Structure of Materials - physics.sharif.edu

Structure of Materials

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Democritus and Leucippus, materials made by unchangeable solid parts, atoms fifth century B.C.

Lavoisier and his wife, the conservation of matter in many careful chemical experiments. ~1790

Dalton, atomicity of nature in the law of multiple proportions of compounds. ~1840

Avogadro, same number of molecules per unit volume. ~ 1850.

Maxwell,1 who showed with his molecular-kinetic theory of gases. ~1870.


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How do the electrons

move around the

nucleus to form a

stable atom, and how

does their motion

account for the

observed spectrallines?


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Matter wave

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In his doctoral dissertation he postulated that because photons have wave and particle

characteristics, perhaps all forms of matter have wave as well as particle properties. This

was a radical idea with no experimental confirmation at that time.

de Broglie

Matter wave

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Wavelike behavior of particles

Is this dual particle-wave nature a property only of light or of material objects as well?

For typical moving automobile ? For shout bullet ?

For us ?For photoelectrons ?

For electrons in SEM gun ?For electrons with K=100 MeV?

Because of the smallness of h, only for particles of atomic or nuclear size will the wave behavior be observable

1924

Associated with the particle as it moves, there is a de Broglie wave of wavelength λ, which shows itself

when a wave-type experiment is performed on it.

Which sort of wave? Amplitude? Wave Mechanics7

Experimental evidence for wavelike behavior of particles

Particle diffraction

In an electron diffraction experiment, a beam of electrons is

accelerated from rest through a potential difference V, acquiring a

nonrelativistic kinetic energy K = eV and a momentum p =√2mK. Wave

mechanics would describe the beam of electrons as a wave of

wavelength λ = h/p.

19061937


1926, Bell lab, Davisson and Germer

Ni

∆V = 54 V

Each of the atoms of the crystal can act as ascatterer, so the scattered electron waves caninterfere, and we have a crystal diffractiongrating for the electrons.

Because the electrons were of low energy, theydid not penetrate very far into the crystal, and itis sufficient to consider the diffraction to takeplace in the plane of atoms on the surface.

reflection-type diffraction grating

For Ni crystal


diffraction of neutrons by a salt crystal

Scattering of 1-GeV

kinetic energy

protons by oxygen

nuclei

Determination of

nuclear sizes from

such diffraction

patterns

Crystalline Polycrystalline Amorphous

Story of oxidized Nickel


The first double-slit experiment with electrons was done in 1961, Jonsson


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double slit of separation

8μm and width 1μm

Experimental application for wavelike behavior of particles

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Double-Slit Experiment with Light

When one of the slits is covered

Interference pattern is due to light passing through both slits : a wave phenomenon

If the light intensity is reduced, and we observe the pattern on ascreen, we learn that the light arriving on the screen producesflashes at various points, entirely representative of particlebehavior! If we take pictures of the screen after varying lengths oftime, eventually the interference pattern characteristic of wavebehavior emerges. There is therefore no contradiction in thisexperiment. If we want to know the precise location of the light(photon), we must use the particle description and not the wavedescription.


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Through Which Slit Does the Particle Pass? If we fired the particles through the slits at a slow enough rate, we

could track each particle as it passed through one slit or the other

and then appeared on the screen.

No matter what sort of device we use to determine through which

slit the particle passes, the interference pattern will be destroyed.

The classical particle must pass through one slit or the other; only

a wave can reveal interference, which depends on parts of the

wave front passing through both slits and then recombining.

imaginary experiment

Asking which slit : particle aspect

principle of complementarity

The particle and wave natures cannot be observed

simultaneously, and the type of behavior that we observe

depends on the kind of experiment we are doing: a particle-

type experiment shows only particle like behavior, and a

wave-type experiment shows only wavelike behavior.


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Thought experiment

The attempt to identify which slit the electron is passing

through will in itself change the interference pattern

Bohr’s principle of complementarity: It is not possible todescribe physical observables simultaneously in terms of bothparticles and waves. We need both!

In trying to determine which slit the electron went through, weare examining the particle-like behavior of the electron. Whenwe are examining the interference pattern of the electron, weare using the wavelike behavior of the electron.

Wave motion

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when two waves are added together, we obtain regions ofrelatively large (and small) displacement. If we add many wavesof different amplitudes and frequencies in particular ways, it ispossible to obtain what is called a wave packet. The importantproperty of the wave packet is that its net amplitude differsfrom zero only over a small region x.

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The individual waves ψ1 and ψ2 each move with their own phase velocity

The combined wave has a phase velocity

Only the group velocity, which describes the speed of the envelope

In order to know precisely the position of the wave packet

envelope (x small), we must have a large range of wave numbers (k

large). Similarly, to know precisely when the wave is at a given

point (t small), we must have a large range of frequencies (f large).

Wave motion

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Wave packets

Localized particle in a region

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Materials in which the phase velocity varies with wavelength are said to exhibit dispersion. (glass)

Wave packets

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Wave packets

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Motion of wave packets

We are now in a position to apply our general theory of wave groups to electrons.the wave packet and the particle move at the same velocity.

show dispersion even in empty space and always travel at a speed that is greater than or at least equal to c.

matter wave envelope should move at the same speed as the particle

De Broglie waves in general have different phase speeds, so that their wave packets expand as they travel

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Motion of wave packets

De Broglie waves in general have different phase speeds, so that their wave packets expand as they travel

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Wave packets

We find that to restrict the size of the wave packet we must sacrifice the precise knowledge of the wavelength

For two waves

To construct a wave packet with a finite width, we must replace the first cosine term with a function that is large in the region where we want to confine the particle but that falls to zero as x→±∞.

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Wave packets for particles

the simplest function that has this property is 1/x, so we might imagine a wave packet whose mathematical form is

Another function that has this property is the Gaussian modulating function

Let’s arbitrarily define the width of the wave packet as the distance over which the amplitude of the central region falls by

1/2. That occurs roughly where the argument of the sine has the value ±π/2, which gives ∆x∆λ ∼ λ2o consistent with our

classical uncertainty estimate

Uncertainty : Classic waves

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The wave extends from −∞ to +∞, so the particle might be found anywhere in that region

a narrow wave pulse for locating the particle in a small region of space, but this wave does not have an easily identifiable

wavelength.

the better we know its momentum (or wavelength), the less we know about its location

The smaller the size of the wave packet, the greater the uncertainty in our knowledge of

the wavelength

Uncertainty : Classic waves

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The “size” of the wave packet is now its duration in time, which is

roughly one period T for this wave packet

We have some difficulty locating exactly the start and end of one cycle, so we have an uncertainty ∆T in measuring the period

Heisenberg Uncertainty Relationship

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Quantum mechanics provides a formal procedure for calculating ∆x and ∆p for wave packets corresponding todifferent physical situations and for different schemes for confining a particle. One outcome of these calculationsgives the wave packet with the smallest possible value of the product ∆x∆p, which turns out to be h/4π

The more we try to confine a particle, the less we know about its momentum

we will do worse than this limit

Nobel 1932

Because of the small value of h, the uncertainty principle becomes important only on the atomic level

if we are uncertain as to the exactposition of a particle, for example,an electron somewhere inside anatom of diameter /, the particlecan’t have zero kinetic energy

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Statistical Interpretation of Uncertainty

which gives in effect a root-mean-square value of px. This can be taken to be a rough measure of the magnitude of ∆Px. Thus it is often said that ∆Px gives a measure of the magnitude of the momentum of the particle.

Consider large number of particles passes (one at a time)

through the slit, and we measure the transverse (x component)

momentum of each particle after it passes through the slit. We

can do this experiment simply by placing a detector at different

locations on the screen where we observe the diffraction

pattern. Because the detector actually accepts particles over a

finite region on the screen, it measures in a range of deflection

angles or equivalently in a range of transverse momentum.

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If the electrons initially have no component of their momentum in the x direction, we know Px exactly (it is exactly zero)

very wide beam of electrons, only a small fraction of which pass through the slit

know quite a bit more about their x location their x location is no larger than a

Larger than the spread of angles into which most of the particles are diffracted

the distribution of transverse momentum given by the uncertainty principleis roughly equivalent to the spreading of the beam into the centraldiffraction peak, and it illustrates again the close connection between wavebehavior and uncertainty in particle location.

As we make the slit narrower, Px increases and the beam spreads evenmore. In trying to obtain more precise knowledge of the location of theparticle by making the slit narrower, we have lost knowledge of thedirection of its travel.

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If a particle has a very short lifetime between its creation and decay (∆t → 0), a measurement of its restenergy (and thus its mass) will be very imprecise (∆E → ∞). Conversely, the rest energy of a stable particle(one with an infinite lifetime, so that ∆t = ∞) can in principle be measured with unlimited precision (∆E = 0).

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Spreading wave packets

The more successful we are at confining a wave packet, the more quickly it spreads

we make the slit, the more the waves diverge after passing through the slit

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PROBABILITY AND RANDOMNESS

We cannot predict the outcome of any single measurement of the position of the electron in the atom we prepared, but if we do a large number of measurements, we ought to find a statistical distribution of results.

We cannot develop a mathematical theory that predicts the result of a single measurement, but we do have a mathematical

theory that predicts the statistical behavior of a system (or of a large number of identical systems)

we would be dealing with a large number of atoms, and so our concept of statistical averages is very useful

Quantum mechanics uses similar language. For example, if we say that the electron in a hydrogen atom has a 50%probability of circulating in a clockwise direction, we mean that in observing a large collection of similarly preparedatoms we find 50% to be circulating clockwise. Of course a single measurement shows either clockwise orcounterclockwise circulation.

Of course, one could argue that the flip of a coin or the roll of the dice is not a random process, but that the apparently random nature of the outcome simply reflects our lack of knowledge of the state of the system.

According to this interpretation, we could predict exactly the behavior of the electron in our atom if only we knew thenature of a set of so-called “hidden variables” that determine its motion. However, experimental evidence disagrees withthis theory, and so we must conclude that the random behavior of a system governed by the laws of quantum physics is afundamental aspect of nature and not a result of our limited knowledge of the properties of the system.

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The Probability Amplitude

What does the amplitude of the de Broglie wave represent? In any wave phenomenon, a physical quantity such as displacement or pressure varies with location and time. What is the physical property that varies as the de Broglie wave propagates?

The probability of finding the particle at any point depends on the amplitude of its de Broglie wave at that point

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Wave mechanics

The confused state of affairs surrounding ψ at that time was nicely described in a poem by Walter Huckel:

The currently held view is that a particle is described by a function ψ(x, y, z, t ) called the wave function. Thequantity ψ*ψ = ІψІ2 represents the probability per unit volume of finding the particle at a time t in a small volumeof space centered on (x, y, z). Probability of finding a particle is directly proportional to ІψІ2 .

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Wave mechanics

the act of measurement disturbs the electron’s path enough to destroy the delicate interference pattern

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