g10 1d crystal waves - sheffield
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Microsoft Word - G10_1Dcrystal_v1.0.docSeptember 2010 V1.0 Page
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Department of Physics and Astronomy 2nd Year Laboratory
G10 1D crystal waves
Scientific aims and objectives
• To simulate experimentally the normal modes of oscillation in one dimensional monatomic and diatomic crystals using a linear air track
• To experimentally determine the spring constant and therefore bonding force constant in the experimental crystals
Learning Outcomes
• To use excel to “model” a complex system • To know the method to solve the propagation of a wave in a lattice, and to know
the form of the dispersion relation for the normal modes of one dimensional monatomic and diatomic lattices
Apparatus
• Air track carriages • Springs • Masses
Safety instructions
2nd year laboratory script G10 1D Crystal waves
September 2010 V1.0 Page 2
Task 1 - Pre-session questions The air track is an experimental one dimensional crystal that can sustain different types of oscillations, some of which correspond to the “normal modes” of a real, bound, finite one- dimensional crystal. The pre task questions require a relevant section of text from a solid state physics book by Hook and Hall to be read. The questions are directly linked to the text and will help you understand both the experiment and its analysis. A detailed understanding of this text is necessary and you should complete the questions before progressing.
Task 2 – The experiment The linear air track can simulate either a monatomic lattice when the same mass is placed on each carriage on the track, or a diatomic lattice with alternating masses on the carriages. The carriages should be connected together by springs and driven at one end by a motor whose angular frequency is controlled by a voltage dial. The angular frequency needs to be calibrated with respect to the voltage dial. You should first study the monatomic lattice with the objective of determining the spring constant of the system. The mass of the carriages can be measured directly and used along with the dispersion relationship ω = f(k) for the normal modes to determine the spring constant. The normal modes (the standing wave modes of the driven lattice) can be found by systematically varying the voltage and observing, firstly whether any stationary points are present, and secondly how many stationary points are present. The number of stationary points is the mode number of the standing wave. These normal modes occur when a fixed number of wavelengths fit into the lattice length: Na=nλ/2.
You can also investigate a diatomic lattice by placing additional mass on alternating carriages on the track. Hook and Hall covers the diatomic lattice theoretically and you should refer to the text to understand the expected dispersion relations. One aim could be to experimentally verify the derived dispersion relation in Hook and Hall. The interesting and important feature of the diatomic lattice is the presence of two solutions to ω for every value of k and how this feature relates to physical properties of the lattice such as the propagation of sound waves.
Task 3 – Reporting In contrast to other experiments in the second year lab your lab book and lab reports should give a summary account of the theoretical calculation of the dispersion relation of the 1D lattices you have studied. Your summary should outline the important assumptions and steps in the calculation as well as graphical descriptions of the functions that are derived. You should also show plots of your experimental data and fitted curves to determine K. If you have investigated the diatomic lattice as well then you can show qualitatively how the theoretical functions match with the experimental results.
2nd year laboratory script G10 1D Crystal waves
September 2010 V1.0 Page 3
Appendix of supporting information
1. Read the extract from Hook and Hall
“Section 2.3 Lattice vibrations of one-dimensional crystals”
2.3.1 Chain of Identical atoms
2.3.2 Chain of two types of atom
2. Pre-lab questions
Although these are called “pre-lab” questions, please use the experimental apparatus freely, alongside the text from Hook and Hall when attempting to answer the questions. Think of these questions as a guide to understanding the experiment.
a) When considering both the theoretical and experimental monatomic 1D lattice which of the following statements are true:
1. The theoretical lattice is bound at each end and can be considered to have a periodic boundary condition – this is exactly the same as the experiment
2. The theoretical lattice is bound at each end and can be considered to have a periodic boundary condition – this is different to the experiment where one end of the lattice is being actively driven
3. In the experiment only the conditions where standing waves are observed obey the dispersion relationship.
4. In the experiment the non-standing wave conditions correspond to solutions of a system of which the experiment is a part. i.e. a normal mode of a lattice that is larger than the air track and where the driving point corresponds to an arbitrary point on the wave and not a stationary point..
5. In the experiment the nodes of the normal standing wave modes must occur at the carriages in the lattice
6. In the experiment the nodes of the normal standing wave modes must never occur at the carriages in the lattice
7. In the experiment the nodes of the normal standing wave modes sometimes occur at the carriages and sometimes between the carriages
8. As the frequency of the normal mode oscillation is increased the wavelength of the oscillation continues to reduce indefinitely
2nd year laboratory script G10 1D Crystal waves
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9. As the frequency of the normal mode oscillation is increased it reaches a limit beyond which the system does not oscillate.
10. The relationship between normal mode frequency and wavenumber is continuous
11. The relationship between the normal mode frequency and wavenumber is quantised and only takes discrete values, limited by the lattice size, since a half integer number of wavelengths must fit precisely into the lattice
12. The solution to the equation of motion of the lattice is found by using a wavelike trial solution for one atom and seeing if it works.
13. Only nearest neighbour interactions are considered in the theory.
14. For the nearest neighbour interactions only the first term in the interaction expansion is considered.
b) When ω gets quite large it becomes increasingly difficult to observe the normal modes. It will be difficult to experimentally observe more that 4 or 5 normal modes for both the monatomic and diatomic chains. Why is this? What factors have not been accounted for in the theory calculations? Why do these factors become more important for the higher modes?
--
2.3 LATTICE VIBRATIONS OF 'ONE-DIMENSIONAL CRYSTALS
2.3.1 Chain of identical atoms
The simplest crystal is the one-dimensional chain of identical atoms of equilibrium lattice spacing a shown in Fig. 2.2. We assume that the atoms can move only in the direction parallel to the chain, that they interact via an interatomic potential of the form shown in Fig. 1.23 and that the forces between them are of such short range that only nearest neighbour forces are important. In this limit the small amplitude motions of the chain of atoms are identical to those of the chain of identical masses M connected by identical springs of spring constant K shown on Fig. 2.3, as we shall now demonstrate. The interaction 1/(1') between nearest neighbours of separation r may, for a small deviation of r from its equilibrium value a, be expanded as a Taylor series about r = a. Thus
a)2 (d 21/)(1' "1/(1') = "I/(a) + -2 + .... (2.4) 2 dr r=a
No term linear in I' - a appears in the Taylor series because the first derivative of ''1' must vanish at the equilibrium spacing where "1/ is a minimum. If the
~ ~ ~ ~ ~ a
Fig. 2.2 One-dimensional crystal consisting of a chain of identical atoms
( 0)
'/1
u un-2 U U un-1 n, n+l n+2
Fig. 2.3 A chain of identical masses M connected by springs : (a) at equilibrium spacings
x~ = na; (b) at displaced positions x, = na + Un
2.3 Lattice vibrations of one-dimensional crystals 37
higher-order terms are ignored (this is the harmonic limit referred to in the introduction to this chapter), Eq. (2.4) looks like the potential energy associated with a spring of spring constant
d 2"1/)K= -2- . (2.5)( dr r=a
The spring constant can be simply related to the elastic modulus C of the one-dimensional crystal, defined by writing the force required to increase the interatomic distance from a to I' as qr - a)/a (i.e. force = elastic modulus x strain). The model of Fig. 2.3 predicts a force K(r - a) and this identifies the following relation between C and K:
C=Ka. (2.6)
We now proceed to calculate the lattice vibrations of the one-dimensional chain of Fig. 2.3, which exhibit many of the important qualitative features of lattice vibrations in general. We will use the laws of classical mechanics and postpone until section 2.5 a discussion of the difference in the results of a quantum mechanical calculation. We shall suppose that the chain consists of a very large number of atoms and that the last is joined to the first so as to make a ring. This last assumption is simply a device to make the chain endless so that all the atoms have an identical environment, and has no important effect on the problem for a long chain, where end effects are unimportant anyway.
If the displacements of the atoms from their equilibrium positions are u; as shown in Fig. 2.3, the force on the nth atom consists of:
(i) K(u - Un-I) to the left , from the spring on its left; andn
(ii) K(u + 1 - Un) to the right, from the spring on its right.n
Equating the total force to the right, (ii) - (i), to the product of mass and acceleration we have
MUn = K(u n + l - 2un + Un-I)' (2.7)
The equations of motion of all atoms are of this form, only the value of n varies. To solve Eqs. (2.7) we try a wavelike solution in which all the atoms oscillate at the same amplitude A. Thus we substitute]
Un = A exp [i(kx~ - cut)] (2.8)
where x~ = na is the undisplaced position of the nth atom, to obtain
_ (j)2MA ei(kna- wt) = KA(ei[k(n+l)a-wt]_ 2e i(kna-QJI) + ei[k(n-l)a- w,]),
t As usual when solving vibration problems by means of complex exponentials, it is the real part of u, that we interpret physically as the atomic displacement.
38 Crystal dynamics Chap. 2
or, on cancelling Aei(k na - wl) from each term ,
ika )- w 2 M = K(eika _ 2 + e
= 2K [cos (ka) - 1] .
w 2M = 4K sin? (~ka) . (2.9)
Fig. 2.4 shows the disper sion relation (relation between frequency wand wavenumber k) given by Eq. (2.9) for our wavelike lattice vibrations. The maximum value of s i n(~ka) is I so that the maximum possible frequency of the waves is 2(K/ M) 1/2 . Th is is known as the cut-off frequency of the lattice.
We notice that n has cancelled out in Eq. (2.9), so that the equ ations of mot ion of all atoms lead to the same algebraic relation between wand k. Thi s sho ws that our trial function for Un is indeed a solution of Eqs. (2.7). It is also imp ortant to not ice that we sta rted from the equations of motion of N coupled harm on ic oscillato rs (Eqs. (2.7)). If one atom starts vibrating it does not continue with constant amplitude, but tran sfers energy to the others in a complicated way; the vibrations of individual at oms are not simple harmonic because of this energy exchange amo ng them. Our wavelike solutions on the other hand are uncoupled oscillations called normal modes; each k has a definite w given by Eq. (2.9) and oscillates independently of the other modes. We should expect the number of modes to be the same as the number of equations N that we sta rted with; let us now see whether this is the case.
To do so we must esta blish which wavenumbers are possible for our one dimension al chain. Not all values are allowed because the nth atom is the same as the (N + n)th as the chain is joined on itself. This means that the wave (2.8) must sa tisfy the periodic boundary condition
un = uN +n (2.10)
which requires that there sho uld be an integral number of wavelengths in the
w
Fig. 2.4 Normal mode frequencies for a chain of identical atoms. Note that the modes with wavenumbers at A, B and C all have the same fre quency and correspond to the same instantaneous atomic displacements (see Fig. 2.5). Point B represents a wave moving to
_2!. o 7r 2JT k the right, points A and C a wave a a a moving to the left
2.3 Lattice vibra tions of one-dimensional crystals 39
length of our ring of a toms
Na = pI, .
k = 2n = 2np (2.11) Ie Na
where p is an integer. Th ere are thu s N possible k values in a range Zn]« of k, say the range
- n/a < k ~ x]a.
Fig. 2.4 shows that this restricted range of k does indeed include all possibl e values of the frequency wand the gro up velocity (dw/dk); it also gives th e N normal modes we expect for N atoms. What , if anything, is the physical significance of wavenumbers outside this rang e?
To understand th is, consider the instantane ous atomic displ acements shown in Fig. 2.5; we are really considering longitudinal waves, but the displacement s are shown as transverse in Fig. 2.5 because this makes their wavelike nature easier to visualize. Fig. 2.5(a) shows the displacement s for k = »[a, which gives
x
( b )
Fig. 2.5 (a) Atomic displacements (shown as transverse for clarity) for wavelength ..1. = 2a, wavenumber k = nla. (b) The atomic displacements for a wave with ), = 7a/4, k = 8n/7a, as given by the full curve, are identical to those for a wave with ..1.= 7a/3,
k = 6n/7a, as given by the broken curve
o ( 0)
40 Crystal dynamics Chap. 2
the maximum frequency; alternate atoms oscillate in antiphase and the waves at this value of k are essentially standing waves. The midpoint of each spring is at rest and each mass therefore behaves as if held by two springs each of spring constant 2K, giving the frequency 2(KjM)1/2 that we have calculated.
Now consider the displacements, shown by the full curve in Fig . 2.5(b), for the slightly larger value 8nj7a of k corresponding to the point A on Fig. 2.4. The displacements can also be represented by the longer wave, shown broken in Fig. 2.5(b), for which Ikl = 6nj7a; this corresponds to points B or C in Fig. 2.4. Thus, points A, Band C correspond to the same instantaneous atomic displacements as well as the same frequency. At B the group velocity dOJjdk > 0, so we have a wave travelling to the right; A and C both represent a wave travelling to the left and are thus completely equivalent. The k values of points A and C differ by 2nja and we therefore conclude that adding any multiple of 2nja to k does not alter the atomic displacements or the group velocity and is without physical significance. We need only consider the range - nja < k ~ nfa, which contains just the N modes we expected.
Further insight into what is special about the k values ± nta is gained by writing down Bragg's law (Eq. (1.3» for the one-dimensional crystal:
nA = n2njk = 2d sin e= 2a
or
k = nn la; (2.12)
where we have taken e= 90° and d = a as appropriate to waves travelling along a one-dimensional chain. Waves with k = ±nja will thus undergo Bragg reflection. The standing wave pattern that occurs at these two k values can be pictured as occurring because of Bragg reflection of running waves.
We note that in the long-wavelength limit, ka ~ 1, Eq. (2.9) reduces to
MOJ2 .: Kk2a2
so that in this limit the waves are dispersionless with group velocity and phase velocity (OJjk) both being equal to
v = a(K jM)1/2 (2.13)
These waves are long-wavelength sound waves and a calculation of their velocity from the macroscopic elastic properties of the crystal by the method used in the previous section yields a velocity (cf Eq. (2.3»
vs = (Cjp)1 /2 (2.14)
where p (= M ja) and C are respectively the mass per unit length and the elastic modulus of the crystal. Using Eq . (2.6), we confirm that Eqs. (2.13) and (2.14) are identical and thus that our more general calculation of lattice vibrations gives the correct answer in the long-wavelength limit. Note that, since there is
2.3 Lattice vibrations of one-dimensional crystals 41
only one possible propagation direction and one polarization direction, the one dimensional crystal has only one sound velocity. Given the velocity of sound and the lattice spacing it is possible to draw the dispersion relation for our simple crystal. This illustrates our statement in the introduction to this chapter that many of the properties oflattice vibrations can be deduced without detailed knowledge of the interatomic forces.
The inclusion of only nearest neighbour forces in our calculation appears very restrictive. Although it is a good approximation for the inert-gas solids, it is not a good assumption for many solids. The effects of removing this restriction can be investigated by using a model in which each atom is attached by springs of different spring constant to neighbours at different distances (see problem 2.1). When this is done, many of the features of the above calculation are preserved. The wave solution of Eq . (2.8) still satisfies the equations of motion. The detailed form of the dispersion relation is changed but OJ is still a periodic function of k with period 2nja and the group velocity vanishes at k = ±nja. There are still N distinct normal modes, which can be represented by the N possible k values in the range - nja < k ~ + nla. Furthermore the motion at long wavelengths corresponds to sound waves with a velocity given by Eq. (2.14).
2.3.2 Chain of two types of atom
We now consider the lattice vibrations of a chain containing two types of atom, of masses M and m, connected by identical springs of spring constant K as shown in Fig. 2.6. This is the simplest possible model of an ionic crystal,
(a )
(b)
a
Fig. 2.6 A chain containing two unequal masses connected by springs: (a) at equilibri um positions x~ = nal2; (b) at displaced positi ons x, = nal2 + un' Here a is the length of
the unit cell as indicated
.........
42 Crystal dynamics Ch ap. 2 2.3 Lattice vibrat ions of one-dimensional crystals 43
altho ugh the assumption of onl y nearest neighbour forces , impli cit in the model , is a poor approxima tion for ionic crystals because of the long ran ge of the Co ulomb interaction between ions. Fortunately the simple model again pro duc es the importa nt qu alit at ive features of the lattice vibrations of ioni c so lids . In section 9.1.4 we discuss the changes to the vibrations when the lon g-ran ge effects of the Co ulomb for ce are included.
To emphasize the more complicated motions that are possible when there is more than one type of a tom we show in Fig. 2.6(b) a configurati on in which the two types of at om ar e displaced in opposite directions. N ote that we use a to den ote the un it cell length (lattice spacing) of the crystal ; the nearest neighbour sepa ra tion of the undi splaced ato ms is a12.
Th e equa tions of moti on can be written do wn in the sam e way as Eq. (2.7) but there are now two distinct types of equation : those for masses M ,
MUn = K(un+I - 'Iu; + Un-I ) (2.15)
and those for masses m,
mUn_ 1 = Ktu; - 2un_1 + un- 2)· (2.16)
For the masses M we may ass ume as before a solution of the form (2.8), i.e.
Un = A exp [i(kx ~ - wt)]
where x~ = nal 2 is the undi splaced a tomic pos ition. There is now an extra unknown qu antity, the relat ive amplitude and phase of the vibra tions of the two types of atom; this we allow for by taking for the masses m
Un = ct.A exp [i (kx~ - wt) ] (2.17)
where ct. is a complex number giving the relative amplitude and phase. Substitution in Eqs. (2.15) and (2.16) then gives
_ w2Mei(kna /2 - wt ) = K(ct.ei[k(n +l )a/2 - wt] _ 2ei(kna/2- WI) + ct.ei[k(n- l)a/2-wt] )
and
_ ct.w 2mei[k(n- l)a/2-wtl = K(ei(kna/2- wt) _ 2ct.ei[k(n-l )a/2- wt] + ei1k(n- 2)a/2- wt])
or, by cancelling commo n factors as before,
- w2M = 2K[ct. cos (! ka) - I] (2.18)
- ct.w2m = 2K[cos (! ka) - ct.]
correct form. Eq s. (2.18) may be rewritten in the form
2K cos (!ka) 2K - w2M ct. = = -::-::-,--------,,.,...,---,- (2.19) 22K - w m 2K cos (! ka) ,
from which by cro ss-multiplication we obta in a qu adratic equat ion for w2 ,
4mMw - 2K (M + m)w2 + 4K 2 sin? (!ka) = 0, (2.20)
with solutions
2 K (M + m) [ ( M + m)2 4 . 2 ( Ik )J I/2 w = + K --- - - Sin "2 a . (2.21) Mm - Mm M m
The two roots are sketched in Fig. 2.7. As there are two values of w for each value of k the dispersion relati on is sa id to have two branches and the upper and lower branches in Fig. 2.7 corresponds to the + and - signs in Eq. (2.21) respectively. We see that chai ns contai ning two types of ato m sha re with tho se containing one the property that the dispersion relations are peri odic in k with period 2nla = 2nl(unit cell length); this result remain s va lid for a cha in containing an arbitra ry number of a toms per unit cell.
If the crystal contain s N unit cells we would expect to find 2N normal modes of vibration as this is the tot al number of ato ms and hence the total number of equ ations of motion (Eqs. (2.15) and (2.16» . Joining the end s of the crystal to form a rin g requires the ato mic displacements to sa tisfy the peri odic boundar y condition Un = U2N +n, leading to the same express ion for the possible k valu es,
k = 2npINa,
Thus instead of a single a lgebra ic equ ation for w as a function of k, we now have Fig. 2.7 Norm al mode frequencies of a cha in of two types of ato m. At A, the two types
a pair of algebraic equations for ct. and w as functions of k. As before th e fact that are oscillating in antiphase with their cen tre of mass at rest ; at B, the lighter mass m is /l does not appear in Eqs. (2.18) indicates th at our assumed solution is of the osci llating and M is at rest ; at C, M is osc illating and m is a t rest
44 Crystal dynamics Chap. 2 2.3 Latt ice vibrations of one-dimensio nal crys ta ls 45
as for the crysta l co nta ining a sin gle type of a to m. Thus th ere are again exactly N allowed values of k in the range - ttfa < k :::; n]a; also as in th e previous section, addi ng a ny multiple of Zn]« to k does not alter the a to mic d isplace ments, a nd we deduce that a ll the allowed motion s can be described by k va lues in th is range. Henc e the two br anches of the dispersion relation co ntai n 2N norm al modes as req uired.
It is instructi ve to examine the limiting solutions of Eq. (2.21) near the poin ts 0 , A, B and C in Fig. 2.7. For ka ~ 1, sin (tko,) ~ t ka and
w2 ~ K(M + 111) [1-:- (I_. ----'!!1~ _ k2a2)1/2J Mm - (M + m)2
K( M + 111)[( 111M 22)J ~ - -- 1 + I - - - - k a lV[ //1 - 2(M + m?
2K(M + m) Kk2a2
~ or . Mm 2(M + m)
By substituting these values of oi in Eq. (2.19) and ucing cos(~ka) ~ I for ka ~ I we find the co rrcspcnding va lues of (J. as
0: ~ - Mlm or I.
T he first so lution corresponds to point A in F ig. 2:7; this value of a co rresponds to M and m osci lla ting in a ntiphase with their cen tre of mass at rest, and th e freq uency is therefore given by the spring constant 2K a nd the red uced mass M* = MI11/(M + m). The second solu tion represents lon g-wav elength so und waves in the neighbourhood of point 0 in Fig . 2.7; th e two types of atom osci lla te with the sa me amplitude and phase, and the veloci ty of sound is
W ( K )1 /2 V - - a s - k - 2(M + m) .
This sound velocity mu st ag ree with that, namely (Cfp) 1/2 (E q. (2. 14» , pr ed icted fro m the macroscop ic elas tic properties of the cryst al ; substituting values of (M + m)/a and Ka/2 (cf. Eq. (2.6), recalling that the definition of a has changed) respecti vely for the mass per unit length p and th e elas tic modulus C into Eq. (2.14) co nfirms th at th is is so.
T he othe r limiting so lutions of Eq . (2.2 1) are for ka = n, i.e. sin(t ka) = I. In thi s case
2_ K(M + m) [(M + m) 2 4 JI/2 W - +K -- -
Mm - Mm Mm
= 2K/m or 2K/M
with , from Eq. (2.19), the co rresponding amplitude ratios 0: = co o r 0: = 0 respectively. In this limit the half-wa velen gth is a, the spacing bet ween a to ms of the same kind. In the first so lution m oscilla tes a nd M is a t rest (poin t B on Fig. 2.7 if M > m), a nd the freq uency therefore depends only on m; in the seco nd solution M oscillat es a nd 111 is a t rest (poi nt C on Fig. 2.7).
It is instructive to com par e our present results with those of sectio n 2.3.1 for a chain of one type of a to m. In F ig. 2.8 we plot the lower-frequency br an ch of Fig. 2.7 in the region k < nla and the higher-frequenc y branch in the region nfa < k < Zn]«. If we now let m -7 M the points B and C in Fig. 2.8 com e together and we recover Fig. 2.4, as we mu st. (D o not forget that the value of a in Fig. 2.4 is half th at in Fig . 2.8.)
There is thu s a certa in a rbit ra riness about how we assign k valu es to th e modes of a diat omi c lattice. The most direct compari son with a mon at omi c lattice is obtai ned with th e ass ignment sho wn in Fig. 2.8, where there is only one W for each k and there are 2N mod es in the ran ge - 2n/a < k :::; 2n/a. It is more usual, however, to ass ign th e lowest possible k to eac h mode as in Fi g. 2.7; th ere are now two br anches with N modes on eac h br anch in the range - n/a < k :::; nta. This latter approach is th e usual one and has the useful feature th at the ran ge of k va lues is 2n/( unit cell side), ind ependently of th e number of ato ms in a unit cell.
Although th e disp ersion relat ion is no lon ger given by Eq . (2.2 1) when th e restriction to nearest neighbour forces is removed, mo st of th e qu alitati ve conclusions con cerning the nature of the d ispersion relation that we have deduced a bove rem a in valid . In particular the dispersion relati on has two
w
2
o 11:
Fig. 2.8 Thi s shows Fig. 2.7 replotted for comparison with Fig. 2.4; in the limit m --> M points Band C come togeth er and the dispersion rela tion is given by the broken line. The
a tomic displacement s at A' are the same as those a t point A on Fig. 2.7
46 Crystal dynamics Chap. 2
branches, both periodic in k with period Zn]«. Only one of the branches has the limiting form of sound waves at long wavelength (w/k ---> constant as k ---> 0). This branch, the lower branch on Fig. 2.7, is consequently known as the acoustic branch. The other branch is called the optical branch because as k ---> 0 on this branch the vibrations of the two types of atom are in antiphase and the resulting charge oscillations in an ionic crystal give a strong coupling to electromagnetic waves at the frequency of point A on Fig. 2.7. An estimate of this frequency is obtained from Fig . 2.8 by extrapolating the linear w/k relation near k = 0 up to k = Zn]«; this is essentially the method we used to estimate the frequency of lattice vibrations at the end of section 2.2, and therefore gives an answer in the infrared region of the electromagnetic spectrum.
2.4 LATTICE VIBRATIONS OF THREE-DIMENSIONAL CRYSTALS
Lattice vibrations.pdf
Department of Physics and Astronomy 2nd Year Laboratory
G10 1D crystal waves
Scientific aims and objectives
• To simulate experimentally the normal modes of oscillation in one dimensional monatomic and diatomic crystals using a linear air track
• To experimentally determine the spring constant and therefore bonding force constant in the experimental crystals
Learning Outcomes
• To use excel to “model” a complex system • To know the method to solve the propagation of a wave in a lattice, and to know
the form of the dispersion relation for the normal modes of one dimensional monatomic and diatomic lattices
Apparatus
• Air track carriages • Springs • Masses
Safety instructions
2nd year laboratory script G10 1D Crystal waves
September 2010 V1.0 Page 2
Task 1 - Pre-session questions The air track is an experimental one dimensional crystal that can sustain different types of oscillations, some of which correspond to the “normal modes” of a real, bound, finite one- dimensional crystal. The pre task questions require a relevant section of text from a solid state physics book by Hook and Hall to be read. The questions are directly linked to the text and will help you understand both the experiment and its analysis. A detailed understanding of this text is necessary and you should complete the questions before progressing.
Task 2 – The experiment The linear air track can simulate either a monatomic lattice when the same mass is placed on each carriage on the track, or a diatomic lattice with alternating masses on the carriages. The carriages should be connected together by springs and driven at one end by a motor whose angular frequency is controlled by a voltage dial. The angular frequency needs to be calibrated with respect to the voltage dial. You should first study the monatomic lattice with the objective of determining the spring constant of the system. The mass of the carriages can be measured directly and used along with the dispersion relationship ω = f(k) for the normal modes to determine the spring constant. The normal modes (the standing wave modes of the driven lattice) can be found by systematically varying the voltage and observing, firstly whether any stationary points are present, and secondly how many stationary points are present. The number of stationary points is the mode number of the standing wave. These normal modes occur when a fixed number of wavelengths fit into the lattice length: Na=nλ/2.
You can also investigate a diatomic lattice by placing additional mass on alternating carriages on the track. Hook and Hall covers the diatomic lattice theoretically and you should refer to the text to understand the expected dispersion relations. One aim could be to experimentally verify the derived dispersion relation in Hook and Hall. The interesting and important feature of the diatomic lattice is the presence of two solutions to ω for every value of k and how this feature relates to physical properties of the lattice such as the propagation of sound waves.
Task 3 – Reporting In contrast to other experiments in the second year lab your lab book and lab reports should give a summary account of the theoretical calculation of the dispersion relation of the 1D lattices you have studied. Your summary should outline the important assumptions and steps in the calculation as well as graphical descriptions of the functions that are derived. You should also show plots of your experimental data and fitted curves to determine K. If you have investigated the diatomic lattice as well then you can show qualitatively how the theoretical functions match with the experimental results.
2nd year laboratory script G10 1D Crystal waves
September 2010 V1.0 Page 3
Appendix of supporting information
1. Read the extract from Hook and Hall
“Section 2.3 Lattice vibrations of one-dimensional crystals”
2.3.1 Chain of Identical atoms
2.3.2 Chain of two types of atom
2. Pre-lab questions
Although these are called “pre-lab” questions, please use the experimental apparatus freely, alongside the text from Hook and Hall when attempting to answer the questions. Think of these questions as a guide to understanding the experiment.
a) When considering both the theoretical and experimental monatomic 1D lattice which of the following statements are true:
1. The theoretical lattice is bound at each end and can be considered to have a periodic boundary condition – this is exactly the same as the experiment
2. The theoretical lattice is bound at each end and can be considered to have a periodic boundary condition – this is different to the experiment where one end of the lattice is being actively driven
3. In the experiment only the conditions where standing waves are observed obey the dispersion relationship.
4. In the experiment the non-standing wave conditions correspond to solutions of a system of which the experiment is a part. i.e. a normal mode of a lattice that is larger than the air track and where the driving point corresponds to an arbitrary point on the wave and not a stationary point..
5. In the experiment the nodes of the normal standing wave modes must occur at the carriages in the lattice
6. In the experiment the nodes of the normal standing wave modes must never occur at the carriages in the lattice
7. In the experiment the nodes of the normal standing wave modes sometimes occur at the carriages and sometimes between the carriages
8. As the frequency of the normal mode oscillation is increased the wavelength of the oscillation continues to reduce indefinitely
2nd year laboratory script G10 1D Crystal waves
September 2010 V1.0 Page 4
9. As the frequency of the normal mode oscillation is increased it reaches a limit beyond which the system does not oscillate.
10. The relationship between normal mode frequency and wavenumber is continuous
11. The relationship between the normal mode frequency and wavenumber is quantised and only takes discrete values, limited by the lattice size, since a half integer number of wavelengths must fit precisely into the lattice
12. The solution to the equation of motion of the lattice is found by using a wavelike trial solution for one atom and seeing if it works.
13. Only nearest neighbour interactions are considered in the theory.
14. For the nearest neighbour interactions only the first term in the interaction expansion is considered.
b) When ω gets quite large it becomes increasingly difficult to observe the normal modes. It will be difficult to experimentally observe more that 4 or 5 normal modes for both the monatomic and diatomic chains. Why is this? What factors have not been accounted for in the theory calculations? Why do these factors become more important for the higher modes?
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2.3 LATTICE VIBRATIONS OF 'ONE-DIMENSIONAL CRYSTALS
2.3.1 Chain of identical atoms
The simplest crystal is the one-dimensional chain of identical atoms of equilibrium lattice spacing a shown in Fig. 2.2. We assume that the atoms can move only in the direction parallel to the chain, that they interact via an interatomic potential of the form shown in Fig. 1.23 and that the forces between them are of such short range that only nearest neighbour forces are important. In this limit the small amplitude motions of the chain of atoms are identical to those of the chain of identical masses M connected by identical springs of spring constant K shown on Fig. 2.3, as we shall now demonstrate. The interaction 1/(1') between nearest neighbours of separation r may, for a small deviation of r from its equilibrium value a, be expanded as a Taylor series about r = a. Thus
a)2 (d 21/)(1' "1/(1') = "I/(a) + -2 + .... (2.4) 2 dr r=a
No term linear in I' - a appears in the Taylor series because the first derivative of ''1' must vanish at the equilibrium spacing where "1/ is a minimum. If the
~ ~ ~ ~ ~ a
Fig. 2.2 One-dimensional crystal consisting of a chain of identical atoms
( 0)
'/1
u un-2 U U un-1 n, n+l n+2
Fig. 2.3 A chain of identical masses M connected by springs : (a) at equilibrium spacings
x~ = na; (b) at displaced positions x, = na + Un
2.3 Lattice vibrations of one-dimensional crystals 37
higher-order terms are ignored (this is the harmonic limit referred to in the introduction to this chapter), Eq. (2.4) looks like the potential energy associated with a spring of spring constant
d 2"1/)K= -2- . (2.5)( dr r=a
The spring constant can be simply related to the elastic modulus C of the one-dimensional crystal, defined by writing the force required to increase the interatomic distance from a to I' as qr - a)/a (i.e. force = elastic modulus x strain). The model of Fig. 2.3 predicts a force K(r - a) and this identifies the following relation between C and K:
C=Ka. (2.6)
We now proceed to calculate the lattice vibrations of the one-dimensional chain of Fig. 2.3, which exhibit many of the important qualitative features of lattice vibrations in general. We will use the laws of classical mechanics and postpone until section 2.5 a discussion of the difference in the results of a quantum mechanical calculation. We shall suppose that the chain consists of a very large number of atoms and that the last is joined to the first so as to make a ring. This last assumption is simply a device to make the chain endless so that all the atoms have an identical environment, and has no important effect on the problem for a long chain, where end effects are unimportant anyway.
If the displacements of the atoms from their equilibrium positions are u; as shown in Fig. 2.3, the force on the nth atom consists of:
(i) K(u - Un-I) to the left , from the spring on its left; andn
(ii) K(u + 1 - Un) to the right, from the spring on its right.n
Equating the total force to the right, (ii) - (i), to the product of mass and acceleration we have
MUn = K(u n + l - 2un + Un-I)' (2.7)
The equations of motion of all atoms are of this form, only the value of n varies. To solve Eqs. (2.7) we try a wavelike solution in which all the atoms oscillate at the same amplitude A. Thus we substitute]
Un = A exp [i(kx~ - cut)] (2.8)
where x~ = na is the undisplaced position of the nth atom, to obtain
_ (j)2MA ei(kna- wt) = KA(ei[k(n+l)a-wt]_ 2e i(kna-QJI) + ei[k(n-l)a- w,]),
t As usual when solving vibration problems by means of complex exponentials, it is the real part of u, that we interpret physically as the atomic displacement.
38 Crystal dynamics Chap. 2
or, on cancelling Aei(k na - wl) from each term ,
ika )- w 2 M = K(eika _ 2 + e
= 2K [cos (ka) - 1] .
w 2M = 4K sin? (~ka) . (2.9)
Fig. 2.4 shows the disper sion relation (relation between frequency wand wavenumber k) given by Eq. (2.9) for our wavelike lattice vibrations. The maximum value of s i n(~ka) is I so that the maximum possible frequency of the waves is 2(K/ M) 1/2 . Th is is known as the cut-off frequency of the lattice.
We notice that n has cancelled out in Eq. (2.9), so that the equ ations of mot ion of all atoms lead to the same algebraic relation between wand k. Thi s sho ws that our trial function for Un is indeed a solution of Eqs. (2.7). It is also imp ortant to not ice that we sta rted from the equations of motion of N coupled harm on ic oscillato rs (Eqs. (2.7)). If one atom starts vibrating it does not continue with constant amplitude, but tran sfers energy to the others in a complicated way; the vibrations of individual at oms are not simple harmonic because of this energy exchange amo ng them. Our wavelike solutions on the other hand are uncoupled oscillations called normal modes; each k has a definite w given by Eq. (2.9) and oscillates independently of the other modes. We should expect the number of modes to be the same as the number of equations N that we sta rted with; let us now see whether this is the case.
To do so we must esta blish which wavenumbers are possible for our one dimension al chain. Not all values are allowed because the nth atom is the same as the (N + n)th as the chain is joined on itself. This means that the wave (2.8) must sa tisfy the periodic boundary condition
un = uN +n (2.10)
which requires that there sho uld be an integral number of wavelengths in the
w
Fig. 2.4 Normal mode frequencies for a chain of identical atoms. Note that the modes with wavenumbers at A, B and C all have the same fre quency and correspond to the same instantaneous atomic displacements (see Fig. 2.5). Point B represents a wave moving to
_2!. o 7r 2JT k the right, points A and C a wave a a a moving to the left
2.3 Lattice vibra tions of one-dimensional crystals 39
length of our ring of a toms
Na = pI, .
k = 2n = 2np (2.11) Ie Na
where p is an integer. Th ere are thu s N possible k values in a range Zn]« of k, say the range
- n/a < k ~ x]a.
Fig. 2.4 shows that this restricted range of k does indeed include all possibl e values of the frequency wand the gro up velocity (dw/dk); it also gives th e N normal modes we expect for N atoms. What , if anything, is the physical significance of wavenumbers outside this rang e?
To understand th is, consider the instantane ous atomic displ acements shown in Fig. 2.5; we are really considering longitudinal waves, but the displacement s are shown as transverse in Fig. 2.5 because this makes their wavelike nature easier to visualize. Fig. 2.5(a) shows the displacement s for k = »[a, which gives
x
( b )
Fig. 2.5 (a) Atomic displacements (shown as transverse for clarity) for wavelength ..1. = 2a, wavenumber k = nla. (b) The atomic displacements for a wave with ), = 7a/4, k = 8n/7a, as given by the full curve, are identical to those for a wave with ..1.= 7a/3,
k = 6n/7a, as given by the broken curve
o ( 0)
40 Crystal dynamics Chap. 2
the maximum frequency; alternate atoms oscillate in antiphase and the waves at this value of k are essentially standing waves. The midpoint of each spring is at rest and each mass therefore behaves as if held by two springs each of spring constant 2K, giving the frequency 2(KjM)1/2 that we have calculated.
Now consider the displacements, shown by the full curve in Fig . 2.5(b), for the slightly larger value 8nj7a of k corresponding to the point A on Fig. 2.4. The displacements can also be represented by the longer wave, shown broken in Fig. 2.5(b), for which Ikl = 6nj7a; this corresponds to points B or C in Fig. 2.4. Thus, points A, Band C correspond to the same instantaneous atomic displacements as well as the same frequency. At B the group velocity dOJjdk > 0, so we have a wave travelling to the right; A and C both represent a wave travelling to the left and are thus completely equivalent. The k values of points A and C differ by 2nja and we therefore conclude that adding any multiple of 2nja to k does not alter the atomic displacements or the group velocity and is without physical significance. We need only consider the range - nja < k ~ nfa, which contains just the N modes we expected.
Further insight into what is special about the k values ± nta is gained by writing down Bragg's law (Eq. (1.3» for the one-dimensional crystal:
nA = n2njk = 2d sin e= 2a
or
k = nn la; (2.12)
where we have taken e= 90° and d = a as appropriate to waves travelling along a one-dimensional chain. Waves with k = ±nja will thus undergo Bragg reflection. The standing wave pattern that occurs at these two k values can be pictured as occurring because of Bragg reflection of running waves.
We note that in the long-wavelength limit, ka ~ 1, Eq. (2.9) reduces to
MOJ2 .: Kk2a2
so that in this limit the waves are dispersionless with group velocity and phase velocity (OJjk) both being equal to
v = a(K jM)1/2 (2.13)
These waves are long-wavelength sound waves and a calculation of their velocity from the macroscopic elastic properties of the crystal by the method used in the previous section yields a velocity (cf Eq. (2.3»
vs = (Cjp)1 /2 (2.14)
where p (= M ja) and C are respectively the mass per unit length and the elastic modulus of the crystal. Using Eq . (2.6), we confirm that Eqs. (2.13) and (2.14) are identical and thus that our more general calculation of lattice vibrations gives the correct answer in the long-wavelength limit. Note that, since there is
2.3 Lattice vibrations of one-dimensional crystals 41
only one possible propagation direction and one polarization direction, the one dimensional crystal has only one sound velocity. Given the velocity of sound and the lattice spacing it is possible to draw the dispersion relation for our simple crystal. This illustrates our statement in the introduction to this chapter that many of the properties oflattice vibrations can be deduced without detailed knowledge of the interatomic forces.
The inclusion of only nearest neighbour forces in our calculation appears very restrictive. Although it is a good approximation for the inert-gas solids, it is not a good assumption for many solids. The effects of removing this restriction can be investigated by using a model in which each atom is attached by springs of different spring constant to neighbours at different distances (see problem 2.1). When this is done, many of the features of the above calculation are preserved. The wave solution of Eq . (2.8) still satisfies the equations of motion. The detailed form of the dispersion relation is changed but OJ is still a periodic function of k with period 2nja and the group velocity vanishes at k = ±nja. There are still N distinct normal modes, which can be represented by the N possible k values in the range - nja < k ~ + nla. Furthermore the motion at long wavelengths corresponds to sound waves with a velocity given by Eq. (2.14).
2.3.2 Chain of two types of atom
We now consider the lattice vibrations of a chain containing two types of atom, of masses M and m, connected by identical springs of spring constant K as shown in Fig. 2.6. This is the simplest possible model of an ionic crystal,
(a )
(b)
a
Fig. 2.6 A chain containing two unequal masses connected by springs: (a) at equilibri um positions x~ = nal2; (b) at displaced positi ons x, = nal2 + un' Here a is the length of
the unit cell as indicated
.........
42 Crystal dynamics Ch ap. 2 2.3 Lattice vibrat ions of one-dimensional crystals 43
altho ugh the assumption of onl y nearest neighbour forces , impli cit in the model , is a poor approxima tion for ionic crystals because of the long ran ge of the Co ulomb interaction between ions. Fortunately the simple model again pro duc es the importa nt qu alit at ive features of the lattice vibrations of ioni c so lids . In section 9.1.4 we discuss the changes to the vibrations when the lon g-ran ge effects of the Co ulomb for ce are included.
To emphasize the more complicated motions that are possible when there is more than one type of a tom we show in Fig. 2.6(b) a configurati on in which the two types of at om ar e displaced in opposite directions. N ote that we use a to den ote the un it cell length (lattice spacing) of the crystal ; the nearest neighbour sepa ra tion of the undi splaced ato ms is a12.
Th e equa tions of moti on can be written do wn in the sam e way as Eq. (2.7) but there are now two distinct types of equation : those for masses M ,
MUn = K(un+I - 'Iu; + Un-I ) (2.15)
and those for masses m,
mUn_ 1 = Ktu; - 2un_1 + un- 2)· (2.16)
For the masses M we may ass ume as before a solution of the form (2.8), i.e.
Un = A exp [i(kx ~ - wt)]
where x~ = nal 2 is the undi splaced a tomic pos ition. There is now an extra unknown qu antity, the relat ive amplitude and phase of the vibra tions of the two types of atom; this we allow for by taking for the masses m
Un = ct.A exp [i (kx~ - wt) ] (2.17)
where ct. is a complex number giving the relative amplitude and phase. Substitution in Eqs. (2.15) and (2.16) then gives
_ w2Mei(kna /2 - wt ) = K(ct.ei[k(n +l )a/2 - wt] _ 2ei(kna/2- WI) + ct.ei[k(n- l)a/2-wt] )
and
_ ct.w 2mei[k(n- l)a/2-wtl = K(ei(kna/2- wt) _ 2ct.ei[k(n-l )a/2- wt] + ei1k(n- 2)a/2- wt])
or, by cancelling commo n factors as before,
- w2M = 2K[ct. cos (! ka) - I] (2.18)
- ct.w2m = 2K[cos (! ka) - ct.]
correct form. Eq s. (2.18) may be rewritten in the form
2K cos (!ka) 2K - w2M ct. = = -::-::-,--------,,.,...,---,- (2.19) 22K - w m 2K cos (! ka) ,
from which by cro ss-multiplication we obta in a qu adratic equat ion for w2 ,
4mMw - 2K (M + m)w2 + 4K 2 sin? (!ka) = 0, (2.20)
with solutions
2 K (M + m) [ ( M + m)2 4 . 2 ( Ik )J I/2 w = + K --- - - Sin "2 a . (2.21) Mm - Mm M m
The two roots are sketched in Fig. 2.7. As there are two values of w for each value of k the dispersion relati on is sa id to have two branches and the upper and lower branches in Fig. 2.7 corresponds to the + and - signs in Eq. (2.21) respectively. We see that chai ns contai ning two types of ato m sha re with tho se containing one the property that the dispersion relations are peri odic in k with period 2nla = 2nl(unit cell length); this result remain s va lid for a cha in containing an arbitra ry number of a toms per unit cell.
If the crystal contain s N unit cells we would expect to find 2N normal modes of vibration as this is the tot al number of ato ms and hence the total number of equ ations of motion (Eqs. (2.15) and (2.16» . Joining the end s of the crystal to form a rin g requires the ato mic displacements to sa tisfy the peri odic boundar y condition Un = U2N +n, leading to the same express ion for the possible k valu es,
k = 2npINa,
Thus instead of a single a lgebra ic equ ation for w as a function of k, we now have Fig. 2.7 Norm al mode frequencies of a cha in of two types of ato m. At A, the two types
a pair of algebraic equations for ct. and w as functions of k. As before th e fact that are oscillating in antiphase with their cen tre of mass at rest ; at B, the lighter mass m is /l does not appear in Eqs. (2.18) indicates th at our assumed solution is of the osci llating and M is at rest ; at C, M is osc illating and m is a t rest
44 Crystal dynamics Chap. 2 2.3 Latt ice vibrations of one-dimensio nal crys ta ls 45
as for the crysta l co nta ining a sin gle type of a to m. Thus th ere are again exactly N allowed values of k in the range - ttfa < k :::; n]a; also as in th e previous section, addi ng a ny multiple of Zn]« to k does not alter the a to mic d isplace ments, a nd we deduce that a ll the allowed motion s can be described by k va lues in th is range. Henc e the two br anches of the dispersion relation co ntai n 2N norm al modes as req uired.
It is instructi ve to examine the limiting solutions of Eq. (2.21) near the poin ts 0 , A, B and C in Fig. 2.7. For ka ~ 1, sin (tko,) ~ t ka and
w2 ~ K(M + 111) [1-:- (I_. ----'!!1~ _ k2a2)1/2J Mm - (M + m)2
K( M + 111)[( 111M 22)J ~ - -- 1 + I - - - - k a lV[ //1 - 2(M + m?
2K(M + m) Kk2a2
~ or . Mm 2(M + m)
By substituting these values of oi in Eq. (2.19) and ucing cos(~ka) ~ I for ka ~ I we find the co rrcspcnding va lues of (J. as
0: ~ - Mlm or I.
T he first so lution corresponds to point A in F ig. 2:7; this value of a co rresponds to M and m osci lla ting in a ntiphase with their cen tre of mass at rest, and th e freq uency is therefore given by the spring constant 2K a nd the red uced mass M* = MI11/(M + m). The second solu tion represents lon g-wav elength so und waves in the neighbourhood of point 0 in Fig . 2.7; th e two types of atom osci lla te with the sa me amplitude and phase, and the veloci ty of sound is
W ( K )1 /2 V - - a s - k - 2(M + m) .
This sound velocity mu st ag ree with that, namely (Cfp) 1/2 (E q. (2. 14» , pr ed icted fro m the macroscop ic elas tic properties of the cryst al ; substituting values of (M + m)/a and Ka/2 (cf. Eq. (2.6), recalling that the definition of a has changed) respecti vely for the mass per unit length p and th e elas tic modulus C into Eq. (2.14) co nfirms th at th is is so.
T he othe r limiting so lutions of Eq . (2.2 1) are for ka = n, i.e. sin(t ka) = I. In thi s case
2_ K(M + m) [(M + m) 2 4 JI/2 W - +K -- -
Mm - Mm Mm
= 2K/m or 2K/M
with , from Eq. (2.19), the co rresponding amplitude ratios 0: = co o r 0: = 0 respectively. In this limit the half-wa velen gth is a, the spacing bet ween a to ms of the same kind. In the first so lution m oscilla tes a nd M is a t rest (poin t B on Fig. 2.7 if M > m), a nd the freq uency therefore depends only on m; in the seco nd solution M oscillat es a nd 111 is a t rest (poi nt C on Fig. 2.7).
It is instructive to com par e our present results with those of sectio n 2.3.1 for a chain of one type of a to m. In F ig. 2.8 we plot the lower-frequency br an ch of Fig. 2.7 in the region k < nla and the higher-frequenc y branch in the region nfa < k < Zn]«. If we now let m -7 M the points B and C in Fig. 2.8 com e together and we recover Fig. 2.4, as we mu st. (D o not forget that the value of a in Fig. 2.4 is half th at in Fig . 2.8.)
There is thu s a certa in a rbit ra riness about how we assign k valu es to th e modes of a diat omi c lattice. The most direct compari son with a mon at omi c lattice is obtai ned with th e ass ignment sho wn in Fig. 2.8, where there is only one W for each k and there are 2N mod es in the ran ge - 2n/a < k :::; 2n/a. It is more usual, however, to ass ign th e lowest possible k to eac h mode as in Fi g. 2.7; th ere are now two br anches with N modes on eac h br anch in the range - n/a < k :::; nta. This latter approach is th e usual one and has the useful feature th at the ran ge of k va lues is 2n/( unit cell side), ind ependently of th e number of ato ms in a unit cell.
Although th e disp ersion relat ion is no lon ger given by Eq . (2.2 1) when th e restriction to nearest neighbour forces is removed, mo st of th e qu alitati ve conclusions con cerning the nature of the d ispersion relation that we have deduced a bove rem a in valid . In particular the dispersion relati on has two
w
2
o 11:
Fig. 2.8 Thi s shows Fig. 2.7 replotted for comparison with Fig. 2.4; in the limit m --> M points Band C come togeth er and the dispersion rela tion is given by the broken line. The
a tomic displacement s at A' are the same as those a t point A on Fig. 2.7
46 Crystal dynamics Chap. 2
branches, both periodic in k with period Zn]«. Only one of the branches has the limiting form of sound waves at long wavelength (w/k ---> constant as k ---> 0). This branch, the lower branch on Fig. 2.7, is consequently known as the acoustic branch. The other branch is called the optical branch because as k ---> 0 on this branch the vibrations of the two types of atom are in antiphase and the resulting charge oscillations in an ionic crystal give a strong coupling to electromagnetic waves at the frequency of point A on Fig. 2.7. An estimate of this frequency is obtained from Fig . 2.8 by extrapolating the linear w/k relation near k = 0 up to k = Zn]«; this is essentially the method we used to estimate the frequency of lattice vibrations at the end of section 2.2, and therefore gives an answer in the infrared region of the electromagnetic spectrum.
2.4 LATTICE VIBRATIONS OF THREE-DIMENSIONAL CRYSTALS
Lattice vibrations.pdf