Problems
Application 2.1 Moving Point Charges
2.3 Microscopic vs. Macroscopic
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38 THE MAXWELL EQUATIONS: HISTORY, AVERAGING, AND LIMITATIONS
(b) (a)
Figure 2.5: Contour plot of the valence charge density of crystalline silicon: (a)ρ(r) extracted from X-ray diffraction data; (b)ρ(r) computed from quantum mechanical calculations and the Maxwell equations. Periodic repetition of either plot in the vertical and horizontal directions generates one plane of the crystal. The rectangular contours reflect “bond” charge between the atoms (black circles). The white regions of very low valence charge density reflect nodes in thesp3wave functions. Figure from Tanaka, Takata, and Sakata (2002).
2.3 Microscopic vs. Macroscopic 39
1 3
2
Figure 2.6: Spherical volumes centered at arbitrary points in spacer1,r2, andr3.
as measured by X-ray diffraction with quantum mechanical calculations of the same quantity. First- principles calculations for diamagnetic molecules and ferromagnetic solids show similar agreement with experiment when the relevant magnetic fields are computed using microscopic magnetostatics.
Indeed, all such evidence suggests that the vacuum Maxwell equations (2.33) and (2.34) are valid for all length scales down to the Compton wavelength of the electronλc=h/mc2.43×10−12m (see Section 2.5). We take this to be the spatial resolution scale for the theory.
In contrast to the spatial degrees of freedom, there is no compelling reason to average over time in the microscopic Maxwell equations. The period of typical electron motion in atoms is 10−15–10−16s.
If this motion were averaged away, no consistent Maxwell theory of ultraviolet or X-ray radiation could be contemplated. This is clearly undesirable.
2.3.1 Lorentz Averaging
Lorentz spatial averaging is a mathematical procedure which produces slowly varyingmacroscopic sources and fields from rapidly varyingmicroscopicsources and fields. No single averaging scheme applies to every physical situation and the desired resolution scale typically differs from problem to problem. Moreover, the average is almost never carried out explicitly. Nevertheless, it is important to grasp the basic ideas because Lorentz averaging produces certain characteristic features of the macroscopic theory which are absent from the microscopic theory (see Section 2.3.2).
Quantum mechanical calculations show that microscopic quantities like the electric fieldEmicro(r, t) can vary appreciably on the scale of the Bohr radiusaB. These spatial variations average out when viewed at the much larger resolution scale of a macroscopic observer. This suggests the following three-step procedure: (1) center a sphere with volumeaB3 at every microscopic point r (see Figure 2.6); (2) carry out a spatial average ofEmicro(r, t) over the volume of the sphere,
E(r, t) = 1
d3sEmicro(s+r, t); (2.37)
and (3) exploit the fact thatE(r, t) varies slowly on the resolution scale of the continuous variabler and replaceE(r, t) byE(R, t) whereRis a low-resolution spatial variable. By “low resolution” we mean that the distance between two “adjacent” points in the continuousR-space is approximately the diameter of the averaging sphere (as measured inr-space). Thus, the “distant” pointsr1andr3 in Figure 2.6 could serve as adjacent points inR-space.E(R, t) is the macroscopic electric field we seek.
For a gas, a good rule of thumb for the averaging sphere is to choosein Figure 2.6 and in (2.37) equal to the inverse density of atoms. For a crystal,is better chosen as the volume of a unit cell.
If necessary or desirable, one can perform an additional spatial average of the macroscopic variable
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40 THE MAXWELL EQUATIONS: HISTORY, AVERAGING, AND LIMITATIONS
E(R, t) over a larger length scale determined by the spatial resolution of an experimental probe or by the spatial extent of density or compositional inhomogeneities.
Some authors offer much more elaborate and mathematically formal discussions of Lorentz aver- aging than we have done here. However, experience shows that it matters little whether one uses (2.37) or some alternative scheme (e.g., one which employs a smoothly varying weight function rather than a sharp cutoff to the averaging volume) to eliminate the spatial variations that occur within the averaging volume. What matters most is that the spatial averaging algorithm be alinearoperation.
This implies that the space and time derivatives which appear in the Maxwell equations commute with spatial averaging. For example, if we use angle brackets to denote the complete Lorentz averaging procedure, it is not difficult to confirm using (2.37) that
∇R×E(R)= ∇r×Emicro(r) (2.38)
and
∂E
∂t =
∂Emicro
∂t
. (2.39)
Similar results apply to∇R·Eand to the space and time derivatives ofB. The important conclusion to be drawn from identities like (2.38) and (2.39) is that the Maxwell equations have exactly the sameformwhether they are written in microscopic variables or macroscopic variables. Therefore, to simplify notation, we will generally write simplyE(r, t) andB(r, t) and rely on context to inform the reader whether these symbols refer to microscopic or macroscopic quantities.
A small cloud appears on the Lorentz averaging horizon when we turn to quantities which are bilinear in the fields and sources. An example is the force on the charge densityρ(r, t) and current densityj(r, t) in a volumeV due to electromagnetic fieldsE(r, t) andB(r, t):
F=
V
d3r{ρE+j×B}. (2.40)
Interpreted as a microscopic formula, direct substitution ofρandjfrom (2.6) and (2.17) confirms that (2.40) reproduces the Lorentz force law (2.1) for each microscopic particle. However, it is generally the case thatρE = ρEandj×B = j × B. How, then, shall we compute the force on a macroscopic body? The answer, not often stated explicitly, is that we simplyassumethat (2.40) remains valid when all the variables are interpreted macroscopically. No ambiguities arise as long asFis the total force on an isolated sample of macroscopic matter in vacuum.
2.3.2 The Macroscopic Surface
Lorentz averaging unavoidably produces singularities and discontinuities in macroscopic quantities when the averaging is performed in the immediate vicinity of a surface or interface. This is so because the input microscopic quantities (which are perfectly smooth and continuous at every point in space) exhibit rapid spatial variations near surfaces which are uncharacteristic of the variations which occur elsewhere. To illustrate this, the top panel of Figure 2.7 shows a contour map of the microscopic, ground state, valence electron charge densityρ0(r) near a flat, crystalline surface of metallic Ag in vacuum. The density was calculated by quantum mechanical methods similar to those used to obtain Figure 2.5(b).
The corrugation of the contour lines of ρ0(r) adjacent to the vacuum region (to the right) is characteristic of vacuum interfaces. Another characteristic feature—the “spilling-out” of the electron distribution into the vacuum—becomes most apparent when we averageρ0(r) over planes parallel to
2.3 Microscopic vs. Macroscopic 41
Figure 2.7: Side view of the free surface of crystalline Ag metal. The vacuum lies to the right. Top panel:
contour plot of the valence electron charge densityρ0(r) in one crystalline plane perpendicular to the surface.
The white circles are regions of very high electron density centered on the atomic nuclei. Bottom panel: the solid curve labeled ¯ρ0(z) isρ0(r) averaged over planes parallel to the surface. The local maxima of this curve are rounded off and do not accurately reflect the electron density near the nuclei; the dashed curve labeledσ λ(z) is the planar average of thechangein electron density induced by an electric field directed to the left; the solid curve labeledEz(z) is the planar average of the electric field perpendicular to the macroscopic surface. The vertical scale is different for each curve. The solid vertical line is the locationz=0 of the macroscopic surface.
Tick marks on the horizontal scale are separated by 2aBohr1 ˚A. Figure adapted from Ishida and Liebsch (2002). Copyright 2002 by the American Physical Society.
the surface with areaA. The function that results,
¯ ρ0(z)= 1
A
A
dxdy ρ0(r), (2.41)
is plotted in the bottom panel of Figure 2.7. The essential point is that ¯ρ0(z) falls to zero in the vacuum over a distance comparable to the interatomic spacing. The change of scale which accompanies Lorentz averaging in thez-direction destroys the fine resolution needed to see this behavior. This obliges us to define the “edge” of the macroscopic surface (solid vertical line) as a plane where the macroscopic charge distribution falls discontinuously to zero.
Now apply a horizontal electric field perpendicular to the surface and directed to the left. This perturbation induces a distortion of the electron wave functions in the immediate vicinity of the Ag surface. If ¯ρ(z) is the planar average of the electronic charge densityin the presence of the fieldandσ is the induced charge per unit area of surface, a functionλ(z) is defined by
¯
ρ(z)=ρ¯0(z)+σ λ(z). (2.42)
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The dashed curve ofσ λ(z) in the lower panel of Figure 2.7 shows that the main effect of the electric field is to pull some electronic charge away from the last plane of nuclei and farther out into the vacuum. The horizontal scale (about 1 ˚A between tick marks) shows that the width of the dashed curve cannot be resolved macroscopically. Therefore, after Lorentz averaging in thez-direction, all the induced charge resides in a single, two-dimensional plane. Quantitatively, Lorentz averaging replaces the smooth microscopic functionσ λ(z) by a singular macroscopic delta function:9
σ λ(z)→σ δ(z) (Lorentz averaging). (2.43) The curve labeledEz(z) in the lower panel of Figure 2.7 is the planar average of the microscopic electric field (see Section 2.4.2) in the direction normal to the surface.Ez(z) is non-zero in the vacuum, but falls very rapidly to zero in the near-surface region. Therefore, a Lorentz average in thez-direction recovers the familiar result that the macroscopic electric field justoutsidea perfect conductor drops discontinuously to zero justinsidea perfect conductor (see Section 5.2.2).
2.3.3 Matching Conditions
There is an analytic connection between the discontinuous electric field and the singular distribution of surface charge discussed in the previous section. To discover it, let the planez=0 separate two regions labeledLandR.EL(r) andρL(r) are the macroscopic electric field and charge density in regionL.ER(r) andρR(r) are the macroscopic electric field and charge density in regionR. Now, the step function(z) is defined (Section 1.5.3) by
(z)=
⎧⎨
⎩
0 z <0, 1 z >0.
(2.44) Using this function, we can write the electric field at every point in space as
E(r)=ER(r)(z)+EL(r)(−z). (2.45)
The charge density can be written similarly except that we must allow for the possibility of a surface charge densityσ(x, y) localized exactly atz=0. In light of (2.43), we write
ρ(r)=ρR(r)(z)+ρL(r)(−z)+σ δ(z). (2.46) Motivated by Gauss’ law,∇ ·E=ρ/0, the divergence of (2.45) is10
∇ ·E=[∇ ·ER](z)+[∇ ·EL](−z)+zˆ·(ER−EL)δ(z). (2.47) On the other hand,∇ ·EL=ρL/0and∇ ·ER=ρR/0. Therefore, if we set (2.46) equal to (2.47) times0and use square brackets to denote the evaluation ofELandERat points which are infinitesi- mally close to one another on opposite sides ofz=0, the final result is
ˆ
z·[ER−EL]=σ/0. (2.48)
This is amatching conditionwhich relates the discontinuity in the normal component of the macro- scopic electric field to the magnitude of the singular charge density at the interface. Our derivation based on the differential form of Gauss’ law draws explicit attention to the non-analytic nature of the
9 The centroid ofλ(z) does not exactly coincide with macroscopic termination of the zero-field charge distribution. We may locate the macroscopic induced charge density atz=0 nevertheless because the error we make is undetectable at the macroscopic scale.
10 ∇ ·[f(±z)]=(±z)∇ ·f+f· ∇(±z)=(±z)∇ ·f±δ(z)f·ˆz.