribution dn = ne(t,v) de Fig 1.21

45 Maxwell-Boltzmann Distribution for Molecular Speeds n = 4πN v m 2πkT 3 / 2 v 2 exp mv2 2kT n v = velocity density function, N = total number of molecules, m = molecular mass, k = Boltzmann constant, T = temperature, v = velocity

46 Relative number of molecules per unit velocity (s/km) v * v a v v r m s v * v a K ( 2 5 C ) v v r m S p e e d ( m / s ) s K ( C ) Maxwell-Boltzmann distribution of molecular speeds in nitrogen gas at two temperatures. The ordinate is dn/(ndv),the fractional number of molecules per unit speed interval in (km/s) -1 Fig 1.22

47 Maxwell-Boltzmann Distribution for Translational Kinetic Energies n = 2 E π N 1 kt 3 / 2 E 1/ 2 exp E kt n E = number of atoms per unit volume per unit energy at an energy E, N = total number of molecules per unit volume, k = Boltzmann constant, T = temperature.

48 N u m b e r o f a t o m s p e r u n i t e n e r g y, n E A v e r a g e K E a t T 1. T 1 A v e r a g e K E a t T 2 T > 2 T 1 E A E n e r g y, E Energy distribution of gas molecules at two different temperatures. The number of molecules that have energies greater than E A is the shaded area. This area depends strongly on the temperature as exp( E A /kt). Fig 1.23

49 Boltzmann Energy Distribution n E N E =C exp kt n E = number of atoms per unit volume per unit energy at an energy E, N = total number of atoms per unit volume in the system, C = a constant that depends on the specific system (weak energy dependence), k = Boltzmann constant, T = temperature

50 Effects of the Distributions The fact that the particles have a range of velocities (randomly distributed according the the MB equation) produces many effects As an example random fluctuations of the atoms will cause macroscopic (but small) variations in position (Brownian motion). The higher the temperature the larger the variations. Random electron motion in conductors causes noise in the currents and voltages key factor in electronics Both are T dependent due to the distribution of energy's/velocities

51 S O L I D G A S M V v m G a s A t o m Solid in equilibrium in air. During collisions between the gas and solid atoms, kinetic energy is exchanged. Fig 1.24

52 E q u i l i b r i u m C o m p r e s s i o n E x t e n s i o n x m x = 0 C o m p r e s s i o n m x < 0 E x t e n s i o n x > 0 m t Fluctuations of a mass attached to a spring due to random bombardment by air molecules Fig 1.25

53 Root Mean Square Fluctuations of a Body Attached to a Spring of Stiffness K ( x) = kt rms K K = spring constant, T = temperature, ( x) rms = rms value of the fluctuations of the mass about its equilibrium position.

54 Random motion of conduction electrons in a conductor results in electrical noise. Fig 1.26

55 Charging and discharging of a capacitor by a conductor due to the random thermal motions of the conduction electrons. Fig 1.27

56 Root Mean Square Noise Voltage Across a Resistance v = 4kTRB rms R = resistance, B = bandwidth of the electrical system in which noise is being measured, v rms = root mean square noise voltage, k = Boltzmann constant, T = temperature

57 Thermal activated processes Distributions very important in determining processes which involve energy barriers. Only particles with enough energy will be able to get over the barrier and initiate the process chemical reactions, electronic devices, transport phenomena. Use the energy distribution to calculate the number with a large enough energy to go over the barrier. Produces an Arrhenius rate equation Process have a characteristic exponential term sometimes a number of steps produces more then one term. Plot on an log graph to see them.

58 Diffusion of an interstitial impurity atom in a crystal from one void to a neighboring void. The impurity atom at position A must posses an energy E A to push the host atoms away and move into the neighboring void at B. Fig 1.29

59 Rate for a Thermally Activated Process ϑ = Av ο exp( E A /kt) E A = U A* U A ϑ = frequency of jumps, A = a dimensionless constant that has only a weak temperature dependence, v o = vibrational frequency, E A = activation energy, k = Boltzmann constant, T = temperature, U A* = potential energy at the activated state A*, U A = potential energy at state A.

60 O ' A f t e r N j u m p s L Y a q = 9 0 y O X q = q = 0 x q = An impurity atom has four site choices for diffusion to a neighboring interstitial vacancy. After N jumps, the impurity atom would have been displaced from the original position at O. Fig 1.30

61 Mean Square Displacement L 2 = a 2 ϑt = 2Dt L = distance diffused after time t, a = closest void to void separation (jump distance), ϑ = frequency of jumps, t = time, D = diffusion coefficient Diffusion coefficient is thermally activated D = 1 2 E a = D exp A 2 ϑ o kt D = diffusion coefficient, D O = constant, E A = activation energy, k = Boltzmann constant, T = temperature

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