angle as a function of its impact parameter b and then average the result over all values of b, considered random. 30
In quantum mechanics, due to the particle/wave duality, a relatively broad, parallel beam of incident particles of the same energy E may be fairly represented with a plane de Broglie wave (1.88):
exp i
,
(3.60)
i
i
k r
i
with the free-space wave number k i = k = (2 mE)1/2/. As a result, the particle scattering becomes a synonym of the de Broglie wave diffraction, and (somewhat counter-intuitively) the description of the effect becomes simpler, excluding the notion of the impact parameter. Indeed, the wave (60) corresponds to a constant probability current density (1.49):
2
j
k ,
(3.61)
i
i
i
m
which is exactly the flux of incident particles per unit area that is used in the denominator of Eq. (58), while the numerator of that fraction may be simply expressed via the probability current density js of the scattered de Broglie waves:
2
d
j r
s
,
at r .
a
(3.62)
d
j i
Hence our task is reduced to the calculation of j s at sufficiently large distances r from the scatterer. For that, let us rewrite the stationary Schrödinger equation for the elastic scattering problem (when the energy E of the scattered particles is the same as that of the incident particles) in the form 2
2
2
k 2
E H ˆ
r
,
(3.63)
0
U ( ) ,
H ˆ
with
and
,
E
0
2 m
2 m
where the potential energy U(r) describes the effect of the scatterer. Looking for the solution of Eq. (62) in the natural form
,
(3.64)
i
s
where i is the incident wave (60) and s has the sense of the scattered wave, and taking into account that the former wave satisfies the free-space Schrödinger equation
ˆ
H E ,
(3.65)
0
i
i
we may reduce Eq. (63) to either of the following equivalent forms:
m
2
2
2
ˆ
E H
U r
k
U r
.
(3.66)
0 s
i
s ,
(
)
s
2
For applications, an integral form of this equation is more convenient. To derive it, we may look at the second of Eqs. (66) as a linear, inhomogeneous differential equation for the function s, thinking of its right-hand side as a known “source”. The solution of such an equation obeys the linear superposition principle, i.e. we may represent it as the sum of the waves outcoming from all elementary volumes d 3 r’ of the scatterer. Mathematically, this sum may be expressed as either 30 See, e.g., CM Sec. 3.5.
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2 m
(r)
U ( '
r ) ( '
r G
) (r, '
r ) d 3 r' ,
(3.67a)
s
2
or, equivalently, as31
2 m
(r) r
(r ) (r ) (r, r )
,
(3.67b)
i
U '
' G
' d 3 r'
2
where G(r, r ’) is the spatial Green’s function, defined as such an elementary, spherically-symmetric response of the 3D Helmholtz equation to a point source, i.e. the outward-propagating solution of the following equation32
2
2
k G (r ' r) .
(3.68)
But we already know such solution of this equation – see Eq. (7) and its discussion:
f
G r
( , '
r ) eikR ,
where R
r '
r ,
(3.69)
R
so that we need just to calculate the coefficient f+ for Eq. (68). This can be done in several ways, for example by noticing that at R << k-1, the second term on the left-hand side of Eq. (68) is negligible, so that it is reduced to the well-known Poisson equation with a delta-functional right-hand side, which describes, for example, the electrostatic potential induced by a point electric charge. Either recalling the Coulomb law or applying the Gauss theorem,33 we readily get the asymptote
1
G
,
at kR ,
1
(3.70)
4 R
which is compatible with Eq. (69) only if f+ = –1/4, i.e. if
1
ikR
G(r, '
r )
e
.
(3.71)
R
4
Plugging this result into Eq. (67a), we get the following formal solution of Eq. (66):
Scattering
m
( '
r )
problem:
(r)
U ( '
r )
eikRd 3 r' .
(3.72)
s
formal solution
2
2
R
Note that if the function U(r) is smooth, the singularity in the denominator is integrable (i.e. not dangerous); indeed, the contribution of a sphere with some radius R 0, with the center at point r ’, into this integral scales as
31 This formula is sometimes called the Lipmann-Schwinger equation, though more frequently this term is reserved for either its operator form or the resulting equation for the spatial Fourier components of and i.
32 Please notice both the similarity and difference between this Green’s function and the propagator discussed in Sec. 2.1. In both cases, we use the linear superposition principle to solve wave equations, but while Eq. (67) gives the solution of the inhomogeneous equation (66), Eq. (2.44) does that for a homogeneous Schrödinger equation.
In the latter case, the elementary wave sources are the elementary parts of the initial wavefunction, rather than of the equation’s right-hand side as in our current problem.
33 See, e.g., EM Sec. 1.2.
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3
R 2
R
d R
R dR
4
4 RdR 2
2
R
.
0
(3.73)
R
R
R
0
0
R
So far, our result (72) is exact, but its apparent simplicity is deceiving, because the wavefunction
on its right-hand side generally includes not only the incident wave I but also the scattered wave s
– see Eq. (64). The most straightforward, and most common simplification of this problem, called the Born approximation,34 is possible if the scattering potential U(r) is in some sense small. (We will derive the quantitative condition of this smallness in a minute.) Since at U(r) = 0 the scattering wave s has to disappear, at small but non-zero U(r), s has to be much smaller than i . In this case, on the right-hand side of Eq. (73) we may ignore s in comparison with i, getting
m
exp ik
'
i
(r)
r
U ( '
r )
eikRd 3 r' .
(3.74) Born
s
2
i
2
approximation
R
Actually, Eq. (74) gives us even more than we wanted: it evaluates the scattered wave at any point, including those within of the scattering object, while to spell out Eq. (62), we only need to find the wave far from the scatterer, at r . However, before going to that limit, we can use this general formula to find a quantitative criterion of the Born approximation’s validity. For that, let us estimate the magnitude of the right-hand side of this equation for a scatterer of a linear size ~ a, and the potential magnitude’s scale U 0. The results are different in the following two limits:
(i) If ka << 1, then inside the scatterer (i.e., at distances r’ ~ a), both exp{ ikr ’} and the second exponent under the integral in Eq. (74) change little, so that a crude but fair estimate of the solution’s magnitude is
m
2
~
U a .
(3.75)
s
2
i
0
2
(ii) In the opposite limit ka >>1, the function under the integral is nearly periodic in one of the spatial directions (that of the scattered wave propagation), so that the net integral accumulates only on distances of the order of the de Broglie wavelength, ~ k-1, and the integral is correspondingly smaller: m
a 2
~
U
.
(3.76)
s
2
i
0
2
ka
These relations allow us to spell out the Born approximation’s condition, s << I , as 2
U
max[ ka, ]
1 .
(3.77)
0
2
ma
In the fraction on the right-hand side, we may readily recognize the scale of the kinetic (quantum-confinement) energy Ea of the particle inside a potential well of a size of the order of a, so that the Born approximation is valid essentially if the potential energy of particle’s interaction with the scatterer is 34 Named after M. Born, who was the first to apply this approximation in quantum mechanics. However, the basic idea of this approach had been developed much earlier (in 1881) by Lord Rayleigh in the context of electromagnetic wave scattering – see, e.g., EM Sec. 8.3. Note also that the contents of that section repeat some aspects of our current discussion – perhaps regrettably but unavoidably so, because the Born approximation is a centerpiece of the theory of scattering/diffraction for both the electromagnetic waves and the de Broglie waves.
Hence I felt I had to cover it in this course for the benefit of the readers who skipped the EM part of my series.
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smaller than Ea. Note, however, that the estimates (75) and (76) are not valid in some special situations when the effects of scattering accumulate in some direction. This is frequently the case for small angles
of scattering by extended objects, when ka >> 1, but ka ≾ 1.
Now let us proceed to large distances r >> r’ ~ a, and simplify Eq. (74) using an approximation similar to the dipole expansion in electrodynamics.35 Namely, in the denominator’s R, we may ignore r’
in comparison with the much larger r, but the exponents require more care, because even if r’ ~ a << r, the product kr’ ~ ka may still be of the order of 1. In the first approximation in r’, we can take (Fig. 9a): R r '
r r n '
r ,
(3.78)
r
and since directions of the vectors k and r coincide, i.e. k = kn r, we get ikR
ikr ik '
kR kr k '
r ,
e
that
so
e e
r .
(3.79)
(a)
(b)
R
'
r
detector
k
q
r
a
0
0
ki
n '
r
r
Fig. 3.9. (a) The long-range expansion of R, and (b) the definitions of q, , and .
With this replacement, Eq. (74) yields
m
i
(r)
eikr U ( '
r ) exp (k k ) r
.
(3.80)
s
i
'
2
i
d 3 r'
2 r
This relation is a particular case of a more general formula36
Scattering
f (k, k )
function:
i
ikr
e
,
(3.81)
s
i
definition
r
where f(k, ki) is called the scattering function.37 The physical sense of this function becomes clear from the calculation of the corresponding probability current density j s. For that, generally, we need to use Eq. (1.47) with the gradient operator having all spherical-coordinate components.38 However, at kr >> 1, the main contribution to s , proportional to k >> 1/ r, is provided by differentiating the factor eikr, which changes in the common direction of vectors r and k, so that
35 See, e.g., EM Sec. 8.2.
36 It is easy to prove that this form is an asymptotic form of any solution s of the scattering problem (even that beyond the Born approximation) at sufficiently large distances r >> a, k-1.
37 Note that the function f has the dimension of length, and does not account for the incident wave. This is why sometimes a dimensionless function, S = 1 + 2 ikf, is used instead. This function S is called the scattering matrix, because it may be considered a natural generalization of the 1D matrix S defined by Eq. (2.124), to higher dimensionality.
38 See, e.g., MA Eq. (10.8).
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n
k
,
at kr 1,
(3.82)
s
r
s
s
r
and Eq. (1.47) yields
2
f k
( , k 2
)
j
i
( )
k .
(3.83)
s
i
2
m
r
Plugging this expression, and also Eq. (61) into Eq. (62), for the differential cross-section we get simply d
2
f (k, k ) ,
(3.84)
i
d
while the total cross-section is
f (k, k ) 2 d Ω
,
(3.85)
i
so that the scattering function f(k, ki) gives us everything we need – and in fact more, because the function also contains information about the phase of the scattered wave.
According to Eq. (80), in the Born approximation the scattering function is reduced to the so-called Born integral
m
i
f (k, k )
U (r) e q r d 3 r ,
(3.86) Born
i
2
2
integral
where for the notation simplicity r ’ is replaced with r, and the following scattering vector is introduced: q k k ,
(3.87)
i
with the length q = 2 k sin(/2), where is the scattering angle between the vectors k and ki – see Fig.
9b. For the differential cross-section, Eqs. (84) and (86) yield39
Differential
2
2
d
m
cross-
iq r 3
U (r)
e
d r .
(3.88) section:
2
d
2
Born
approximation
This is the main result of this section; it may be further simplified for spherically-symmetric scatterers, with
U (r) U ( r).
(3.89)
In this case, it is convenient to represent the exponent in the Born integral as exp{- iqr’ cos}, where is the angle between the vectors k (i.e. the direction n r toward the detector) and q (rather than the incident wave vector ki!) – see Fig. 9b. Now, for a fixed q, we can take this vector’s direction for the polar axis of a spherical coordinate system, and reduce Eq. (86) to a 1D integral:
2
m
f (k, k )
r drU ( r) d
sin d exp{ iqr' cos }
i
2
2
2 0
0
0
39 Note that according to Eq. (88), in the Born approximation the scattering intensity does not depend on the sign of the potential U, and also that scattering in a certain direction is completely determined by a specific Fourier component of the function U(r), namely by its harmonic with the wave vector equal to the scattering vector q.
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m
2
2sin qr
2 m
r drU ( r) 2
U ( r)sin( qr) rdr.
(3.90)
2 2
2
qr
q
0
0
As a simple example, let us use the Born approximation to analyze scattering on the following spherically-symmetric potential:
2
Ur
r
U exp
.
(3.91)
0
2
2 a
In this particular case, it is better to avoid the temptation to exploit the spherical symmetry by using Eq.
(90), and instead, use the general Eq. (88), because it may be represented as a product of three similar Cartesian factors:
mU
x 2
f (k, k )
0 I I I ,
with I exp
,
(3.92)
i
2
x y z
x
iq x
2
x
dx
2
2 a
and similar integrals for Iy and Iz. From Chapter 2, we already know that the Gaussian integrals like Ix may be readily worked out by complementing the exponent to the full square, in our current case giving
I
x
2
2
q a
2 1/2 a exp
x
etc.,
,
2
(3.93)
2
2
2
2 2
d
mU
mU a
0
I I I
2 2
0
q a
a
e
.
d
2 2 x y z
2
Now, the total cross-section is an integral of d/ d over all directions of the vector k. Since in our case the scattering intensity does not depend on the azimuthal angle , the only nontrivial integration is over the scattering angle – see Fig. 9b:
2
2
2
d
d
mU a
d 2
sin d 4 2 2
0
π a
exp
2 k sin
2
a sin
d
2
d
d
2
0
0
(3.94)
2
2
2
2
2
2 2
mU a
π mU a
k a
4 2 2
0
π a
exp 2 2 2
k a 1 cos
2
d 1
( cos )
0
1
4
e
.
2
2
2
k
0
Let us analyze these results. In the low-energy limit, ka << 1 (and hence qa << 1 for any scattering angle), the scattered wave is virtually isotropic: d/ d const – a very typical feature of a scalar-wave scattering40 by small objects, in any approximation. Note that according to Eq. (77), the Born expression for , following from Eq. (94) in this limit,
2
2
mU a
8 2 2
0
a
,
(3.95)
2
40 Note that this is only true for scalar (e.g., the de Broglie) waves, and different for vector ones, in particular the electromagnetic waves, where the intensity of the dipole radiation, and hence the scattering by small objects vanishes in the direction of the incident field’s polarization – see, e.g., EM Eqs. (8.26) and (8.139).
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is only valid if is much smaller than the scale a 2 of the physical cross-section of the scatterer. In the opposite, high-energy limit ka >>1, the scattering is dominated by small angles q/ k ~ 1/ ka ~ / a: 2
2
d
mU a
2
0
2 a
2 2 2
exp k a
.
(3.96)
2
d
This is, again, very typical for diffraction. Note, however, that due to the smooth character of the Gaussian potential (91), the diffraction pattern (98) exhibits no oscillations of d/ d as a function of the diffraction angle .
Such oscillations naturally appear for scatterers with sharp borders. Indeed, let us consider a uniform spherical scatterer, described by the potential
U , for r R
,
U (r) 0
(3.97)
,
0
otherwise.
In this case, integration by parts of Eq. (90) readily yields
2
2 mU
d
2 mU
0
f (k, k )
qR cos qR sin qR ,
that
so
qR cos qR sin qR . (3.98)
i
0
2
2
3
2 3
q
d
q
According to this result, the scattered wave’s intensity drops very fast with q, so that one needs a semi-log plot (such as shown in Fig. 10) to reveal small diffraction fringes,41 with the n th destructive interference (zero-intensity) point tending to qR = ( n + ½) at n . Since, as Fig. 9b shows, q may only change from 0 to 2 k, these intensity minima are only observable at sufficiently large values of the parameter kR, when they correspond to real values of the scattering angle . (At kR >> 1, approximately kR/ of these minima, i.e. “dark rings” of low scattering probability, are observable.) On the contrary, at kR << 1 all allowed values of qR are much smaller than 1, and is this limit, the differential cross-section does not depend on qR, i.e. the scattering by the sphere (as by any object in this limit) is isotropic.
0.1
1
d
0.01
u 2 d
0
g
1 1
0 3
Fig. 3.10. The differential cross-section of
the Born scattering of a particle by a
“hard”
(sharp-border)
sphere
(97),
1 1
0 4
normalized to its geometric cross-section
g R 2 and the square of the potential’s
1 1
0 5
magnitude parameter u
0 U 0/(2/2 mR 2), as
a function of the normalized magnitude of
1 1
0 6
the scattering vector q.
0
1
2
3
4
5
qR /
This example shows that in quantum mechanics the notions of particle scattering and diffraction are essentially inseparable.
41 Their physics is very similar to that of the Fraunhofer diffraction on a 1D scatterer – see, e.g., EM Sec. 8.4.
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The Born approximation, while being very simple and used more than any other scattering theory, is not without substantial shortcomings, as becomes clear from the following example. It is not too difficult to prove the so-called optical theorem, strictly valid for an arbitrary scatterer: Optical
k
Im f k , k
.
(3.99)
theorem
i
i
4
However, Eq. (86) shows that in the Born approximation, the function f is purely real at q = 0 (i.e. for k
= ki), and hence cannot satisfy the optical theorem. Even more evidently, it cannot describe such a simple effect as a dark shadow ( 0) cast by a virtually opaque object (say, with U >> E). There are several ways to improve the Born approximation, while still keeping its general idea of an approximate treatment of U.
(i) Instead of the main assumption s U 0, we may use a complete perturbation series:
...
(3.100)
s
1
2
with
n
n U 0 , and find successive approximations n one by one. In the 1st approximation we return to the Born formula, but already the 2nd approximation yields
k
Im f k , k
,
(3.101)
2 i
i
1
4
where 1 is the total cross-section calculated in the 1st approximation, so that the optical theorem (99) is
“almost satisfied”.
(ii) As was mentioned above, the Born approximation does not work very well for the objects stretching along the direction (say, x) of the initial wave vector ki. This deficiency may be corrected by the so-called eikonal 42 approximation, which replaces the plane-wave representation (60) of the incident wave with a WKB-like exponent, though still in the 1st approximation in U 0:
x
x
1/ 2
m E U '
r
exp ik x
i k ' r dx'
i
dx'
0
2
( )
exp
( )
exp
0
0
(3.102)
x
m
exp i k x
U ( '
r )
dx' .
0
2
k
0 0
Results of this approach satisfy the optical theorem (99) already in the 1st approximation.
Another way toward quantitative results in the theory of scattering, beyond the Born approximation, may be pursued for spherically-symmetric potentials (89); I will discuss it in Sec. 8, after a general discussion of particle motion in such potentials in Sec. 7.
3.4. Energy bands in higher dimensions
In Sec. 2.7, we have discussed the 1D band theory for potential profiles U( x) that obey the periodicity condition (2.192). For what follows, let us notice that the condition may be rewritten as U ( x X ) U ( x) ,
(3.103)
42 From the Greek word , meaning “image”. In our current context, this term is purely historic.
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where X = a, with being an arbitrary integer. One may say that the set of points X forms a periodic 1D lattice in the direct (r-) space. We have also seen that each Bloch state (i.e., each eigenstate of the Schrödinger equation for such periodic potential) is characterized by the quasimomentum q, and its energy does not change if q is changed by a multiple of 2/ a. Hence if we form, in the reciprocal (q-) space, a 1D lattice of points Q = lb, with b 2/ a and integer l, any pair of points from these two mutually reciprocal lattices satisfies the following rule:
2
exp iQX exp
2
il
a
i
l
e
1.
(3.104)
a
In this form, the results of Sec. 2.7 may be readily extended to d-dimensional periodic potentials whose translational symmetry obeys the following natural generalization of Eq. (103):
U (r R) U (r) ,
(3.105)
where the points R, which may be numbered by d integers j, form the so-called Bravais lattice:43
d
Bravais
R a ,
(3.106)
j
j
lattice
j1
with d primitive vectors a j. The simplest example of a 3D Bravais lattice is given by the simple cubic lattice (Fig. 11a), which may be described by a system of mutually perpendicular primitive vectors a j of equal length. However, not in any lattice these vectors are perpendicular; for example, Figs. 11b and 11c show possible sets of the primitive vectors describing, respectively, the face-centered cubic (fcc) lattice and the body-centered cubic (bcc) lattice. In 3D, the science of crystallography, based on group theory, distinguishes, by their symmetry properties, 14 Bravais lattices grouped into 7 different lattice systems.44
(a)
(b)
(c)
a
a
3
3
a
1
a3
a2
a1
a
a
2
2
a1
Fig. 3.11. The simplest (and most common) 3D Bravais lattices: (a) simple cubic, (b) face-centered cubic (fcc), and (c) body-centered cubic (bcc), and possible choices of their primitive vector sets (blue arrows).
Note, however, not all highly symmetric sets of points form Bravais lattices. As probably the most striking example, the nodes of a very simple 2D honeycomb lattice (Fig. 12a)45 cannot be 43 Named after Auguste Bravais, the crystallographer who introduced this notion in 1850.
44 A very clear, well-illustrated introduction to the Bravais lattices is given in Chapters 4 and 7 of the famous textbook by N. Ashcroft and N. Mermin, Solid State Physics, Saunders College, 1976.
45 This structure describes, for example, the now-famous graphene: isolated monolayer sheets of carbon atoms arranged in a honeycomb lattice with an interatomic distance of 0.142 nm.
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described by a Bravais lattice – while those of the 2D hexagonal lattice shown in Fig. 12b, can. The most prominent 3D case of such a lattice is the diamond structure (Fig. 12c), which describes, in particular, silicon crystals.46 In cases like these, the band theory is much facilitated by the fact that the Bravais lattices using some point groups (called primitive unit cells) may describe these systems.47 For example, Fig. 12a shows a possible choice of the primitive vectors for the honeycomb lattice, with the primitive unit cell formed by any two adjacent points of the original lattice (say, within the dashed ovals on that panel). Similarly, the diamond lattice may be described as an fcc Bravais lattice with a two-point primitive unit cell – see Fig. 12c.
(a)
(b)
(c)
a
a
1
2
a3
a2
a
a
1
2
a1
Fig. 3
.12. Two important periodic structures that require two-point primitive cells for their Bravais lattice representation: (a) 2D honeycomb lattice and (c) 3D diamond lattice, and their primitive vectors. For contrast, panel (b) shows the 2D hexagonal structure which forms a Bravais lattice with a single-point primitive cell.
Now we are ready for the following generalization of the 1D Bloch theorem, given by Eqs.
(2.193) and (2.210), to higher dimensions: any eigenfunction of the Schrödinger equation describing particle’s motion in the spatially-unlimited periodic potential (105) may be represented either as iq R
r
( R) r
( ) e
,
(3.107)
3D Bloch or as
theorem
(r) u(r) q
i r
e
, with u(r R) u(r),
(3.108)
where the quasimomentum q is again a constant of motion, but now it is a vector. The key notion of the band theory in d dimensions is the reciprocal lattice in the wave-vector (q) space, formed as Reciprocal
d
lattice in
Q l b ,
(3.109)
j
j
q-space
j1
with integer lj, and vectors b j selected in such a way that the following natural generalization of Eq.
(104) is valid for any pair of points of the direct and reciprocal lattices:
46 This diamond structure may be best understood as an overlap of two fcc lattices of side a, mutually shifted by the vector {1, 1, 1} a/4, so that the distances between each point of the combined lattice and its 4 nearest neighbors (see the solid gray lines in Fig. 12c) are all equal.
47 A harder case is presented by so-called quasicrystals (whose idea may be traced down to medieval Islamic tilings, but was discovered in natural crystals, by D. Shechtman et al., only in 1984), which obey a high (say, the 5-fold) rotational symmetry, but cannot be described by a Bravais lattice with any finite primitive unit cell. For a popular review of quasicrystals see, for example, P. Stephens and A. Goldman, Sci. Amer. 264, #4, 24 (1991).
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Q
i R
e
1.
(3.110)
One way to describe the physical sense of the lattice Q is to say that according to Eqs. (80) and/or (86), it gives the set of the vectors q k – ki for that the interference of the waves scattered by all Bravais lattice points is constructive, and hence strongly enhanced.48 Another way to look at the reciprocal lattice follows from the first formulation of the Bloch theorem, given by Eq. (107): if we add to the quasimomentum q of a particle any vector Q of the reciprocal lattice, the wavefunction does not change. This means, in particular, that all information about the system’s eigenfunctions is contained in just one elementary cell of the reciprocal space q. Its most frequent choice, called the 1st Brillouin zone, is the set of all points q that are closer to the origin than to any other point of the lattice Q. (Evidently, the 1st Brillouin zone in one dimension, discussed in Sec. 2.7, falls under this definition – see, e.g., Figs.
2.26 and 2.28.)
It is easy to see that the primitive vectors b j of the reciprocal lattice may be constructed as Reciprocal
a a
a a
a a
2
3
3
1
1
2
b 2
, b 2
, b 2
.
(3.111) lattice:
1
a a a
a a a
a a a
primitive
1 2
2
3
1 2
3
3
1 2
3
vectors
Indeed, from the “operand rotation rule” of the vector algebra49 it is evident that a jb j’ = 2 jj’. Hence, with the account of Eq. (109), the exponent on the left-hand side of Eq. (110) is reduced to eiQ R
exp 2 i l l l
.
(3.112)
1 1
2 2
3 3
Since all lj and all j are integers, the expression in the parentheses is also an integer, so that the exponent indeed equals 1, thus satisfying the definition of the reciprocal lattice given by Eq. (110).
As the simplest example, let us return to the simple cubic lattice of a period a (Fig. 11a), oriented in space so that
a an , a an , a an , (3.113)
1
x
2
y
3
z
According to Eq. (111), its reciprocal lattice is also cubic:
2
Q
( l n l n l n ) ,
(3.114)
x
x
y
y
z
z
a
so that the 1st Brillouin zone is a cube with the side b = 2/ a.
Almost equally simple calculations show that the reciprocal lattice of fcc is bcc, and vice versa.
Figure 13 shows the resulting 1st Brillouin zone of the fcc lattice.
The notion of the reciprocal lattice makes the multi-dimensional band theory not much more complex than that in 1D, especially for numerical calculations, at least for the single-point Bravais lattices. Indeed, repeating all the steps that have led us to Eq. (2.218), but now with a d-dimensional Fourier expansion of the functions U(r) and u(r), we readily get its generalization: 48 This is why the notion of the Q-lattice is also the main starting point of X-ray diffraction studies of crystals.
Indeed, it allows rewriting the well-known Bragg condition for diffraction peaks in an extremely simple form: k =
ki + Q, where ki and k are the wave vectors of the, respectively, incident and diffracted waves – see, e.g., EM Sec.
8.4 (where it was more convenient for me to use the notation k0 for ki ).
49 See, e.g., MA Eq. (7.6).
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U u ( E E u
) ,
(3.115)
l ' l l '
l
l
l ' l
where l is now a d-dimensional vector of integer indices lj. The summation in Eq. (115) should be carried over all essential components of this vector (i.e. over all relevant nodes of the reciprocal lattice), so that writing a corresponding computer code requires a bit more care than in 1D. However, this is just a homogeneous system of linear equations, and numerous routines of finding its eigenvalues E are readily available from both public sources and commercial software packages.
qz
Fig. 3.13. The 1st Brillouin zone of the fcc
q
y
lattice, and the traditional notation of its
main
directions.
Adapted
from
qx
http://en.wikipedia.org/wiki/Band_structure,
as a public domain material.
What is indeed more complex than in 1D is representation (and hence comprehension :-), of the calculated results and experimental data. Typically, the representation is limited to plotting the Bloch state eigenenergy as a function of components of the vector q along certain special directions the reciprocal space of quasimomentum (see, e.g., the red lines in Fig. 13), typically on a single panel. Fig.
14 shows perhaps the most famous (and certainly the most practically important) of such plots, the band structure of electrons in crystalline silicon. The dashed horizontal lines mark the so-called indirect gap of the width ~1.12 eV between the “valence” (nominally occupied) and the next “conduction”
(nominally unoccupied) energy bands.
Fig. 3.14. The band structure of silicon, plotted along
the special directions shown in Fig. 13. (Adapted from
http://www.tf.uni-kiel.de/matwis/amat/semi_en/.)
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In order to understand the reason for such complexity, let us see how would we start to calculate such a picture in the weak-potential approximation, for the simplest case of a 2D square lattice – which is a subset of the cubic lattice (106), with 3 = 0. Its 1st Brillouin zone is of course also a square, of the area (2/ a)2 – see the dashed lines in Fig. 15. Let us draw the lines of the constant energy of a free particle ( U = 0) in this zone. Repeating the arguments of Sec. 2.7 (see especially Fig. 2.28 and its discussion), we may conclude that Eq. (2.216) should be now generalized as follows,
2
2
2
2
2
k
2 l
2 l
x
y
E
q
,
(3.116)
x
q
2 m
2 m
y
a
a
with all possible integers lx and ly. Considering this result only within the 1st Brillouin zone, we see that as the particle’s energy E grows, the lines of equal energy, for the lowest energy band, evolve as shown in Fig. 15. Just like in 1D, the weak-potential effects are only important at the Brillouin zone boundaries, and may be crudely considered as the appearance of narrow energy gaps, but one can see that the band structure in q-space is complex enough even without these effects – and becomes even more involved at higher E.
qy
(a)
(b)
(c)
Fig. 3.15. The lines of constant
2
energy E of a free particle, within
the 1st Brillouin zone of a square
a
0
qx
Bravais lattice, for: (a) E/ E 1 0.95,
(b) E/ E 1 1.05; and (c) E/ E 1 2.05,
where E 1 22/2 ma 2.
2 / a
The tight-binding approximation is usually easier to follow. For example, for the same square 2D
lattice, we may repeat the arguments that have led us to Eq. (2.203), to write 50
i a
a
a
a
a
,
(3.117)
0,0
n ,
1 0
,
1 0
0, 1
0, 1
where the indices correspond to the deviations of the integers x and y from an arbitrarily selected minimum of the potential energy – and hence of the wavefunction’s “hump”, quasi-localized at this minimum. Now, looking for the stationary solution of these equations, that would obey the Bloch theorem (107), instead of Eq. (2.206) we get
iq a
iq a
iq a
iq a
E E E e x e
x
e y e
y
E 2
(3.118)
n
n
n
n
n
n cos q a
cos q a
x
y
.
Figure 16 shows this result, within the 1st Brillouin zone, in two forms: as color-coded lines of equal energy, and as a 3D plot (also enhanced by color). It is evident that the plots of this function along different lines on the q-plane, for example along one of the axes (say, qx) and along a diagonal of the 1st Brillouin zone (say, with qx = qy) give different curves E( q), qualitatively similar to those of silicon (Fig.
50 Actually, using the same values of n in both directions ( x and y) implies some sort of symmetry of the quasi-localized states. For example, the s-states of axially-symmetric potentials (see the next section) always have such symmetry.
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14). However, the latter structure is further complicated by the fact that the primitive cell of its Bravais lattice contains 2 atoms – see Fig. 12c and its discussion. In this case, even the tight-binding picture becomes more complex. Indeed, even if the atoms at different positions of the primitive unit cell are similar (as they are, for example, in both graphene and silicon), and hence the potential wells near those points and the corresponding local wavefunctions u(r) are similar as well, the Bloch theorem (which only pertains to Bravais lattices!) does not forbid them to have different complex probability amplitudes a( t) whose time evolution should be described by a specific differential equation.
qy
n
4 n
2
q
0
q
y
x
a
0
Fig. 3.16. The allowed band
energy n E – En for a square
4 n
2D lattice, in the tight-binding
qx
approximation.
2 / a
As the simplest example, to describe the honeycomb lattice shown in Fig. 12a, we have to prescribe different probability amplitudes to the “top” and “bottom” points of its primitive cell – say,
and , correspondingly. Since each of these points is surrounded (and hence weakly interacts) with three neighbors of the opposite type, instead of Eq. (117) we have to write two equations:
3
3
i
i
,
(3.119)
n
,
j
n
j'
j1
j' 1
where each summation is over three next-neighbor points. (In these two sums, I am using different summation indices just to emphasize that these directions are different for the “top” and “bottom” points of the primitive cell – see Fig. 12a.) Now using the Bloch theorem (107) in the form similar to Eq.
(2.205), we get two coupled systems of linear algebraic equations:
3
3
i
q r
i '
q r
E E
e
E E
e
(3.120)
n
n
j ,
n n j' ,
j 1
j' 1
where r j and r ’j’ are the next-neighbor positions, as seen from the top and bottom points, respectively.
Writing the condition of consistency of this system of homogeneous linear equations, we get two equal and opposite values for energy correction for each value of q:
3
'
1/ 2
iq r
j
r j'
E E ,
where
.
(3.121)
e
n
n
j, j'1
According to Eq. (120), these two energy bands correspond to the phase shifts (on the top of the regular Bloch shift qr) of either 0 or between the adjacent quasi-localized wavefunctions u(r ).
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The most interesting corollary of such energy symmetry, augmented by the honeycomb lattice’s symmetry, is that for certain values qD of the vector q (that turn out to be in each of six corners of the honeycomb-shaped 1st Brillouin zone), the double sum vanishes, i.e. the two band surfaces E(q) touch each other. As a result, in the vicinities of these so-called Dirac points,51 the dispersion relation is linear:
~
~
E
E v q ,
where q q q ,
(3.122)
q q
n
n
D
D
with vn n being a constant with the dimension of velocity – for graphene, close to 106 m/s. Such a linear dispersion relation ensures several interesting transport properties of graphene, in particular of the quantum Hall effect in it – as was already mentioned in Sec. 2. For their more detailed discussion, I have to refer the reader to special literature.52
3.5. Axially-symmetric systems
I cannot conclude this chapter (and hence our review of wave mechanics) without addressing the exact solutions of the stationary Schrödinger equation53 possible in the cases of highly symmetric functions U(r). Such solutions are very important, in particular, for atomic and nuclear physics, and will be used in the later chapters of this course.
In some rare cases, such symmetries may be exploited by the separation of variables in Cartesian coordinates. The most famous (and rather important) example is the d-dimensional harmonic oscillator –
a particle moving inside the potential
2
d
m
U
0 r 2 .
(3.123)
j
2
j1
Separating the variables exactly as we did in Sec. 1.7 for the rectangular hard-wall box (1.77), for each degree of freedom we get the Schrödinger equation (2.261) of a 1D oscillator, whose eigenfunctions are 51 This term is based on a (rather indirect) analogy with the Dirac theory of relativistic quantum mechanics, to be discussed in Chapter 9 below.
52 See, e.g., the reviews by A. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009) and by X. Lu et al., Appl. Phys.
Rev. 4, 021306 (2017). Note that the transport properties of graphene are determined by coupling of 2 p- state
electrons of its carbon atoms (see Secs. 6 and 7 below), whose wavefunctions are proportional to exp{ i} rather than are axially-symmetric as implied by Eqs. (120). However, due to the lattice symmetry, this fact does not affect the above dispersion relation E(q).
53 This is my best chance to mention, in passing, that the eigenfunctions n(r) of any such problem do not feature the instabilities typical for the deterministic chaos effects of classical mechanics – see, e.g., CM Chapter 9. (This is why the term quantum mechanics of classically chaotic systems is preferable to the occasionally used term
“quantum chaos”.) It is curious that at the initial stages of the time evolution of the wavefunctions of such systems, their certain correlation functions still grow exponentially, reminding the Lyapunov exponents of their classical chaotic dynamics. This growth stops at the so-call Ehrefect times t E ~ -1ln( S/), where S is the action scale of the problem – see, e.g., I. Aleiner and A. Larkin, Phys. Rev. E 55, R1243 (1997). In a stationary quantum state, the most essential trace of the classical chaos in a system is an unusual statistics of its eigenvalues, in particular of the energy spectra. We will have a chance for a brief look at such statistics in Chapter 5, but unfortunately, I will not have time/space to discuss this field in much detail. Perhaps the best available book for further reading is the monograph by M. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer, 1991.
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given by Eq. (2.284), and the energy spectrum is described by Eq. (2.162). As a result, the total energy spectrum may be indexed by a vector n = { n 1, n 2,…, nd} of d independent integer quantum numbers nj:
d
d
E
,
(3.124)
n
n
0
j
j1
2
each ranging from 0 to . Note that every energy level of this system, with the only exception of its ground state,
d
1
d
1
( )
exp
,
(3.125)
g
r
r 2
0
j
d / 4
d / 2
2 j
j1
x
2
0
x 0 j1
is degenerate: several different wavefunctions, each with its own different set of quantum numbers nj, but the same value of their sum, have the same energy.
However, the harmonic oscillator problem is an exception: for other central- and spherically-symmetric problems the solution is made easier by using more appropriate curvilinear coordinates. Let us start with the simplest axially-symmetric problem: the so-called planar rigid rotator (or “rotor”), i.e.
a particle of mass m,54 constrained to move along a plane circle of radius R (Fig. 17).55
n m
R
l
R
0
Fig. 3.17. A planar rigid rotator.
The classical planar rotator may be described by just one degree of freedom, say the angle displacement (or equivalently the arc displacement l R) from some reference direction, with the energy (and the Hamiltonian function) H = p 2/2m, where p mv = mn( dl/ dt), n being the unit vector in the azimuthal direction – see Fig. 17. This function is similar to that of a free 1D particle (with the replacement x l R), and hence the rotator’s quantum properties may be described by a similar Hamiltonian operator:
p ˆ2
ˆ
H
,
p
with ˆ in
i n
,
(3.126)
2m
l
R
whose eigenfunctions have a similar structure:
ikl
ikR
Ce
Ce
.
(3.127)
54 From this point on (until the chapter’s end), I will use this exotic font for the particle’s mass, to avoid any chance of its confusion with the impending “magnetic” quantum number m, traditionally used in axially-symmetric problems.
55 This is a reasonable model for the confinement of light atoms, notably hydrogen, in some organic compounds, but I am addressing this system mostly as the basis for the following, more complex problems.
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The “only” new feature is that in the rotator, all observables should be 2-periodic functions of the angle . Hence, as we have already discussed in the context of the magnetic flux quantization (see Fig. 4 and its discussion), as the particle makes one turn around the central point 0, its wavefunction’s phase kR may only change by 2 m, with an arbitrary integer m (ranging from – to +): 2 im
(
2 ) ( ) e
.
(3.128)
m
m
With the eigenfunctions (127), this periodicity condition immediately gives 2 kR = 2 m. Thus, the wave number k can take only quantized values km = m/ R, so that the eigenfunctions should be indexed by this magnetic quantum number m:
l
C exp im
C exp
,
(3.129) Planar rotator:
m
m
m
im
R
eigenfunctions
and the energy spectrum is discrete:
2
2
2
2
2
p
k
m
m
E
m
.
(3.130) Planar rotator:
m
2
2m
2m
2 R
m
eigenenergies
This simple model allows exact analysis of the external magnetic field effects on a confined motion of an electrically charged particle. Indeed, in the simplest case when this field is axially symmetric (or just uniform) and directed normally to the rotator’s plane, it does not violate the axial symmetry of the system. According to Eq. (26), in this case, we have to generalize Eq. (126) as 2
2
1
1
ˆ
H
in
A
q
i n
A
q
.
(3.131)
2m
l
2
m
R
Here, in contrast to the Cartesian gauge choice (44), which was so instrumental for the solution of the Landau level problem, it is beneficial to take the vector potential in the axially-symmetric form A =
A()n, where { x, y} is the 2D radius-vector, with the magnitude = ( x 2 + y 2)1/2. Using the well-known expression for the curl operator in the cylindrical coordinates,56 we can readily check that the requirement A = Bn z, with B = const, is satisfied by the following function:
B
A n
.
(3.132)
2
For the planar rotator, ρ = R = const, so that the stationary Schrödinger equation becomes 1
2
R
B
i
q
E .
(3.133)
2
m
R
2
m
n
m
A little bit surprisingly, this equation is still satisfied with the eigenfunctions (127). Moreover, since the periodicity condition (128) is also unaffected by the applied magnetic field, we return to the periodic eigenfunctions (129), independent of B. However, the field does affect the system’s eigenenergies:
56 See, e.g., MA Eq. (10.5).
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2
2
2
2
Planar rotator:
1 1
R
B
1 m
R
B
Φ
magnetic
E
i
q
,
(3.134)
m
m
q
m
2
field’s effect
2
m
m
R
2
2m R
2
2
R
m
Φ
' 0
where R 2B is the magnetic flux through the area limited by the particle’s trajectory, and 0 ’
2/ q is the “normal” magnetic flux quantum we have already met in the AB effect’s context – see Eq.
(34) and its discussion. The field also changes the electric current of the particle in each eigenstate:
*
iqR B
2
I q
c.c.
.
(3.135)
m
m
m
q
Cm m
2 i m R
2
m R
' 0
Normalizing the wavefunction (129) to have Wm = 1, we get Cm 2 = 1/2 R, so that Eq. (135) becomes
Φ
q
I m
I ,
with I
.
(3.136)
m
Φ
0
0
'
2
2
m R
0
The functions Em() and Im () are shown in Fig. 18. Note that since 0 ’ 1/ q, for any sign of the particle’s charge q, dIm/ d < 0. It is easy to verify that this means that the current is diamagnetic for any sign of q:57 the field-induced current flows in the direction that its own magnetic field tries to compensate for the external magnetic flux applied to the loop. This result may be interpreted as a different manifestation of the AB effect.58 In contrast to the interference experiment that was discussed in Sec. 1, in the situation shown in Fig. 17 the particle is not absorbed by the detector but travels around the ring continuously. As a result, its wavefunction is “rigid”: due to the periodicity condition (128), the quantum number m is discrete, and the applied magnetic field cannot change the wavefunction gradually. In this sense, the system is similar to a superconducting loop – see Fig. 4 and its discussion.
The difference between these systems is two-fold:
Em
m 1
m 0
m 1
0
1
/ ' 0
Im
Fig. 3.18. The magnetic field effect on a
charged planar rotator. Dashed arrows show
possible inelastic transitions between
metastable and ground states, due to weak
0
1
/ '
0
interaction with the environment, as the
m 1
m 0
m 1
external magnetic field is slowly increased.
57 This effect, whose qualitative features remain the same for all 2D or 3D localized states (see Chapter 6 below), is frequently referred to as orbital diamagnetism. In magnetic materials consisting of particles with uncompensated spins, this effect competes with an opposite effect, spin paramagnetism – see, e.g., EM Sec. 5.5.
58 It is straightforward to check that the final forms of Eqs. (134)-(136) remain valid even if the magnetic field is localized well inside the rotator’s circumference so that its lines do not touch the particle’s trajectory.
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(i) For a single charged particle, in macroscopic systems with practicable values of q, R, and m, the scale I 0 of the induced current is very small. For example, for m = m e, q = – e, and R = 1 m, Eq.
(136) yields I 0 3 pA.59 With the ring’s inductance L of the order of 0 R,60 the contribution I = L I ~
0 RI 0 ~ 10-24 Wb of such a small current into the net magnetic flux is negligible in comparison with
0 ’ ~ 10-15 Wb, so that the wavefunction quantization does not lead to the constancy of the total magnetic flux.
(ii) As soon as the magnetic field raises the eigenstate energy Em above that of another eigenstate Em’, the former state becomes metastable, and a weak interaction of the system with its environment (which is neglected in our simple model, but will be discussed in Chapter 7) may induce a quantum transition of the system to the lower-energy state, thus reducing the diamagnetic current’s magnitude –
see the dashed lines in Fig. 18. The flux quantization in superconductors is much more robust to such perturbations.61
Now let us return, once again, to the key Eq. (129), and see what does it give for one more important observable, the particle’s angular momentum
L r p ,
(3.137)
In this particular geometry, the vector L has just one component, normal to the rotator plane: L Rp .
(3.138)
z
In classical mechanics, Lz of the rotator should be conserved (due to the absence of external torque), but it may take arbitrary values. In quantum mechanics, the situation changes: with p = k, our result km =
m/ R for the m th eigenstate may be rewritten as
Angular
( L ) R k m .
(3.139) momentum
z m
m
quantization
Thus, the angular momentum is quantized: it may be only a multiple of the Planck constant –
confirming the N. Bohr’s guess – see Eq. (1.8). As we will see in Chapter 5, this result is very general (though it may be modified by spin effects), and the wavefunctions (129) may be interpreted as eigenfunctions of the angular momentum operator.
Let us see whether this quantization persists in more general, but still axial-symmetric systems.
To implement the planar rotator in our 3D world, we needed to provide rigid confinement of the particle both in the motion plane and along the 2D radius . Let us consider a more general situation when only the former confinement is strict, i.e. to the case when a 2D particle moves in an arbitrary centrally-symmetric potential
U (ρ) U () .
(3.140)
59 Such weak persistent, macroscopic diamagnetic currents in non-superconducting systems have been experimentally observed by measuring the weak magnetic field induced by the currents, in systems of a large number (~107) of similar conducting rings – see, e.g., L. Lévy et al., Phys. Rev. Lett. 64, 2074 (1990). Due to the dephasing effects of electron scattering by phonons and other electrons (unaccounted for in our simple theory), the effect’s observation requires submicron rings and millikelvin temperatures.
60 See, e.g., EM Sec. 5.3.
61 Interrupting a superconducting ring with a weak link (Josephson junction), i.e. forming a SQUID, we may get a switching behavior similar to that shown with dashed arrows in Fig. 18 – see, e.g., EM Sec. 6.5.
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Using the well-known expression for the 2D Laplace operator in polar coordinates,62 we may represent the 2D stationary Schrödinger equation in the form
2
1
1 2
.
(3.141)
2
2
U (
)
E
2m
Separating the radial and angular variables as63
R () F () ,
(3.142)
we get, after the division of all terms by and their multiplication by ρ 2, the following equation: 2
d dR 1 d 2
