Fundamentals of the Physics of Solids / 08-Methods of Structure Determination

8.1 The Theory of Di raction

251

e−iK·(Rm −Rn ) = N 2	δK,G ,	(8.1.39)
Rm ,Rn	G

where we have used the relation (C.1.39). Indeed, di raction peaks appear only for scattering processes that satisfy the Laue (Bragg) condition, however the intensity of such Bragg peaks is determined by |fK |2, which depends on atomic scattering amplitudes as well as the geometry of the basis.

Next, we shall demonstrate that the result for the cross section of di raction obtained by the above calculation,

dσ	=		Rn	rj	fj,K e−iK·(Rn +rj )		2	(8.1.40)
dΩ							,

does not fundamentally rely on the assumption that the scattered beam is a plane wave: the same result is recovered using the more natural assumption that scattered waves are spherical.

If the scattering center is at the origin then the wavefunction of the scattered particle and the unscattered part of the beam is given by

ψ(r) = eik·r + f (Ω)	eikr		(8.1.41)
		,
	r

where Ω is the solid angle defined by the direction of r. Theoretical considerations show that the prefactor of the second term, f (Ω) is just the scattering amplitude in that direction, since the scattering cross section for a beam given in the form (8.1.41) is just

dσ	= |f (Ω)|2 .	(8.1.42)
dΩ

If the scatterer is not at the origin but at Rn +rj and the scattering amplitude is fj then an extra phase factor appears in the wavefunction. As the beam travels only a distance |r − Rn − rj | until it reaches the detector,

	eik r−Rn−rj
ψ = eik·(Rn +rj ) eik·(r−Rn−rj ) + fj (K)	| |	.	(8.1.43)
	|r − Rn − rj |

Since the size of sample is small compared to the distance of the detector,

k r R

n −

r

j | ≈

kr

−

k

r · (Rn + rj )

= kr

−

k

·

(R

+ r

) ,

(8.1.44)

| −

r

n

j

as k points in the direction of the detector located at r, and its magnitude is equal to that of k. Thus the sum of the incoming plane wave and the outgoing spherical wave takes the form

ψ = eik·r + fj (Ω)	eikr+iK ·(Rn+rj )	.	(8.1.45)
	r

252 8 Methods of Structure Determination

For several scatterers we have	fj (Ω) eikr+iK·(Rn +rj )
ψ = eik·r +			,	(8.1.46)
		r
Rn	rj
so the total scattering amplitude is given by
f (Ω) =	fj (Ω)eiK·(Rn +rj ) .			(8.1.47)
Rn	rj

When this is substituted into the cross-section formula (8.1.42), expression (8.1.40) is recovered.

Barring the case of large ideal crystals, when a beam of wave number k is incident upon a sample, the intensity of the scattered beam of wave number k will be shown to be proportional to |fK |2. To determine the crystal structure, the distribution of the di racted beam and the relative intensity of the Bragg peaks are measured, and then the arrangement of atoms is deduced. When X-ray or electron di raction is used, the interaction potential is proportional to the electron density – and then their spatial distribution can also be determined.

According to (C.1.36), lattice-periodic charge distributions can be ex-

panded into Fourier series using vectors of the reciprocal lattice,
1
ρtot(r)		fGeiG·r ,	(8.1.48)
	v
		G

where v is the volume of the primitive cell. This implies that if the structure amplitudes were known for each reciprocal-lattice vector, charge density could, in principle, be obtained by an inverse Fourier transformation. The evaluation of di raction measurements is encumbered by the fact that the structure factor |fK |2 rather than the structure amplitude is amenable to direct measurement, and thus the information contained in the complex phase of the structure amplitude is lost. To overcome this problem, specific assumptions are made about the structure, and iterative methods are used until an atomic configuration is found that is in agreement with measurements. Even then the structure cannot be reconstructed without some ambiguity. In recent years successful attempts have been made to extend holography to the X-ray region, opening the way to X-ray holograms that contain phase information, and provide a three-dimensional image of the neighborhood of the atom.5

8.1.4 The Shape and Intensity of Di raction Peaks

The above considerations are suitable for the determination of the direction of di raction peaks for infinitely large samples. For finite samples interference

5Outstanding contributions to X-ray holography have been made by G. Faigel and M. Tegze. However, there is still a long way to go before practical applications are available.

8.1 The Theory of Di raction

253

is expected to remain constructive, and thus the scattered intensity will be maximal in these directions, however, in other directions cancellation will be only partial, leading to small but nonzero intensities. In analogy to optics, one expects that by increasing the size of the sample (that is, the number of scatterers), the di raction peak becomes sharper, and practically complete cancellation is observed in any direction that does not satisfy the Bragg condition. To determine the angular dependence of intensity, we shall return to the formulation asserting that the transition probability is proportional to the absolute square of the transition matrix element

k|U (r)|k = U (r)e−i(k−k )·r dr , (8.1.49)

where U (r) is the full potential. In terms of the scattering amplitude f (r) instead of the potential, the amplitude of the scattered beam is proportional to

fK = f (r)e−i(k−k )·r dr . (8.1.50)

In crystalline material atoms are arranged in a periodic array, thus the scattering amplitude is also periodic and satisfies condition (5.1.2). It is therefore possible to expand it into a Fourier series in terms of the reciprocal-lattice vectors. Using the form

	1
f (r) =		fGeiG·r ,	(8.1.51)
	v	G

where v is the volume of the primitive cell, the amplitude of the scattered beam is found to be proportional to

fK = v	G fG	e−i(k−k −G)·r dr .	(8.1.52)
1

For infinite crystals scattered beams arise only in the directions that satisfy the condition k = k − G; cancellation is perfect in all other directions.

In finite crystals, an arbitrary position r is written as the vector sum of the lattice point Rn associated with the primitive cell and the position vector u within the cell:

r = Rn + u = n1a1 + n2a2 + n3a3 + u1a1 + u2a2 + u3a3 . (8.1.53)

The coordinates ui of the position within the primitive cell satisfy the condition

0 ≤ uj ≤ 1 , j = 1, 2, 3.

(8.1.54)

Next, the vector K = k −k is expressed in terms of reciprocal-lattice vectors. From the foregoing it is intuitively obvious that scattering is maximal if K is precisely equal to a vector of the reciprocal lattice. Since we are interested

254 8 Methods of Structure Determination

in the shape of the peak, K is now allowed to di er by an amount q from a reciprocal-lattice vector G. Expressed in terms of the primitive vectors b1, b2, and b3 of the reciprocal lattice,

K = G + q = hb1 + kb2 + lb3 + q1b1 + q2b2 + q3b3 .

(8.1.55)

As the scattered beam is confined to a small cone around the direction of k = k − G, the fractional numbers qi are chosen to fall in the interval

−21 ≤ qj ≤ 21 , j = 1, 2, 3.

(8.1.56)

To evaluate the integral in (8.1.52) for a fixed q, it is separated into an integral over the primitive cell and a sum over primitive cells:

fq = v fG		Rn	e−iq·(Rn+u) du .	(8.1.57)
1		v

Suppose that the shape of the sample is such that there are N1, N2, and N3 primitive cells along the directions of the primitive vectors a1, a2, and a3. As the scalar product of a directand a reciprocal-lattice vector satisfies (5.2.12),

fq = fGM1M2M3 ,			(8.1.58)
where	1
Nj −1
Mj = nj =0		e−2πi(nj +uj )qj duj .	(8.1.59)
	0

Summation and integration can be separated. Each operation is then elementary, leading to

Mj = Sj Mj ,

(8.1.60)

where

= Nj −1 e−2πinj qj =

Sj

e−2πiNj qj

− 1

,

(8.1.61)

e−2πiqj

−

1

nj =0

and

1

1

e−2πiqj

Mj

= 0

e−2πiuj qj duj =

−

,

(8.1.62)

2πiqj

that is

1 − e−2πiNj qj

Mj =

.

(8.1.63)

2πiqj

The intensity I of the scattered beam contains the absolute square of the amplitude. Using

	1	2πiqj		π2qj2		(8.1.64)

1 1 1 .

4 4 4

				8.1		The Theory of Di raction				255
the intensity distribution is given by
2	1		sin2 πq1N1 sin2 πq2N2 sin2 πq3N3
I(q) = |fG|									(8.1.65)
	v2		π2q12		π2q22			π2q32

for the di raction peak associated with the reciprocal-lattice vector G and the direction characterized by the components qi. This function has a sharp maximum at q = 0, where its value is proportional to the square of the number of scatterers – that is, the square of the sample volume. This result is counterintuitive: the intensity of the scattered beam is expected to be proportional to the volume of the sample (and not its square). If we were to determine not just the peak height but the total scattered intensity, it would be proportional to the volume of the sample, as the width of peaks decreases inversely with the volume of the sample. Nevertheless this result indicates the necessity of a more rigorous discussion of the rays scattered from the sample interior. We shall revisit this point later, when the dynamical theory of di raction has been outlined.

8.1.5 Cancellation in Structures with a Polyatomic Basis

In structures with a monatomic basis, where each primitive cell contains a single atom, the intensity of scattering associated with vector G of the reciprocal lattice is simply proportional to |fG|2. The situation is di erent in crystals with a basis of several atoms. Here intensity relations depend on the geometry of the atoms of the basis as well as the magnitude of the individual scattering amplitudes. The situation is substantially simplified when the atoms of the basis are all identical. The atomic scattering factor can then be factored out, leaving behind a structure factor that conveys information about the structure alone. Consider, for example, a diamond lattice, which is a face-centered cubic structure with a diatomic basis. In terms of the edge vectors of the face-centered cubic Bravais cell the coordinates are 000 and Using the primitive vectors given in (7.2.9) leads to the same coordinates: the second carbon atom is at 14 (a1 + a2 + a3). The structure amplitude for reciprocal-lattice vector G = hb1 + kb2 + lb3 – where bi is defined by (7.2.12)

– is

Ahkl = 1 + e−i(hb1 +kb2 +lb3)·(a1+a2 +a3)/4 = 1 + e−iπ(h+k+l)/2 . (8.1.66)

Depending on the indices hkl of the Bragg peak, this quantity takes the values 2, 0, or 1 ± i. The structure amplitude vanishes when h + k + l is even but not divisible by four (h + k + l = 4j + 2). This means that there are directions in which Bragg peaks are absent even though the Laue (Bragg) condition is satisfied by the reciprocal-lattice vector Ghkl. For other values of hkl there are Bragg peaks, however, when h + k + l is odd (h + k + l = 2j + 1), their intensity is just half of those peaks for which h+k +l is divisible by four (h+k +l = 4j). As Fig. 8.5 shows, those lattice points of the bcc reciprocal lattice for which

256 8 Methods of Structure Determination

the structure amplitudes – and hence the Bragg peak intensities – are equal are located in planes perpendicular to the space diagonal.

b1	b
	2

b3

Fig. 8.5. Reciprocal-lattice vectors of the diamond structure that give rise to di raction peaks of di erent intensities. The structure amplitude for vectors pointing to full circles, shaded circles, and empty circles are 2, 1 ± i, and 0. The intensity ratio of the corresponding Bragg peaks is therefore 4 : 2 : 0

The reason for this is that rays scattered by the two carbon atoms of the basis may interfere constructively or destructively; they may even cancel out perfectly. Observation of these cancellations permits experimentalists to distinguish the diamond structure from face-centered structures with a monatomic basis.

When the basis contains two di erent atoms – as in the sphalerite structure, where Zn and S atoms are at points 000 and 14 14 14 , respectively –, the scattering amplitude of the primitive cell is calculated from atomic scattering amplitudes using weights characteristic of the structure:

fhkl = fZn + fSe−iπ(h+k+l)/2 .

(8.1.67)

This shows that there are no reciprocal-lattice vectors for which cancellation is perfect. From the intensity distribution of the Bragg peaks one may infer the atomic scattering factors and atomic positions within the cell – that is, the nature of the crystal.

Similarly to the case of the diamond structure, a cancellation (albeit an apparent one) is observed when the crystal has a centered Bravais lattice – but this information is not known prior to the evaluation of the di raction measurements. As a simple example consider a face-centered cubic lattice with a lattice constant a. When, based on macroscopic properties, the crystal structure is assumed to possess cubic symmetry, one may attempt to interpret the di raction peaks in terms of the reciprocal lattice of a simple cubic crystal. Using the edge vectors a, b, and c of the Bravais cell (see Fig. 8.6(a)), the

8.1 The Theory of Di raction

257

coordinates of the four atoms of the cell are 000, 12 12 0, 12 0 12 , and 0 21 12 . The counterparts of the direct-lattice edge vectors a, b, c are the reciprocal-space

vectors a , b , and c of length 2π/a. In terms of these the structure amplitude for the vector G = ha + kb + lc is

Ahkl = e−2πi(hxj +kyj +lzj ) = 1 + e−iπ(h+k) + e−iπ(h+l) + e−iπ(k+l) . (8.1.68) j

The value of this quantity is 4 when the Miller indices hkl are all even or all odd. When odd as well as even indices occur, the structure factor vanishes

– and so no scattered beam emerges in the corresponding direction. Certain Bragg peaks are absent – while they should be present if the crystal had a simple cubic structure. Empty circles in Fig. 8.6(b) indicate those vectors G of the reciprocal lattice for which the structure factor vanishes, and full circles those for which the Bragg peak is finite.

			b1
			c*
			b2
c			b*
a1			a*
	a2
	b		b3
(a)		a3	(b)
			a

4 /a

Fig. 8.6. (a) The primitive cell and the Bravais cell for a face-centered cubic crystal. (b) The body-centered cubic reciprocal lattice of the face-centered cubic direct lattice is indicated by full circles. The simple cubic lattice with the additional lattice points (empty circles) is the reciprocal of the lattice spanned by the primitive vectors a, b, c

Compared to the simple cubic lattice, certain Bragg peaks are thus absent in the corresponding face-centered cubic crystal. This can be interpreted as the result of destructive interference between the rays scattered by the atoms at the vertices and face centers. This is illustrated in Fig. 8.7. The planes drawn across the atoms at the vertices of the simple cubic lattice are indicated by solid lines, while the extra planes appearing in an fcc lattice due to centering by dashed lines.

The figure shows that the spacing between the atomic planes of an fcc crystal is half of that for the simple cubic crystal from which it is obtained by centering. Let us consider directions for which the phase di erence between rays reflected from adjacent planes of the simple cubic lattice is an odd multiple of 2π – i.e., the path di erence is an odd multiple of λ. Among themselves,

258 8 Methods of Structure Determination

Fig. 8.7. Projection of the face-centered cubic crystal onto the plane perpendicular to the [001] direction, with the (100) and (110) planes. Full circles, empty circles and symbols indicate atoms at vertices, base centers, and side centers. Solid lines show those members of the family of planes that contain atoms at the vertices of the Bravais cell. Atomic planes drawn with dashed lines are the result of centering

these rays interfere constructively. When, however, the rays scattered from the extra planes of the fcc lattice are also taken into account, cancellation occurs, because these rays are precisely in the opposite phase.

However, this cancellation is only apparent: it occurs only when the Bragg condition is employed for the atomic planes of the simple cubic lattice whose centering gives the real lattice, the face-centered cubic one. Similarly: when using the Laue condition, the absence of scattering for certain vectors of the reciprocal lattice is the consequence of considering a simple rather than a facecentered cubic lattice. The reciprocal lattice should be defined in terms of the primitive vectors of the true primitive cell a1, a2, and a3 rather than those of the Bravais cell (conventional unit cell). Since the (direct-lattice) Bravais cell is larger than the primitive cell, the reciprocal lattice G = ha + kb + lc

– generated by the vectors a , b , c associated with the edge vectors of the Bravais cell – is denser than the true reciprocal lattice, spanned by the primitive vectors (7.2.12). As we have seen, the latter is a body-centered cubic lattice with edge length 4π/a. This lattice is shown by the full circles in Fig. 8.6. Scattered beams emerge only in those directions for which the Laue condition k − k = G is satisfied by the vectors of reciprocal lattice of the face-centered cubic lattice.

8.1.6 The Dynamical Theory of Di raction

When calculating the intensity of the di raction peak it was assumed that the atoms of the sample scatter the coherent incident beam independently of each other, i.e., a photon scattered by an atom does not undergo another scattering. Neglecting multiple scattering processes may be justified for small samples

8.1 The Theory of Di raction

259

and powders of polycrystalline materials, where interference is assumed to be absent for beams scattered by grains of di erent orientation. However, for larger single crystals it is no longer true that rays scattered by atoms deep inside the sample and close to its surface are equally intense. A beam incident upon the sample is scattered by the topmost atomic layer, giving a reflected and a transmitted beam. The latter arrives at the second layer of atoms, which reflects a part of it and transmits another to the third layer. Naturally, radiation reflected from deeper layers may bounce back from upper ones. This is illustrated in Fig. 8.8.

Fig. 8.8. Interference of the waves reflected from and transmitted by subsequent layers of the crystal

Beams undergoing multiple scattering in the sample may interfere with each other, building up an internal field. To determine this field the phase shift of the waves occurring in scattering processes must not be ignored. The theory that takes into account the amplitude and phase relationships of beams scattered in the sample is called the dynamical theory of di raction,6 as opposed to the kinematical theory discussed above. Without going into details, we shall briefly present the crucial elements of the theoretical description, pointing out the di erences with the problem discussed in Chapter 25, the scattering of light by solids (where Fresnel’s equations are used).

The electromagnetic field built up inside the sample obeys Maxwell’s equations. Since conductivity practically vanishes for X-ray frequencies, these simplify to

curl H =	∂D		curl E = −	∂B		(8.1.69)
		,			.
	∂t			∂t

For nonmagnetic materials B = μ0H to a good approximation, while the dielectric constant relating D and E is assumed to be lattice periodic within the sample,7 and can thus be expanded into a Fourier series in terms of the reciprocal-lattice vectors:

6C. G. Darwin (1914) and P. P. Ewald (1916).

7This assumption is not needed in the optical region where the dielectric constant is assumed to be uniform both inside and outside the sample.

260 8 Methods of Structure Determination

	1
r =		GeiG·r .	(8.1.70)
	v	G

Let a plane wave of wave vector k be incident on the sample and scattered by the periodic structure. The scattered wave is a mixture of waves with wave vectors kG = k + G. The interior electric and magnetic fields are therefore written as the linear combinations of plane waves with such wave vectors. Substituting them back into Maxwell’s equations gives relationships among the Fourier coe cients. The problem is substantially simplified by the assumption that the Bragg condition is satisfied for a single reciprocal-lattice vector, i.e., the amplitude is important for a single component k + G apart from k.

The next important step is to satisfy the matching conditions across the boundary surface. In contrast to the optical region, where the directions and amplitudes of the incident, reflected and transmitted beams need to be matched, the beam reflected from the surface is practically absent in the X- ray region as the di erence of r from unity is tiny (on the order of 10−4 or less), and so the amplitude of the incident beam is equal to that of the transmitted beam at the surface. Only wave vectors need to be matched; for this it should be borne in mind that inside the sample k may be complex because of absorption.

Calculations show that if the sample is su ciently thick and the crystal is ideal – that is, subsequent layers are perfectly parallel – then cancellation is exact in the transmitted beam, and so the di racted beam emerges in a very narrow pencil – that is less than an arc minute across, yet finite in diameter

– around the direction satisfying the Bragg condition. The width of the beam is determined by the size of the sample and the Fourier transform fG of the spatial distribution of scatterers. The total intensity of the Bragg peak is proportional to the volume, just like in the kinematical theory, however, here it is proportional to |fG| instead of its square. Of course, intensity also depends on whether the Laue or Bragg case is considered. The former corresponds to a di racted beam emerging on the other side of the sample – and thus attenuated by absorption –, while the latter corresponds to a di racted beam emerging from the same surface on which the incoming beam is incident.

To evaluate the intensity of the scattered beam one has to take into account that atoms of the crystal are not fixed rigidly at their equilibrium positions but oscillate around it. The amplitude of this vibration becomes larger at higher temperatures, and its consequences cannot be ignored in a precise study of di raction. We shall see in Chapter 12 on lattice vibrations that the Bragg condition is still valid for elastic scattering: even when atoms oscillate harmonically, di raction peaks continue to be sharp, however the intensity of the scattered beam decreases with increasing temperature.