12.10 Amorphous Magnets (MET, MS) 639
Substituting (12.53) into (12.54) and maximizing the expression with regard to the hopping range R gives:
σ =σ0 exp[(−T0 / T )1/ 4 ] , |
(12.55) |
where |
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T |
=1.5α3β / N (E |
F |
) , |
(12.56) |
0 |
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and β is a constant, whose value follows from the derivation, but in fact needs to be more precisely evaluated in a more rigorous presentation.
Maximizing also yields
R = constant(1/ T )1/ 4 , |
(12.57) |
so the theory is said to be for variable-range hopping (VRH); the lower the temperature, the longer the hopping range and the less energy is involved.
Equation (12.55), known as Mott’s law, is by no means a universal expression for the hopping conductivity. This law may only be true near the Mott transition, and even then that is not certain. Electron–electron interactions may cause a Coulomb gap (Coulombic correlations may lead the density of states to vanish at the Fermi level), and lead to a different exponent (from one quarter–actually to 1/2 for low-temperature VRH).
12.10 Amorphous Magnets (MET, MS)
Magnetic effects are typically caused by short-range interactions, and so they are preserved in the amorphous state although the Curie temperature is typically lowered. A rapid quench of a liquid metallic alloy can produce an amorphous alloy. When the alloy is also magnetic, this can produce an amorphous magnet. Such amorphous magnets, if isotropic, may have low anisotropy and hence low coercivities. An example is Fe80B30, where the boron is used to lower the melting point, which makes quenching easier. Transition metal amorphous alloys such as Fe75P15C10 may also have very small coercive forces in the amorphous state.
On the other hand, amorphous NdFe may have a high coercivity if the quench is slow so as to yield a multicrystalline material. Rare earth alloys (with transition metals) such as TbFe2 in the amorphous state may also have giant coercive fields (~ 3 kOe). For further details, see [12.20, 12.26, 12.36].
We should mention that bulk amorphous steel has been made. It has approximately twice the strength of conventional steel. See Lu et al [12.44].
Nanomagnetism is also of great importance, but is not discussed here. However, see the relevant chapter references at the end of this book.
640 12 Current Topics in Solid Condensed–Matter Physics
12.11 Soft Condensed Matter (MET, MS)
12.11.1 General Comments
Soft condensed-matter physics occupies an intermediate place between solids and fluids. We can crudely say that soft materials will not hurt your toe if you kick them.
Generally speaking, hard materials are what solid-state physics discusses and the focus of this book was crystalline solids. Another way of contrasting soft and hard materials is that soft ones are typically not describable by harmonic excitations about the ground-state equilibrium positions. Soft materials are also often complex, as well as flexible. Soft materials have a shape but respond more easily to forces than crystalline solids.
Soft condensed-matter physics concerns itself with liquid crystals and polymers, which we will discuss, and fluids as well as other materials that feel soft. Also included under the umbrella of soft condensed matter are colloids, emulsions, and membranes. As a reminder, colloids are solutes in a solution where the solute clings together to form ‘particles,’ and emulsions are two-phase systems with the dissolved phase being minute drops of a liquid. A membrane is a thin, flexible sheet that is often a covering tissue. Membranes are two-dimensional structures built from molecules with a hydrophilic head and a hydrophobic tail. They are important in biology.
For a more extensive coverage the books by Chaikin and Lubensky [12.11], Isihara [12.27], and Jones [12.30] can be consulted.
We will discuss liquid crystals in the next Section and then we have a Section on polymers, including rubbers.
12.11.2 Liquid Crystals (MET, MS)
Liquid crystals involve phases that are intermediate between liquids and crystals. Because of their intermediate character some call them mesomorphic phases. Liquid crystals consist of highly anisotropic weakly coupled (often rod-like) molecules. They are liquid-like but also have some anisotropy. The anisotropic properties of some liquid crystals can be changed by an electric field, which affects their optical properties, and thus watch displays and screens for computer monitors have been developed. J. L. Fergason [12.19] has been one of the pioneers in this as well as other applications.
There are two main classes of liquid crystals: nematic and smectic. In nematic liquid crystals the molecules are partly aligned but their position is essentially random. In smectic liquid crystals, the molecules are in planes that can slide over each other. Nematic and smectic liquid crystals are sketched in Fig. 12.19. An associated form of the nematic phase is the cholesteric. Cholesterics have a director (which is a unit vector along the average axis of orientation of the rod-like molecules) that has a helical twist.
12.11 Soft Condensed Matter (MET, MS) 641
Liquid crystals still tend to be somewhat foreign to many physicists because they involve organic molecules, polymers, and associated structures. For more details see deGennes PG and Prost [12.15] and Isihara [12.27 Chap. 12].
Fig. 12.19. Liquid crystals. (a) Nematic (long-range orientational order but no long-range positional order), and (b) Smectic (long-range orientational order and in one dimension long-range positional order)
12.11.3 Polymers and Rubbers (MET, MS)
Polymers are a classic example of soft condensed matter. In this section, we will discuss polymers4 and treat rubber as a particular example.
A monomer is a simple molecule that can join with itself or similar molecules (many times) to form a giant molecule that is referred to as a polymer. (From the Greek, polys – many and meros – parts). A polymer may be either naturally occurring or synthetic. The number of repeating units in the polymer is called the degree of polymerization (which is typically of order 103 to 105). Most organic substances associated with living matter are polymers, thus examples of polymers are myriad. Plastics, rubbers, fibers, and adhesives are common examples. Bakelite was the first thermosetting plastic found. Rayon, Nylon, and Dacron (polyester) are examples of synthetic fibers. There are crystalline polymer fibers such as cellulose (wood is made of cellulose) that diffract X-rays and by contrast there are amorphous polymers (rubber can be thought of as made of amorphous polymers) that don’t show diffraction peaks.
There are many subfields of polymers of which rubber is one of the most important. A rubber consists of many long chains of polymers connected together somewhat randomly. The chains themselves are linear and flexible. The random linking bonds give shape. Rubbers are like liquids in that they have a well-defined volume, but not a well-defined shape. They are like a solid in that they maintain their shape in the absence of forces. The most notable property of rubbers is that
4As an aside we mention the connection of polymers with fuel cells, which have been much in the news. In 1839 William R. Grove showed the electrochemical union of hydrogen and oxygen generates electricity—the idea of the fuel cell. Hydrogen can be extracted from say methanol, and stored in, for example, metal hydrides. Fuel cells can run as long as hydrogen and oxygen are available. The only waste is water from the fuel-cell reaction. In 1960 synthetic polymers were introduced as electrolytes.
642 12 Current Topics in Solid Condensed–Matter Physics
they have a very long and reversible elasticity. Vulcanizing soft rubber, by adding sulfur and heat treatment makes it harder and increases its strength. The sulfur is involved in linking the chains.
A rubber can be made by repetition of the isoprene group (C5H8, see Fig. 12.20).5 Because the entropy of a polymer is higher for configurations in which the monomers are randomly oriented than for which they are all aligned, one can estimate the length of a long linear polymer in solution by a random-walk analysis. The result for the overall length is the length of the monomer times the square root of their number (see below). The radius of a polymer in a ball is given by a similar law. More complicated analysis treats the problem as a self-avoiding random walk and leads to improved results (such as the radius of the ball being approximately the length of the monomer times their number to the 3/5 power). Another important feature of polymers is their viscosity and diffusion. The concept of reptation (which we will not discuss here, see Doi and Edwards [12.16]), which means snaking, has proved to be very important. It helps explain how one polymer can diffuse through the mass of the others in a melt. One thinks of the Brownian motion of a molecule along its length as aiding in disentangling the polymer.
CH3
[ CH2 C 
CH CH2 ]
Fig. 12.20. Chemical structure of isoprene (the basic unit for natural rubber)
We first give a one-dimensional model to illustrate how the length of a polymer can be estimated from a random-walk analysis. We will then discuss a model for estimating the elastic constant of a rubber.
We suppose N monomers of length a linked together along the x-axis. We suppose the ith monomer to be in the +x direction with probability of 1/2 and in the −x direction with the same probability. The rms length R of the polymer is calculated below.
Let xi = a for the monomer in the +x direction and −a for the −x direction. Then the total length is x = ∑xi and the average squared length is
x2 = ∑ xi |
2 = ∑ xi2 , |
(12.58) |
since the cross terms drop out, so |
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x2 = Na2 , |
(12.59) |
or |
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R = a |
N . |
(12.60) |
5 See, e.g., Brown et al [12.4]. See also Strobl [12.57].
12.11 Soft Condensed Matter (MET, MS) 643
We have already noted that a similar scaling law applies to the radius of a N- monomer polymer coiled in a ball in three dimensions.
In a similar way, we can estimate the tension in the polymer. This model or generalizations of it to two or three dimensions (See, e.g., Callen [12.7]) seem to give the basic idea. Let n+ and n− represent the links in the + and − directions. The length x is
x = (n+ − n−)a , |
(12.61) |
and the total number of monomers is |
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N = (n+ + n−) . |
(12.62) |
Thus |
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N − |
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The number of ways we can arrange N monomers with n+ in the +x direction and n− in the − direction is
Using S = kln(W) and using Stirling’s approximation, we can find the entropy S. Then since dU = TdS + Fdx, where T is the temperature, U the internal energy and F the tension, we find
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∂S |
+ |
∂U |
, |
(12.65) |
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∂x |
∂x |
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so we find (assuming we use a model in which ∂U/∂x can be neglected)
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kT |
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+ x / Na |
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kTx |
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F = |
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ln |
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(12.66) |
2a |
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Na2 |
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1− x / Na |
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The tension F comes out to be proportional to both the temperature and the extension x (it becomes stiffer as the temperature is raised!). Another way to look at this is that the polymer contracts on warming. In 3D, we think of the polymer curling up at high temperatures and the entropy increasing.
644 12 Current Topics in Solid Condensed–Matter Physics
Problems
12.1If the periodicity p = 50 Å and E = 5×104 V/cm, calculate the fundamental frequency for Bloch oscillations. Compare the results to relaxation times τ typical for electrons, i.e. compute ωBτ.
12.2Find the minimum radius of a spherical quantum dot whose electron binding energy is at least 1 eV.
12.3Discuss how the Kronig–Penny model can be used to help understand the motion of electrons in superlattices. Discuss both transverse and in-plane motion. See, e.g., Mitin et al [12.47 pp. 99-106].
12.4Consider a quantum well parallel to the (x,y)-plane of width w in the z direction. For simplicity assume the depth of the quantum well is infinite. Assume also for simplicity that the effective mass is a constant m for motion in all directions, See, e.g., Shik [12.54, Chaps. 2 and 4] .
a) Show the energy of an electron can be written
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E = |
2π 2n2 |
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(kx2 + k y2 ) , |
2mw2 |
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2m |
where px = kx and py = |
ky and n is an integer. |
b) Show the density of states can be written |
D(E) = |
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∑nθ(E − En ) , |
π |
2 |
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where D(E) represents the number of states per unit area per unit energy in the (x,y)-plane and
θ(x) is the step function θ(x) = 0 for x < 0 and = 1 for x > 0.
c) Show also D(E) at E >~ E3 is the same as D3D(E) where D3D represents the density of states in 3D without the quantum well (still per unit area in the (x,y)-plane for a width w in the z direction)
d) Make a sketch showing the results of b) and c) in graphic form.
12.5For the situation of Problem 12.4 impose a magnetic field B in the z direction. Show then that the allowed energies are discrete with values
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En, p = |
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Problems 645
where n, p are integers and ωc = |eB/m| is the cyclotron frequency. Show also the two-dimensional density of states per spin (and per unit energy and area in (x,y)-plane) is
D(E) = |
eB |
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when |
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These results are applicable to a 2D Fermi gas, see, e.g., Shik [12.54, Chaps. 7] as well as 12.7.2 and 12.7.3.
Appendices
A Units
The choice of a system of units to use is sometimes regarded as an emotionally charged subject. Although there are many exceptions, experimental papers often use mksa (or SI) units, and theoretical papers may use Gaussian units (or perhaps a system in which several fundamental constants are set equal to one).
All theories of physics must be checked by comparison to experiment before they can be accepted. For this reason, it is convenient to express final equations in the mksa system. Of course, much of the older literature is still in Gaussian units, so one must have some familiarity with it. The main thing to do is to settle on a system of units and stick to it. Anyone who has reached the graduate level in physics can convert units whenever needed. It just may take a little longer than we wish to spend.
In this appendix, no description of the mksa system will be made. An adequate description can be found in practically any sophomore physics book.1
In solid-state physics, another unit system is often convenient. These units are called Hartree atomic units. Let e be the charge on the electron, and m be the mass of the electron. The easiest way to get the Hartree system of units is to start from the Gaussian (cgs) formulas, and let |e| = Bohr radius of hydrogen = |m| = 1. The results are summarized in Table A.1. The Hartree unit of energy is 27.2 eV. Expressing your answer in terms of the fundamental physical quantities shown in Table A.1 and then using Hartree atomic units leads to simple numerical answers for solid-state quantities. In such units, the solid-state quantities usually do not differ by too many orders of magnitude from one.
1Or see “Guide for Metric Practice,” by Robert A. Nelson at http://www.physicstoday.org/guide/metric.html.
Table A.1. Fundamental physical quantities*
Quantity |
Symbol |
Expression / value in |
Expression / value in |
Value in Hartree |
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Gaussian units |
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Charge on |
e |
1.6 × 10−19 coulomb |
4.80 × 10−10 esu |
1 |
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electron |
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Mass of elec- |
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0.91 × 10−27 g |
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1.054 × 10−27 erg s |
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Compton |
λc |
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2π( /mc) |
2π( /mc) |
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1 |
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137 |
wavelength of |
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2.43 × 10−12 m |
2.43 × 10−10 cm |
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electron |
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4πε0 2/me2 |
2/me2 |
1 |
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hydrogen |
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0.53 × 10−10 m |
0.53 × 10−8 cm |
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Fine structure |
α |
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e2/ c |
e2/ c |
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Speed of light |
c |
3 × 108 m s−1 |
3 × 1010 cm s−1 |
137 |
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Classical |
r0 |
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e2/4πε0mc2 |
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electron ra- |
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2.82 × 10−15 m |
2.82 × 10−13 cm |
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dius |
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Energy of |
E0 |
e4m/32(πε0 )2 |
me4/2 2 |
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ground state |
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13.61† eV |
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of hydrogen |
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Bohr magne- |
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e /2mc |
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ton (calcu- |
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0.927 × 10−23 |
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274 |
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lated from |
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amp meter2 |
erg gauss−1 |
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above) |
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Cyclotron |
ωc, or |
(μ0e/2m)(2H) |
(e/2mc)(2H) |
1 |
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(2H) |
frequency |
ωh |
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274 |
(calculated from above)
* The values given are greatly rounded off from the standard values. The list of fundamental constants has been updated and published yearly in part B of the August issue of Physics Today. See, e.g., Peter J. Mohr and Barry N. Taylor, “The Fundamental Physical Constants,” Physics Today, pp. BG6-BG13, August, 2003. Now see http://www.physicstoday.org/guide/fundcon.html.
† 1 eV = 1.6 × 10−12 erg = 1.6 × 10−19 joule.
Normal Coordinates 649
B Normal Coordinates
The main purpose of this appendix is to review clearly how the normal coordinate transformation arises, and how it leads to a diagonalization of the Hamiltonian. Our development will be made for classical systems, but a similar development can be made for quantum systems. An interesting discussion of normal modes has been given by Starzak.2 The use of normal coordinates is important for collective excitations such as encountered in the discussion of lattice vibrations.
We will assume that our mechanical system is described by the Hamiltonian
H = |
1 |
∑i, j (xi x jδij +υij xi x j ) . |
(B.1) |
2 |
In (B.1) the first term is the kinetic energy and the second term is the potential energy of interaction among the particles. We consider only the case that each particle has the same mass that has been set equal to one. In (B.1) it is also assumed that υij = υji; and that each of the υij is real. The coordinates xi in (B.2) are measured from equilibrium that is assumed to be stable. For a system of N particles in three dimensions, one would need 3N xi to describe the vibration of the system. The dot of x· i of course means differentiation with respect to time, x· i =
dxi/dt.
The Hamiltonian (B.1) implies the following equation of motion for the mechanical system:
∑ j (δij x j +υij x j ) = 0 . |
(B.2) |
The normal coordinate transformation is the transformation that takes us from the coordinates xi to the normal coordinates. A normal coordinate describes the motion of the system in a normal mode. In a normal mode each of the coordinates vibrates with the same frequency. Seeking a normal mode solution is equivalent to seeking solutions of the form
In (B.3), c is a constant that is usually selected so that ∑j|xj|2 = 1, and |caj| is the amplitude of vibration of xj in the mode with frequency ω. The different frequencies ω for the different normal modes are yet to be determined.
Equation (B.2) has solutions of the form (B.3) provided that
∑ j (υij a j −ω2δij a j ) = 0 . |
(B.4) |
Equation (B.4) has nontrivial solutions for the aj (i.e. solutions in which all the aj do not vanish) provided that the determinant of the coefficient matrix of the aj vanishes. This condition determines the different frequencies corresponding to the different normal modes of the mechanical system. If V is the matrix whose
2 See Starzak [A.25 Chap. 5].
