17.1 Introduction

The basis of electrical phenomena is the Coulomb force –– the force exerted by a charged particle on other charged particles. The electric field due to a collection of charged particles is defined as the force on a small “test” particle, such as an electron. The charge of the test particle is assumed to be so small as to allow virtually no perturbation of the electric field due to the test particle’s presence. In practice, this is almost always an excellent approximation for use with low-density electron charge distributions (low density here means that the motion of the electrons is not influenced by that of the other electrons). Typically, electronic devices are built in high-vacuum containers (CRT displays, scanning electron microscopes), and electron motion in a static electric field is an exercise in solving Newton’s equations. We solve the following equation, usually numerically, where is the vector of the position coordinates of the charged particle:

(17.1)

When the electric field forces are strong enough to actually move one of the electrodes, then we cannot use equation (17.1), which assumes that the electric field is constant in time. Instead, we must use the principle of virtual work,¹ which states that for a movement (of an electrode or some other conductor or dielectric that contributes to the stored energy calculation), in the X direction, for example, we obtain the following equation, where U_E is the stored energy in the electric field.

(17.2)

When U_E increases with a movement in the +X direction, equation (17.1) predicts a positive force; therefore, F_x is the force pushing the electrode in the +X direction, and the work done by this force increases the energy stored in the electric field.

In the strictest definition, since the electric field is changing with time, we are no longer dealing with an electrostatic system. In practical-scale systems, however, the speed of mechanical motion of masses is so low in comparison to the speed of light that we are basically still dealing with an electrostatic model with parameters changing very slowly. A numeric value that is often quoted to justify this assertion is the approximate ratio of the speed of light to the speed of sound (elastic wave propagation), which, in most solids, is equal to 10⁵. Sometimes the term quasistatic is used to summarize a situation where the electric field is changing as a result of movement(s) in the structure over the time period of interest, but these movements are so slower than the speed of light that electrostatic analysis is still valid.

17.2 Electron Beam Acceleration and Control

Analyzing an electron (or an electron beam) trajectory in a vacuum is accomplished by first finding the electric field distribution in the structure and then solving equation (17.1) numerically. Equation (17.1) is a vector equation, meaning that there are two equations for a two-dimensional (2d) problem, three equations for a three-dimensional (3d) problem, and so on. Since these equations do not couple, their solutions are handled by solving them separately; consequently we are solving ordinary differential equations (ODEs). MATLAB has many tools available for this task,² and online tutorials are available.^3–5

To keep things simple and not get diverted into a study of numerical ODE solutions, we’ll use the simplest numerical integration technique available, namely, equal forward time steps.

Since

(17.3)

(the x component of) equation (17.1) may be rewritten as

(17.4)

For small time steps Δt, we obtain

(17.5)

and then

(17.6)

Consider the 2d structure shown in Figure 17.1. The outer box consists of an X = 0 wall set at 0 V, an X = h wall set at V_a volts, and two Y walls (the exact dimensions don't matter) on which the voltage is linearly graded between 0 and V_a volts.

Electrons will be inserted at (0,0) (the cathode) at essentially 0 velocity. They will be accelerated to X = h (the anode) by the anode potential V_a (see Figure 17.1). When they strike the anode, each electron will have acquired -eV_a energy, where e is the electron charge.

Setting the energy put into an electron (which started at rest) is equal to its kinetic energy

(17.7)

we can find the velocity of the electron:

(17.8)

If the electron has velocity components in y and/or z as well as in x, then the total velocity must be used to calculate the kinetic energy.

As an aside, kinetic energy in meter–kilogram–second (mks). units is expressed in joules. In practice, an electron’s energy is often expressed in electronvolts (eV). One electronvolt is the amount of energy gained by an electron when it is accelerated through a one-volt potential difference. Numerically, it is one joule multiplied by the charge on one electron: 1 eV = 1.6 × 10^-19 joules (J).

Returning to Figure 17.1, an electron staring at rest at (0,0) will accelerate toward the anode, passing through a small slit in an electrode whose voltage is determined by its placement so as to cause minimal deviation in the electric field:

(17.9)

The purpose of the slit electrode will be discussed below.

Next, the electron passes between two parallel plates (control electrodes) whose voltages are set using a combination of two parameters, V _f and V_s , where

(17.10)

The MATLAB program electrons_1.m solves the electrostatic problem for the structure described above and then calculates and displays the electron’s trajectory:

• [MATLAB script electrons_1.m]—electron trajectory calculation program:

  % electrons_1.m  2d electron steering and focusing
  
  clear; close all;
  
  q = -1.602e-19;   m = 9.109e-31; % electron properties
  
  global x_max;   global y_max;
  y_max = 100; % height of the box, (nr of rows = 1st indec)
  x_max = 500; % with of the box, (nr of cols = 2nd index)
  Va = 1000; % anode voltage
  
  n_max = x_max*y_max;
  a = spalloc(n_max,n_max,5*n_max); % coefficient array
  b = zeros(n_max,1); % forcing function array
  
  % set up internal node coefficient array
  for i = 2 : y_max - 1
   for j = 2 : x_max - 1
    n = get_n(i,j);
    a(n,n) = -4;
    a(n,n-1) = 1;
    a(n,n + 1) = 1;
    a(n,n + x_max) = 1;
    a(n,n-x_max) = 1;
   end
  end
  
  % set up outer box bcs
  for i = 1 : y_max
   n = get_n(i,x_max); % anode end
   a(n,:) = 0; a(n,n) = 1;  b(n) = Va;
   n = get_n(i,1); % cathode end
   a(n,:) = 0; a(n,n) = 1; b(n) = 0;
  end
  
  % Graded walls
  for j = 1 : x_max
   n_low = get_n(1,j);
   n_high = get_n(y_max,j);
   Vj = Va*(j-1)/(x_max-1);
   a(n_low,:) = 0; a(n_low,n_low) = 1; b(n_low) = Vj;
   a(n_high,:) = 0; a(n_high,n_high) = 1; b(n_high) = Vj;
  end
  
  % emission slit
  x_slit = 20;
  y_slit_1 = y_max/2-2; y_slit_2 = y_max/2 + 2;
  V_slit = Va*(x_slit-1)/(x_max-1);
  for i = 2 : y_slit_1
   n = get_n(i,x_slit);
   a(n,:) = 0;
   a(n,n) = 1;
   b(n) = V_slit;
  end
  for i = y_slit_2 : y_max-1
   n = get_n(i,x_slit);
   a(n,:) = 0;
   a(n,n) = 1;
   b(n) = V_slit;
  end
  
  % control electrodes
  x1 = x_slit + 90; x2 = x1 + 50;
  y_up = y_slit_2;
  y_down = y_slit_1;
  V_bal = 220; V_diff = 0.3;
  V_up = V_bal + V_diff/2; V_down = V_bal - V_diff/2.;
  for j = x1 : x2
   n = get_n(y_up,j);
   a(n,:) = 0; a(n,n) = 1; b(n) = V_up;
   n = get_n(y_down,j);
   a(n,:) = 0; a(n,n) = 1; b(n) = V_down;
  end
  
  % solve it
  V = a;
  % An xy profile will be very useful
  
  Vxy = zeros(x_max, y_max);
  for i = 1 : y_max
   for j = 1 : x_max
    n = get_n(i,j);
    Vxy(i,j) = V(n);
   end
  end
  
  % ---------- electron trajectory section
  
  dt = 1.e-8;
  
  % Set up a picture of the structure then add the trajectories
  
  figure(1)
  axis([0, x_max, y_max/2-25,y_max/2 + 25])  % outer box
  hold on
  line ([x_slit,x_slit], [y_max/2-25, y_slit_1], 'Color', [0 0 0])
  line ([x_slit,x_slit], [y_slit_2, y_max/2 + 25], 'Color', [0 0 0])
  line ([x1,x2],[y_up,y_up], 'Color', [0 0 0]);
  line ([x1,x2],[y_down,y_down], 'Color', [0 0 0]);
  xlabel ('X')
  ylabel ('Y')
  
  angs = [-.3, -.2, -.1, .1, .2, .3];
  for angle = angs   % degrees
   angle
  
   x = [1];  y = [y_max/2];  vx = [0];  vy = [0];
   t = 0;
   set_angle_flag = 1;
   while x(end)  <  x_max & y(end)  <  y_max & y(end) > 0
    t = [t, t(end) + dt];
    [Ex,Ey] = get_E(x(end),y(end),V);
    vx = [vx, vx(end) + q*Ex*dt/m];
    x = [x, x(end) + vx(end)*dt];
    vy = [vy, vy(end) + q*Ey*dt/m];
    y = [y, y(end) + vy(end)*dt];
  
    if x(end) > = x_slit & set_angle_flag == 1
     set_angle_flag = 0;
     v = sqrt(vx(end)^2 + vy(end)^2)
     vx(end) = v*cos(angle*pi/180);
     vy(end) = v*sin(angle*pi/180);
    end
   end
  
   plot(x(1:100:end),y(1:100:end),'k')
   y(end)
  end

• [MATLAB function get_n.m]:

  function n = get_n(i,j)
  
   global x_max; global y_max;
  
   n = j + (i-1)*x_max;
  
  end

• [MATLAB function get_ij.m]:

  function [ i_out,j_out ] = get_ij(n)
  
   global x_max; global y_max;
  
   i = ceil(n/x_max - 1);
   j = n - 1 - i*x_max;
   i_out = i + 1;
   j_out = j + 1;
  
  end

• [MATLAB function get_E.m]:

  function [ Ex, Ey ] = get_E(x,y,V)
  % Calculate the electric field based upon position
  
   global x_max; global y_max;
  
   j = floor(x); i = floor(y);
   n1 = get_n(i,j); V1 = V(n1);
   n2 = get_n(i,j + 1); V2 = V(n2);
   n3 = get_n(i + 1,j + 1); V3 = V(n3);
   n4 = get_n(i + 1,j); V4 = V(n4);
  
   x_l = x - j; y_l = y - i;
   Ex = (V1-V2)*(1-y_l) + (V4-V3)*y_l;
   Ey = (V1-V4)*(1-x_l) + (V2-V3)*x_l;
  
  end

The structure parameters, as defined in this program, are

• Outer box: x_max = 500, y_max = 100
• Slit: x_slit = 20, slit_width = 4
• Control electrodes: x1 = x_slit + 110, x2 = x1 + 140, separation = 4

Script electrons_1.m uses a simple finite difference (FD) program and voltage/field interpolation scheme based on the analyses presented in Chapter 10.

For the parameters as listed and Δt = 1 × 10^-8, electrons_1.m predicts an electron velocity at the anode of 1.8755 - 10⁷ (m/s), which is [see equation (17.2)] correct to five decimal places. Note that the dimensions chosen for this example are somewhat unrealistic –– a 500-m-long vacuum chamber is not a common item. These numbers were chosen for convenience in providing an example. In practice, the length of the vacuum chamber could be approximately 0.25 m; everything else (including Δt) would be scaled accordingly.

A real electron source does not generate all electrons at exactly zero energy and at exactly the same place. The purpose of the slit is to provide for an accelerating region (to the left of the slit) where the ideal electron source’s electrons can travel in the +X direction, unperturbed by anything happening at the control electrodes. At the slit, we perturb the Y velocity of the electrons by a chosen amount while maintaining its total velocity. Examining the trajectories of electrons beginning at different small angles, distributed about zero degrees, allows us to gain some insight as to the real issues and techniques involved in an electron beam structure such as a cathode ray tube (CRT).

When V_s , the scanning voltage, is zero, both control electrodes are at the same focus voltage V_f .

In all three diagrams (Figure 17.2a–c), electrons were started with initial angles of ± 0.3°, ± 0.2°, and ± 0.1°. In Figure 17.2a, V_f   =  220 V and has very little effect on the electron trajectories. In Figure 17.2b, V_f  = 140 V; the outer electron trajectories are being bend toward the anode, resulting in a tighter bunching of trajectories at the anode. In Figure 17.2c, V_f  = 120 V; the outer trajectory electrons are being bent too far –– their trajectories cross each other on the way to the anode, resulting in a poorly bunched distribution of electrons.

This electron behavior is in many ways analogous to optical focusing, and is called electron optics.⁶ The control electrodes used here are a crude form of an electron lens that will have a defined focal length. In practice, an electron lens will be cylindrically symmetric and may be composed of several elements.

Once the electron beam has been focused, it must be aimed at a desired location on the anode. This is accomplished here by the voltage V_s , which is applied differentially to the control electrodes. Figure 17.3 is a repeat of Figure 17.1 but with V_f  = 0.3 V. The differential scanning voltage has relocated the electron beam location at the anode in the +Y direction.

An actual electron lens system is much more complicated than the simple example portrayed here. While the focusing structure is cylindrically symmetric, there must be both X and Y deflection electrodes so that any desired position on the anode may be addressed by the electron beam.

A problem with electrostatic beam focusing and steering arises when high anode voltages (25 KV is typical) are used in structures such as CRTs. Since the electrons accelerate to much higher velocities at higher anode voltages, they spend less time in the region of any given electrode structure, and hence these electrode structures lose effectiveness.

A practical answer to this problem is to abandon electrostatic focusing and use magnetic field focusing. In a magnetostatic field

(17.11)

where B is the magnetic field. Since the force on the electron is proportional to its velocity (and the field strength), this system is effective at high anode voltages. Inspection of the back of a CRT television, for example, will reveal a doughnut-shape coil structure called a yoke wrapped around the neck of the CRT. The yoke provides the magnetic field necessary for controlling the electron flow accurately and effectively.

17.3 The Electrostatic Relay (Switch)

In the context here, a relay is an electrically controlled electrical switch. A typical commercial relay is shown (schematically) in Figure 17.4.

As shown in the figure, two electrical contact points are positioned facing each other. One of the contacts is mounted on a rigid arm; the other, on an arm that can pivot (move). The contacts are separated by a spring that is connected to the latter (pivoting) support. When the electromagnet is energized by applying a voltage to the control wires, the pivoting arm is drawn toward the electromagnet until the contacts touch and the switch is closed.

A common variation on this structure is for the pivot mount to be replaced by a fixed mount and the spring eliminated. The upper arm is made of a springy material that at rest keeps the contacts apart. The magnetic force causes the upper arm to bend until the contacts touch.

Historically, electrical relays have always used magnetic force to move one of the contacts. When arm sizes can be measured in millimeters, electrostatic forces at practical voltages are simply too small to be of use.

The advent of microelectromechanical machining (MEMM) has made the electrostatic relay a practical device.^7,8

Figure 17.5 shows, schematically, the essential components of a MEMM relay. Two conducting plates of surface area A face each other. One of the plates is rigidly mounted; the other is suspended by a spring with a spring constant K. At equilibrium, there is zero force compressing or extending the spring, and the two plates are separated by a distance g (gravitational force is ignored). When a voltage V is a applied between the plates, the Coulomb force causes the upper plate to move toward the lower plate. The instantaneous position of the upper plate is u(y):

(17.12)

Assuming that the plate separation is small compared to A, we can approximate the capacitance between the plates using the ideal parallel plate capacitor relationship:

(17.13)

The energy stored in the electric field is

(17.14)

The force on the upper plate is a combination of the spring force and the Coulomb force:

(17.15)

The equilibrium position s is the position where the spring force and the Coulomb force balance:

(17.16)

It is convenient at this point to normalize this equation. Let

(17.17)

and

(17.18)

Substituting equations (17.17) and (17.18) into (17.16), we obtain

(17.19)

Equation (17.19), describing the equilibrium position of the upper plate(s) as a function of voltage (βV), is plotted in Figure 17.6.

Starting at (0,0), the displacement s is zero when the voltage βV is zero. As the voltage increases (note that the sign of the voltage does not matter), the displacement increases until it reaches a peak of . Further increases of displacement until s = -1 (the electrodes are touching) require less voltage, until again zero voltage is required for the electrodes to touch.

This unusual behavior is better understood by reference to Figure 17.7.

Figure 17.7 shows the net force on the movable electrode plotted against the electrode’s position for several voltages. Consider first the upper curve, (βV)² = 0.25. This curve crosses the net force = 0 line twice, at s ≈ -0.08 and s ≈ -0.70; thus, these values of s are equilibrium points for this voltage.

At s ≈ -0.08, the slope of the force curve as it passes through equilibrium is such that the moving electrode is pushed toward this point. If s is not large enough, the force is negative so as to push the electrode down, decreasing s (remember, s is negative). If s is too large (too negative), the force is positive, pushing s toward zero. Hence s ≈ -0.08 is a stable equilibrium point. This is analogous to a ball rolling into a depression in the floor –– small movements result in a force returning the ball to the depression.

At s ≈ -0.70, the forces are directed so as to move the electrode away from the equilibrium point. If the electrode is moved a small amount toward s = 0, the positive force will return the electrode to the stable equilibrium point. If the electrode is moved a small amount toward s = -1, the negative force will send the electrode toward contacting the fixed electrode. This is called an unstable equilibrium condition, analogous to balancing a ball on the tip of a pencil.

As βV increases, the stable equilibrium point moves toward more negative values of s, as shown in both Figures 17.7 and 17.10. At , however, Figure 17.7 shows that there are no longer any equilibrium points; the force is always negative, driving the moving electrode down toward the fixed electrode.

If the two electrodes touch, the voltage across them is, of course, short-circuited and goes to zero, causing the electrodes to spring apart. The mass of the moving electrode in conjunction with the spring have a resonance frequency that determines how quickly the electrodes move apart and then start coming together again (as soon as they’re apart, the voltage is reestablished). The magnetic version of this is known as a “door buzzer.”

If a thin dielectric layer is added to either of the electrodes, the voltage is never short-circuited and at a high enough voltage, the electrodes will “latch” at s = -1, squeezing on the dielectric layer. This device is, of course, no longer a switch –– the electrodes never make contact.

Figure 17.8 shows one way of designing an electrostatic relay that circumvents the abovementioned problem. The structure in Figure 17.8 has separate control and switch wire connections, as did the magnetic relay shown in Figure 17.4. This is an important property for a useful relay –– it is almost always desirable to have the control and switched circuits isolated from each other.

In this device, the moving electrode and spring have been combined into a single cantilevered arm. The springiness of the arm keeps the contacts apart when no voltages are present. When a sufficiently high control voltage is applied, the arm bends downward until the switch contacts touch, preventing further motion. A limitation of this structure is that the voltage being switched must not be so high as to cause significant electrostatic force. Since the force is proportional to the cross-sectional area of the electrodes, making the switch electrodes’ surface area much smaller than the control electrodes’ surface area (assuming both voltages to be approximately equal) is usually adequate to achieve this goal.

The structure in Figure 17.8 may also be used as a variable-voltage capacitor. Adjusting the electrode gaps so that the control voltage can be conveniently set to levels below allows for varying s between 0 and with the capacitance between the switch electrodes varying accordingly.

The basic electrostatic switch structure can be modified to allow for several other applications. Since the moving mass–spring system has a natural mechanical resonance, an electric oscillator circuit can be coupled to the device to create an electromechanical resonator. Such a resonator has the interesting property that the resonance can be varied, or “tuned” by adjusting the control voltage since the electrostatic force due to the control voltage interacts with the spring constant and creates an “effective” spring constant.⁷

If a voltage is applied to the basic electrostatic switch structure to bring the movable electrode to, say, s = , then a stable equilibrium is created. If the voltage source used to establish this bias condition has very high internal impedance, then it behaves like a current source for times which are short as compared to the RC time constant of the source–capacitor system (see Chapter 2 for a discussion of RC networks and time constants).

Now, assume that the movable electrode is a thin membrane or an electrode mounted on a thin membrane. Vibrations in the air near this membrane will cause the membrane to vibrate, which, in turn, will cause the voltage across the capacitor to oscillate. This oscillating voltage can be sensed, and we now have a device known as a DC condenser microphone. [Notes: (1) the abbreviation DC (direct current) is commonly applied for any electrical system with fixed voltages present, (2) condenser is another (old) name for capacitor].

Figure 17.9 shows (schematically) another approach to building an electrostatic capacitor/position sensor.

The structure in Figure 17.9 is an interlaced “comb” of moving and fixed electrodes. Since the capacitance (for motion near the position shown) is due primarily to the sidewall to sidewall proximity of the electrodes, the capacitance is, to first order, a linear function of position.

For Figure 17.9, let X ₀ be the position of the fingers when the spring is at rest. For a system of n fingers of thickness w and a finger–finger gap of g, we obtain

(17.20)

The electric field stored energy is

(17.21)

and the Coulomb force is

(17.22)

The force on the spring is

(17.23)

Setting the sum of these forces equal to zero, we obtain

(17.24)

The response is, of course, always downward (an attractive force result), but in this case x is linear in V² with no unstable region. If a small bias voltage V ₀ is applied and then a small AC. signal is superimposed on this bias voltage, the response (x) will be approximately linear. Conversely, if a small motion is applied (accelerometer or microphone) when there is a bias voltage present, an approximately linear response voltage will be developed.

17.4 Electrets and Piezoelectricity: an Overview

When a conventional dielectric material is polarized (e.g., used as the dielectric layer in a capacitor that has a voltage across it), the dielectric material is polarized––that is, a dipole moment is induced in the dielectric that serves to reduce the electric field in the dielectric. This dipole moment, in turn, produces a surface charge which, in turn, increases the capacitance of the capacitor from its airdielectric value. When the voltage is removed, the polarization and the surface charge vanish.

In some cases, if the dielectric is heated sufficiently and then cooled back to room temperature with the charging voltage applied, the polarization “freezes” into the material. Even after the charging voltage and electrodes are removed, the polarization, charge, and resulting external electric field remain. This material, in this state, is called an electret. An electret’s properties are the electrical analog of the magnetic field and magnetic moment that are“frozen into” a permanent magnet.

If the airgap in a DC condenser microphone is partially replaced with electret material (leaving the diaphragm electrode free to vibrate), then no bias, or polarizing, voltage is necessary. Note that this is only one of several possible configurations for an electret microphone.⁹

Certain crystalline materials contain aligned dipole moments along anisotropic axes. Quartz is an example of such a material. When a voltage is applied across the material (in the proper axis), the dipoles try to align with the applied field and the material stresses. This stress could be microscopic or quite macroscopic, depending on the material. If the same material is mechanically stressed (again, choice of axes is important), the dipoles rotate and a charge is developed at some surfaces. If these surfaces are electroded, a voltage is developed between them.

These materials are called piezoelectric. The fact that the piezoelectric effect is bidirectional (with voltage producing stress and stress producing voltage) is derivable from basic thermodynamic considerations.

There is a second category of materials, made up of some ceramics, which can be polarized into a piezoelectric state in the same manner as electrets are polarized. Barium titanate is an example of these materials. These materials are typically referred to as poled piezoelectric materials.

The piezoelectric effect is used in loudspeakers and micropositioners (voltage to stress) and igniters for flame ignition (stress to voltage). A carefully dimensioned plate or tuning fork of piezoelectric material can be excited to vibrate at one of its natural mechanical resonance frequencies and, coupled to an electrical oscillator circuit, will oscillate and act as a frequency reference. Every quartz clock and watch made contains such a quartz crystal resonator.

The equations governing piezoelectricity are coupled elastic wave equation and electrostatic equations. Since the ratio of the speed of light to the speed of sound in most materials is approximately 10⁵, the time derivatives of mechanical variables (stress, strain) can be significant, while the time derivatives of electrical variables can be totally insignificant, thereby making this a true electrostatic-realm situation.

Elastic wave analysis involves tensor mathematics. These are not necessarily difficult analyses, but the background discussion requires several chapters in a book and are outside the scope of this text. The interested reader is directed to the references.^10–12

17.5 Points on a Sphere

A practical problem that arises often in many disciplines is the need to uniformly disperse some number of points (n > 1) on a sphere. At the outset we must realize that thus far this is an inadequately defined problem. The first issue is that a given distribution of points can be moved (“slid”) around on the surface in two axes. We resolve this issue by first fixing one point arbitrarily, say, at the north pole: (⊖,ϕ) = (0,0). This eliminates the ⊖ sliding problem. Then we constrain the second point in ϕ; say, to φ = 0.

We can now start writing solutions. Some of the solutions can be written easily:

• For n = 2: (⊖,ϕ) = (0,0), (π,0)
• For n = 3: (⊖,ϕ) = (0,0), (π/3,0), (π/3,π)
• For n = 4: (⊖,ϕ) = (0,0), (π,0), (π/2,0), (π/2,π)
• For n = 5: (⊖,ϕ) = (0,0), (π/2,0), (π/2,2π/3), (π/2,4π/3), (π,0)
• For n = 6: (⊖,ϕ) = (0,0), (π/2,0), (π/2, π/2), (π/2, π), (π/2, 3π/2), (π,0)

At n = 7 we find several problems preventing us from writing a solution set. Simple symmetry arguments are no longer of any use. We need a definition of uniformly distributed. There is the possibility that, for a given definition, an exact solution might not exist. If an exact solution does exist, it might not be unique. If an exact solution does not exist, can we define a best approximation, which also might not be unique. Finally, once we have our definition in order, can we create an algorithm to produce the array of points?

One way to approach this problem is to convert it to an electrostatic problem. Assume that each point on the surface of the sphere has a charge of +1 and is free to move about the surface (obeying the two constraints given above). The points will arrange themselves so as to minimize the energy of the system. Since the potential between two points is 1/r, where r is the geodesic distance (the straight line cutting through the sphere), the penalty for being too close to some point outweighs the benefit of being too far from some other point, and the points should settle themselves in a compromise distribution, which we can define as uniformly distributed.

For r_i,j the distance between points i and j, in an arbitrary set of units, we obtain

(17.25)

The potential of point i is

(17.26)

and the total potential energy of the system, again in arbitrary units, is

(17.27)

The –– or possibly a –– distribution that minimizes V _tot is our desired distribution.

Numerical search procedures that find an extreme of a figure of merit such as V _tot in the expression above form a sophisticated field by themselves.¹³

To consider the distributions generated using this definition without being sidetracked by the details of the search algorithm, the program points_on_sphere.m was written using a two-phase random search algorithm:

1. Try 10,000 random combinations of (θ,φ), evaluate V _tot, and save the best results (the distribution yielding the lowest value of V _tot.)
2. Iterate (an additional) 40,000 times using a normal distribution of random numbers with a mean of the latest best solution and a sigma (σ) of 1% of this same mean.

Here is the file points_on_sphere.m –– a program to generate a uniform point distribution on a sphere:

% playing with locating points on a sphere by minimizing electrostatic
% energy

close
global radius
radius = 1.0;

n = 19;   % total number of points

V_max = 1.e20;
nr_tries = 50000

for i_try = 1 : nr_tries

% point one is fixed by default to (theta,phi) = (0,0)
% point two is partially constrained to (theta,0)
 if i_try < 10000
  t = acos(2*rand(n,1) - 1); t(1) = 0;
  p = 2*pi*rand(n,1); p(1) = 0; p(2) = 0;
 else
  t = t_max + .01*t_max.*randn(n,1);
  if t > pi/2, t = pi - t; end;
  p = p_max + .01*p_max.*randn(n,1);
  if p > 2*pi, p = 4*pi - p; end;
 end
 [thetas,phis] = meshgrid(t,p);

 rij = dist(thetas, phis, thetas', phis'),
 rij(1: n + 1 : end) = Inf;   % set diagonal terms to infinity
 Vij = sum(1./rij);
 V_tot = sum(Vij);

if V_tot < V_max
  V_max = V_tot;
  t_max = t;
  p_max = p;
 end

end
V_max

x_plot = radius.*cos(p_max).*sin(t_max);
y_plot = radius.*sin(p_max).*sin(t_max);
z_plot = radius.*cos(t_max);

Tri = fliplr(convhulln([x_plot,y_plot,z_plot]));
trimesh(Tri,x_plot,y_plot,z_plot)
axis square
C = [.5,.5,.5; .5,.5,.5; .5,.5,.5];
colormap(C)

quality = std(Vij)/mean(Vij)/sqrt(length(Vij))

After settling on a distribution, the program creates a triangular meshing of the points found and generates graphics to show the results.

Figure 17.10 shows the results produced by points_on_sphere.m for n = 79 (an arbitrarily chosen number).

As n increases, this program produces poorer and poorer results because of its very crude search algorithm. Semechko’s MATLAB program handles the search procedure much more elegantly than the simple system shown here, with correspondingly better (and much more rapidly generated) results for large values of n.¹⁴ Inspection of his well-documented code shows that while implementing the algorithm takes only a few lines of code, implementing the search procedure is a nontrivial task.

References

1. 1. http://en.wikipedia.org/wiki/Virtual_work.
2. 2. http://www.mathworks.com/support/solutions/en/data/1-JDKMLK/index.html.
3. 3. http://www.jhu.edu/motn/relevantnotes/usingmatlab_ode.pdf.
4. 4. http://www.math.pitt.edu/~sussmanm/2071Spring08/lab01b/index.html.
5. 5. http://laser.ceb.cam.ac.uk/wiki/images/e/e5/NumMeth_Handout_7.pdf.
6. 6. P. Hawkes and E. Casper, Electron Optics, Academic Press, 1989.
7. 7. C. Liu, Foundations of MEMs, 2nd ed. Pearson Education, 2011.
8. 8. L. Dworsky and M. Chason, Electrostatically Switched Integrated Relay and Capacitor, US Patent 5,051,643, 1991.
9. 9. http://en.wikipedia.org/wiki/Electret_microphone.
10. 10. APC International Limited, Piezoelectric Ceramics: Principles and Applications, 2011.
11. 11. W. Nelson, ed., Piezoelectric Materials: Structure, Properties and Applications, Nova Science Publishers, 2010.
12. 12. V. Bottom, Introduction to Quartz Crystal Unit Design, Van Nostrand Reinhold, 1982.
13. 13. S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes, 3rd ed., The Art of Scientific Computing (series), Cambridge Univ. Press, 2007. (earlier versions of this book are available online).
14. 14. http://www.mathworks.com/matlabcentral/fileexchange/37004-uniform-sampling-of-a-sphere.