Vibrations on a Linear Atomic Chain of length LNa

On a chain of N masses of length L, and connccted by springs of constant K, denote the longitudinal displacement of the nth mass from its equilibrium position by un. The differential equation (Newton's Second Law) F =*ma for the nth mass is md2un/dt2 + K(un+i-2 « & (2.4)

A traveling wave solution to this equation is un » u0co$((i)t + kna). (2.5)

Here kna denotes kx^lnxjk, where k is referred to as the wave number. Substitution of this solution into the difference equation reveals the auxiliary condition mfi>2 = 4Ksin2(fctt/2). This "dispersion relation" is the central result for this problem. One sees that the allowed frequencies depend upon the wavenumber k = 2jr/A, as w**2(K/m)yz | sin{fea/2) |. (2.6)

The highest frequency, 2(K/m)1/2 occurs for kaj2 = Ji/2or k =7c/a; where the wavelength A=2a, and nearest neighbors move in opposite directions. The smallest frequency is at k = ji/Na = jt/L, which corresponds to L-/1/2. Here one can use the expansion of sinfx) = a: for small x. This gives m=2(K/ra)1/2ka/2 = a(K/m)l/2k, representing a wave velocity co/k = v = a(K/m)lu.

Comparing this speed with v - (Y/p)1/2 for a thin rod of Young's modulus Y, and mass density p, we deduce that Y/p= Ka2/m, Here K and a are, respectively, the spring constant, and the spacing a of the masses. [1]

Young's modulus can thus be expressed in microscopic quantities as Y =pKa2/m if the atoms have spacing a, mass m, and the interactions can be described by a spring constant K.

Table 2.1 Mechanical properties of some strong solids

Material

Young's modulus Y (CPa)

Strength (GPa)

Melting point (K)

Density p (kg/m3)

Diamond

1050

50

1800

3500

Graphite

686

20

3300

2200

SiC

700

21

2570

3200

Si

182

7

1720

2300

Boron

440

13

2570

2300

ai2o3

532

15

2345

4000

Si3N4

385

14

2200

3100

Tungsten

350

4

3660

19300

A cantilever of length I clamped at one end and free at the other, such as a diving board, resists transverse displacement y (at its free end, x = L) with a force -Ky. The effective spring constant K for the cantilever is of interest to designers of scanning tunneling microscopes and atomic force microscopes, as well as to divers. The resonant frequency of the cantilever varies as I-2 according to the relation q) = 2k (0.56/L2)(YI/pA)1/2. Here p is the mass density, A the cross section area and I is the moment of the area in the direction of the bending motion. If t is the thickness of the cantilever in the y direction, then IA = ¡A(y)y2dy = wt3/12 where w is the width of the cantilever. It can be shown that K = 3YI/L3. For cantilevers used in scanning tunneling microscopes the resonant frequencies are typically 10 kHz - 200 kHz and the force constant K is in the range 0.01 - 100 Newtons/m. It is possible to detect forces of a small fraction of a nanoNewton (nN).

Cantilevers can be fabricated from Silicon using photolithographic methods. Shown in Figure 1.1 is a "nanoharp" having silicon "wires" of thickness 50 nm and lengths I varying from 1000 nm to 8000 nm. The silicon rods are not under tension, as they would be in a musical harp, but function as doubly clamped beams. The resonant frequency for a doubly clamped beam differs from that of the cantilever, but has the same characteristic Independence upon length: aj = (4.7 3 / L)2 (YI/pA)1/2. The measured resonant frequencies in the nanoharp structure range from 15 MHz to 380 MHz.

The largest possible vibration frequencies are those of molecules, for example, the fundamental vibration frequency of the CO molecule is 6.42xl013Hz (64.2 THz). Analyzing this vibration as two masses connected by a spring, the effective spring constant is 1860 N/m.

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