Linear chain

Figure: Linear chain of particles separated by a distance $a$ and connected to their nearest neighbours by springs with constant $k$. The displacements from their equilibrium positions (un-stretched springs) are described by the variables $x_i$. The first particle of the chain is connected to the last, so that there are exactly $N$ elements in the chain.

We now want to find expressions for the vibrational frequencies $\omega_i$, so we can compute the partition function, but before tackling a three dimensional crystal it is useful to discuss a simpler system, made of a linear chain of particles connected by harmonic springs with spring constant $k$. The chain has $N$ elements and closes on itself with period boundary conditions (PBC), so that the first element of the chain is also the last. If the number of elements in the chain is large enough it does not matter if it is left open or closed with PBC, but the latter is mathematically convenient, as we shall see. The potential energy is:

$$U = U_0 + \frac{1}{2} k \sum_{n=0}^{N-1} (x_n - x_{n-1})^2 = U_0 + k \sum_{n=0}^{N-1} (x_n^2 - x_n x_{n-1}),$$ (7.12)

where $U_0$ is the potential energy of the system when all springs are un-stretched, which without loss of generality we can set equal to zero, and $x_n$ is the displacement of the $n^{th}$ element in the chain from its equilibrium position. Eq.  is not in the form of Eq. , however, with a proper transformation of coordinates it can be reduced to that form, and when this is achieved the new coordinates are called the normal modes of the system. Consider the following transformation:

$$X_l = \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} e^{2\pi i \frac{l n}{N}} x_n.$$ (7.13)

The transformed variables have the property:

$$X_{N+l} = X_l.$$ (7.14)

We can express $x_n$ by inverting :

$$x_n = \frac{1}{\sqrt{N}} \sum_{l=0}^{N-1} e^{-2\pi i \frac{l n}{N}} X_l,$$ (7.15)

which can be easily verified using the orthogonality relation:

$$\frac{1}{N}\sum_{l=0}^{N-1} e^{- 2\pi i \frac{l (n^\prime - n) }{N}} = \delta_{n,n^\prime},$$ (7.16)

where $\delta_{n,n^\prime} = 1$ if $n = n^\prime$ and zero otherwise. We see therefore that the $X_l$ represent collective excitations, in the form of waves on the chain with wavelength $\lambda_l = N a / l$, where $a$ is the distance between the particles on the chain. This can be seen by assigning a value to $X_l$ for a particular value of $l$ and zero for all other values and noting that, as a result of , $x_n$ is displaced by $e^{-2\pi i \frac{l n}{N}} X_l$, and so the displacement $x_{n+mN/l}$, with $1\le m \le l$, is the same as the displacement $x_n$ whenever $mN/l$ is an integer.

If we now substitute  into  we obtain:

$$U = k \sum_{n=0}^{N-1} \frac{1}{N} \sum_{l,l^\prime = 0}^{N-1} e^... ...pi i \frac{(l+l^\prime) n}{N}} e^{2\pi i \frac{l^\prime}{N}} X_l X_{l^\prime} =$$

$$k \frac{1}{N} \sum_{n=0}^{N-1} \sum_{l,l^\prime = 0}^{N-1} e^{-2\... ...ime) n}{N}} \left (1- e^{2\pi i \frac{l^\prime}{N}} \right ) X_l X_{l^\prime} =$$

$$k\sum_{l,l^\prime = 0}^{N-1} \left (1- e^{2\pi i \frac{l^\prime}{... ...\underbrace{\frac{1}{N} \sum_{n=0}^{N-1} e^{-2\pi i \frac{(l+l^\prime) n}{N}}}_ \delta_{l,-l^\prime} =$$

$$k\sum_{l = 0}^{N-1} \left (1- e^{-2\pi i \frac{l}{N}} \right ) X_l X_{-l}. %=$$ (7.17)

Because of , the sum in  can be extended over any $N$ consecutive integers, and for example for $l = -(N-1), 0$, so we also have

$$U = k\sum_{l = -(N-1)}^0 \left (1- e^{-2\pi i \frac{l}{N}} \right... ...{-l} = k\sum_{l =0}^{N-1} \left (1- e^{2\pi i \frac{l}{N}} \right ) X_{-l} X_l,$$ (7.18)

where the last equality is obtained by replacing $l$ with $-l$. By summing  and  we obtain:

$$U = \frac{1}{2} k\sum_{l =0}^{N-1} \left (1- e^{-2\pi i \frac{l}{N}} + 1- e^{2\pi i \frac{l}{N}} \right ) X_{-l} X_l =$$

$$k\sum_{l =0}^{N-1} \left [1 - \cos \left (2\pi \frac{l}{N} \right ) \right ] X_l X_{-l}.$$ (7.19)

Now we use the trigonometry identity

$$\cos(\alpha) = \left [ \cos \left(\frac{\alpha }{2}\right ) \right ]^2 - \left [ \sin \left(\frac{\alpha}{2}\right ) \right ]^2,$$ (7.20)

and so

$$1 - \cos \left (2\pi \frac{l}{N} \right ) = 2 \left [ \sin \left(\pi \frac{l}{N}\right ) \right ]^2,$$ (7.21)

which gives

$$U = \sum_{l =0}^{N-1} \frac{1}{2} M \omega_l^2 X_l X_{-l},$$ (7.22)

where $M$ is the mass of the particles and we have defined:

$$\omega_l^2 = 4\left (\frac{k}{M} \right ) \left [\sin\left(\pi \frac{l}{N}\right) \right ]^2.$$ (7.23)

From this we get the dispersion relation:

$$\omega_l = 2\left (\frac{k}{M} \right )^\frac{1}{2} \left \vert\sin\left(\pi \frac{l}{N}\right) \right \vert,$$ (7.24)

where we have taken only the positive solution of the square root in . In  the term $l=0$ is already a normal mode. Using  we see that the displacements $x_n$ are all equal:

$$x_n (0)= \frac{1}{\sqrt{N}} X_0,$$ (7.25)

and therefore the mode corresponds to a rigid shift of the whole chain. This means that the springs are not compressed or elongated, and if the chain is not attached to any external spring the mode has zero frequency. If $N$ is even, also the mode $X_{\frac{N}{2}}$ is already a normal mode. It corresponds to particles vibrating with the same amplitude but alternating directions:

$$x_n\left(\frac{N}{2}\right) = \frac{1}{\sqrt{N}} e^{-2\pi i \frac{n}{2}} X_\frac{N}{2} = \frac{1}{\sqrt{N}} (-1)^n X_\frac{N}{2}.$$ (7.26)

It is a stationary wave on the chain, and it is the mode with the largest possible frequency, equal to $\omega_{N/2} = 2\sqrt{k/M}$. For the other values of $l$ we need to further consider the real and the imaginary part of $X_l$. Let us define $A_l$ and $B_l$ through:

$$X_l = \frac{1}{\sqrt{2}} (A_l + i B_l),$$ (7.27)

$$X_{-l} = \frac{1}{\sqrt{2}} (A_l - i B_l),$$

which gives

$$X_l X_{-l} = \frac{1}{2} (A_l^2 + B_l^2).$$ (7.28)

Noting  we can re-write the potential energy (for $N$ even) as:

$$U = \frac{1}{2}M \omega_{N/2}^2 X_{N/2}^2 + \sum_{l=1}^{N/2-1} \frac{1}{2}M\omega_l^2 (A_l^2 + B_l^2).$$ (7.29)

This expression has the same form as Eq. , and $X_{N/2}, A_l, B_l$ with $1 \le l \le N/2 -1$, are the normal coordinates. They represent oscillations of the type of Eq. , $A_l(t) \propto \cos(\omega_l t + a_l)$ and $B_l(t) \propto \cos(\omega_l t + b_l)$, with the frequency $\omega_l$ determined by the dispersion relation  and $a_l, b_l$ some constant phases. Since $X_{-l} = X_l^*$ (Eq. ), we have $A_{-l}=A_l$ and $B_{-l}=-B_l$, and with no loss of generality we can take $A_l(t) \propto \cos(\omega_l t)$ and $B_l(t) \propto \sin(\omega_l t)$, so that $X_l \propto e^{i \omega_l t}$. For a particular mode $l$, the displacement $x_n$ of the particle at position $n$ on the chain is given by:

$$x_n(l) = \frac{1}{\sqrt{N}} e^{-2\pi i \frac{l n}{N}} X_l \propto \frac{1}{\sqrt{N}} e^{-i \left (2\pi \frac{l n}{N} - \omega_l t\right)},$$ (7.30)

and therefore these displacements oscillate with a time dependent phase given by $\omega_l t$ and a phase shift $-2\pi l n/N$, determined by their position on the chain. Eq.  represents a travelling wave with phase velocity $v_l = \frac{a \omega_l}{2\pi l /N} = \frac{\omega_l}{2\pi}\lambda_l$.

Figure: Dispersion relation for a linear chain of 20 particles, each one connected to its two nearest neighbours by harmonic springs.

This transformation of coordinates is useful, because we have seen that with the potential energy expressed in this simple form it is easy to compute the partition function of the system, which is given by Eqs. , for a quantum system, or by Eq.  in the classical limit. Indeed, all that it is needed to compute the partition function is knowledge of the frequencies . Defining the wavenumber $q_l = 2\pi / \lambda_l = 2\pi l / N a$, we can re-express the dispersion relation as:

$$\omega_l = 2\left (\frac{k}{M} \right )^\frac{1}{2} \left \vert\sin\left(\frac{a q_l}{2}\right) \right \vert,$$ (7.31)

which is plotted in Fig. . as function of wavenumber $q_l$. We see that this is a periodic function of $q_l$, which therefore only needs to be defined over one period. For convenience, it is useful to plot it in the range $-\pi/ a < q_l \le \pi / a$, which is known as the first Brillouin zone. The number of points defined in the dispersion relation is equal to the number of particles in the chain, and so the longer the chain the shorter the distance between the corresponding points in reciprocal space.

The derivative of the frequency w.r.t. the wavenumber gives the group velocity of a wave packet travelling on the chain, $v_g = d\omega_l / d q_l$. In the long wavelength limit $\lambda_l \gg a$ the system behaves as if it were a continuum medium. The waves have frequencies that are proportional to the wavenumber, and the constant of proportionality is their speed. As the wavelength is reduced the system starts to depart from the continuum picture, and the direct proportionality between frequency and wavenumber breaks down. At the shortest possible wavelength, equal to two times the inter-particle distance, the waves have zero velocity. These are described by the zone boundary modes, $X_\frac{N}{2}$, which represent stationary waves.

Note that expression  is not limited to the case of nearest neighbour interactions. Suppose we add a second spring, with spring constant $k_2$, connecting each oscillator on the chain with its second nearest neighbours, then the potential energy function becomes:

$$U = k \sum_{n=0}^{N-1} (x_n^2 - x_n x_{n-1}) + k_2 \sum_{n=0}^{N-1} (x_n^2 - x_n x_{n-2}),$$ (7.32)

and it is straightforward to show that one obtains  again, but now with:

$$\omega_l^2 = \frac{4}{M}\left [k \sin\left(\pi \frac{l}{N}\right) + k_2 \sin\left(2\pi \frac{l}{N}\right) \right ]^2.$$ (7.33)

Note that at zone boundary $l=\pm N/2$ the second term on the r.h.s. in the expression above does not contribute to the dispersion relation. This is not surprising, of course, because that is the mode with the shortest wavelength in the system, and corresponds to a standing wave in which second nearest neighbours particles move in phase. The distance between these particles therefore does not change, and the presence or otherwise of the $k_2$ spring does not make any difference to the dynamics of the system, as that spring remains unstretched.

The result  can easily be generalised to coupled harmonic interactions between any of the possible $N/2$ neighbours (for $N$ even), each with spring constant $k_m$, for which the potential energy function reads:

$$U = \sum_{m=1}^{N/2}\sum_{n=0}^{N-1} k_m (x_n^2 - x_n x_{n-m}).$$ (7.34)

Introducing the normal modes as in  and  this transforms into a sum of independent harmonic oscillators of the type  with

$$\omega_l^2 = \sum_ {m=1}^{N/2} \frac{4}{M}\left [k_m \sin\left(m \pi \frac{l}{N}\right) \right ]^2.$$ (7.35)