Electrometers and Electroscopes

General theory.¹ Expressed in terms of Maxwell's coefficients, the electrostatic energy of any system of conductors at potentials, is given by²

In all instruments this electrical force or torque is balanced by some restoring force. If we know how the c's and the restoring force depend on , then with the help of Eq. 3 we can solve for the voltage sensitivity.

Since most electroscopes and electrometers are used to measure electric charge, it is the charge sensitivity in which we are interested, although the voltage sensitivity is the more easily measured.

Referring back to Eq. 2, let i be the moving system. The charge sensitivity, , is given by

For most electroscopes, is small compared with ck, so that we can usually write

The effective capacity has increased to

Any further increase in the voltage sensitivity results in a more rapidly increasing capacity and a decrease in charge sensitivity.

These conditions more aptly apply to electrometers and will be discussed in that connection in the following section.

Applications to electrometers. All electrometers can be considered as made from three conductors, two of which are stationary and usually similar, while the third is movable. All three are connected electrically to the outside of the instrument. We shall assume in the following discussion that the charge or potential to be measured is applied to the movable system, while the two stationary parts are maintained at equal and opposite potentials. This arrangement is not necessary, and in general the following discussion holds equally well for the case when the charge or potential to be measured is applied to a stationary part and the moving system is kept at the high potential. It is further assumed that the electrical and mechanical zeros of the instrument coincide. This last condition is fulfilled if, while the movable system is grounded with equal and opposite potentials applied to the stationary parts, no deflection takes place.

Of the twenty-seven terms in Eq. 3, twenty-five are small or zero, as compared with the remaining two under the above conditions. If the moving part is symmetrical with respect to the stationary parts, then these two terms are equal, and Eq. 3 becomes

where is the capacity between the moving system and one of the stationary parts, which is at a potential , while is the potential of the moving system.

The charge on the moving system, 3, becomes

and

The electrical torque from Eq. 9 is therefore

Now the resultant torque is

the equation of motion becomes

and the period is

In other words the period has become 40 per cent less than when no potentials are applied. The maximum charge sensitivity in terms of the period, capacity, and moment of inertia of the system becomes

the values of the voltage and charge sensitivities are as follows:

or

and

and

This last is just what was obtained when the dependence of the capacity between the moving and stationary parts was a linear function of the displacement.

The effective capacity at the maximum charge sensitivity is the same as in the simpler case, namely, twice the pure electrostatic capacity.

The total torque may be written as

If g is positive, then the electrometer has what is known as positive control, while if g is negative, it has negative control. In the latter case the net torque may become zero at some point of the deflection, in which case the instrument becomes unstable. This frequently occurs when the sensitivity is high and is especially true with string electrometers, limiting the useful range to deflections near the midpoint. The period becomes longer if g is negative and will become longer the greater the amplitude of vibration.

It will be seen from the characteristics of the instrument in Case II that they are not so desirable as those in Case I, since they depend on the amount of displacement. However, there may be other advantages which make instruments of the second type more desirable, such as portability, ease of operation, and so forth. It should be borne in mind also that the above theory contains many simplifying assumptions, and the actual behavior of the instrument in some cases may be quite different. The chief differences are due to (1) a more complicated dependence of the capacity, between the stationary and moving parts, on the displacement and (2) air damping of the moving system. It is important to realize that where electrical charge is to be measured, there is an optimum value of the potential applied to the stationary parts for which the charge sensitivity has either an optimum or a maximum value, and that there is experimentally an easy way to test for such a condition.

Some types of electroscopes. The familiar gold-leaf electroscope is made either with a vertical stationary metal piece and a single strip of gold leaf fastened near the top, or with two gold leaves mutually repelling each other as shown in Fig. 3. The lead-in is insulated from the metal box with an amber or sulphur bushing. The capacity will be from 3 to 5 cm and the potential necessary to give a 45 degree deflection will be from 300 to 500 volts. When the leaf is observed with a microscope or a telescope, it becomes a quantitative instrument and will serve many purposes where high charge sensitivity is not important. The technique of mounting the gold leaves will be discussed at the end of this chapter.

The Wilson tilted electroscope, designed by C. T. R. Wilson³ and G. W. C. Kaye,⁴ is a hybrid of the electroscope and the electrometer. The narrow gold leaf in Fig. 4(a) hangs normally in a downward position, and is observed by means of a microscope with a micrometer ocular. A potential of about 200 volts is applied to the plate. This plate is adjustable; that is, it can be moved in or out along the axis of its support. The proximity of the plate and the potential applied to it give an electrostatic control which tends to neutralize the effect of gravity on the leaf.

Three cases in general may be cited for the voltage sensitivity as shown in Fig. 4 (b): Case I, where there is little electrostatic control and the voltage sensitivity is linear over the entire scale; Case II, where the leaf is stable over the whole range but the electrostatic control is almost sufficient to neutralize the effect of gravity over part of the range; Case III, where there is an unstable region and consequently two "zeros." Case II is the most useful, and if deflections are taken over the same regions of the scale, there is no trouble about nonlinearity.

The Wilson tilted electroscope, while it may find a use in some types of work, has been largely displaced by more modern instruments, such as quadrant and string electrometers.

The Wülf bifilar electroscope⁵ has frequently been used in cosmic-ray work. It is well suited for a portable instrument but must be read in a fixed position. As is the case with most electroscopes, not only the reading but also the calibration is affected by tilting the instrument. It is usually enclosed in an airtight ionization chamber, in which the gas pressure is often increased to increase the number of ions formed by a given radiation. The charge is renewed on the electroscope either by a mechanical arm working through an airtight bushing in the wall or, what is better, by an internal arm operated by an electromagnet.

The construction of the electroscope proper is shown in Fig. 5. A clean quartz rod is cemented into the metal piece which holds a short 0.5mm rod by means of a setscrew. The small rod is flattened at the lower end. Two metal-coated straight quartz fibers from 5 to 10 cm long and from 10 to 20 microns in diameter are cemented or soldered (with Wood's metal) to the flattened piece. The lower ends of the two fibers are cemented side by side to an insulating quartz bow made from 10 to 20 micron fiber. It is essential that the fibers, when uncharged, hang parallel to each other. Means of straightening quartz fiber will be found in Chapter V. If shellac is used as a cement, there will be sufficient conductivity from the metal to the fibers. The potential of 200 to 400 volts is applied to the upper metal support. If the fibers are 20 microns in diameter and 8 cm long, a spread of 3 mm will be produced by about 300 volts. The capacity will be in the neighborhood of 1 cm.

The plane of motion of the fibers should be perpendicular to the optical axis of the microscope, the necessary adjustment being made by rotating the piece held by the setscrew.

It is possible to increase the sensitivity of the Wülf electroscope either by increasing the magnification of the microscope or by decreasing the diameter of the fibers. However, this is limited by the fact that the-collecting potential for the ions should not drop too low, depending on the nature of the gas and its pressure. It is customary to have a collecting potential of not less than 100 volts.

From Eq. 8 the charge sensitivity is

and if n is the average number of ions per second collected from each cubic centimeter of the gas, then

Regener's electroscope is a single-fiber type shown in Fig. 6. The conducting quartz fiber or Wollaston wire is mounted near a metal piece and is held taut by a fine bow. The whole is supported by a quartz insulator, and the charge is renewed in a way similar to that used with the Wülf type.

Lauritsen has used a small quartz-fiber electroscope with much success not only in small pocket dose-meters for X-ray work but also in measuring radiations found in nuclear investigations. Its outstanding feature is its simplicity. A wire is flattened at one end and bent over at right angles. A 5 micron (0.005-mm) metal-coated quartz fiber about 6 mm long is cemented to this flat piece with shellac or colloidal graphite, making an angle with the wire support as shown in Fig. 7. A short piece of the same size fiber is cemented to the end of the longer fiber, at right angles to the plane of the wire and to the first fiber. This added piece is to form an index for viewing with a microscope. The wire support is mounted in an amber insulator, which in turn is mounted on the end of the microscope. When the electroscope is used inside an ionization chamber, contact is made by a movable arm to the base of the metal support.

In order to obtain reliable readings on cosmic rays in airplanes, a torsion type of electroscope was developed in 1932. It was necessary to have a self-recording instrument of high sensitivity, the readings of which would not be affected by tilt or vibration of the plane. As far as tilt is concerned, this effect on the readings can be reduced to less than 0.001 of the total deflection for a tilt of 90 degrees. As for vibration, satisfactory readings have been obtained with the electroscope mounted within 3 feet of the engine in a pursuit airplane.

A drawing of the electroscope is reproduced in Fig. 8. It is made entirely of fused quartz. The torsion fiber is stretched until its length is increased about 1 per cent. The crossarm is bent at right angles at one end and, in case high magnification is used, it is drawn down to a convenient size. A short bit of fiber serves as a fiducial mark. The shapes of the stationary parts combine to give a linear scale over most of the range of discharge. A piece of platinum cemented to the quartz with a polymerizing cement is the point at which a new charge is placed on the system. With a very small oxygen-gas flame all joints are fused together so that the whole system becomes essentially one piece of quartz. The system from the platinum down is covered with a conducting layer of gold. The vane is balanced by cutting off one end. For many applications this balancing I need not be done with great care and, in fact, becomes rather delicate if a very fine torsion fiber is used. If too much is cut off, mass can be added by applying some thin gold china paint⁶ and heating it with a hot wire.

In general it will be necessary to put a permanent twist in the torsion fiber. This can be done by forcing the vane beyond the stop through the desired angle, relieving the tension by pushing on the bow at the bottom, and heating the fiber at each end with a small pure gas flame. This will soften the quartz just enough. The twist, of course, must be put in before the system is covered with its conducting coating of metal. The following illustration gives some idea of how much twist is needed: With a torsion fiber 5 microns in diameter and 12 mm long and a crossarm or vane 18 mm long, if a 30 degree twist is put in the fiber, the deflection will begin at about 200 volts and the sensitivity will be about radian/volt. The electrostatic capacity will be about 0.5 cm and the charge sensitivity about 1.2 radians/statcoulomb.

with V = 1 statvolt. Since b is a geometrical quantity, it will not depend on the size of the torsion fiber. The above relation may be used to get an approximation to the sensitivity for other values of the torsion constant k.

If a very fine torsion fiber is used, in order to keep the collecting voltage up it may be necessary to twist the torsion fiber around one or more times. If the crossarm is not larger than 20 microns in diameter, this can be done manually after the conducting coat has been put on by using a needle and forcing the ends through, between the torsion fiber and the main quartz support.

Some types of electrometers. The Dolezalek quadrant electrometer⁷ is perhaps the most common type and the most useful. The general plan of the instrument is shown in Fig. 9. It consists of a cylindrical box, or "pillbox," divided into four equal and insulated quadrants. Opposite quadrants are connected together. There are two ways of using quadrant electrometers. One is to keep the needle at the high potential with respect to ground and apply the charge to be measured to one pair of quadrants while the other pair is grounded. The other way is to maintain one pair of quadrants at the potential + V and the other at–V and place the charge to be measured on the needle. The first method is illustrated in Fig. 10(a). V will usually be from 50 to 150 volts, depending on the desired sensitivity.

For the second case the battery connections can be those shown in Fig. 10(b), where the main batteries furnish only a potential and slight adjustments are made by a potentiometer as shown. Or the main batteries may be placed across the high resistances R and R, and adjustments aremadewith the potentiometer R', as represented in Fig. 10(c). The former circuit has the advantage that the life of the high-voltage batteries is essentially their shelf life, while the chief advantage of the second is that | + V| always equals |–V| and that the mechanical and electrical zeros remain together once they are made to coincide. However, modern "B" batteries maintain a remarkably constant potential at no current over long periods of time and have a very low temperature coefficient, so that in many cases the first circuit can be used.

An approximation can be made to the value b for the Dolezalek electrometer in terms of the geometry of the instrument. It will be seen that the vane is of such shape that the capacity between the vane and the box, as a deflection takes place, varies linearly with the change of angle. Let be the deflection, R the radius of the vane, h the distance of the vane from one side of the box, and d the depth of the box. Then b is the increase of capacity between the vane and the conductor into which it moves per unit of angle, or

Since there is an equal and opposite vane on the other side, the total electrical torque, by Eq. 9, is

which at equilibrium is equal to . The voltage sensitivity is then

and the charge sensitivity is

(15)

At the optimum value of ,

and the maximum charge sensitivity becomes The effective capacity at this Bensitivity ifi, of course, 2c, where c is the total electrostatic capacity on which the charge q is placed.

Eqs. 14 and 15 predict a constant voltage and charge sensitivity for given values of the potential on the auadrants or on the needle and for given geometrical conditions. Actually, however, this is not the case, for it will be found that as the potential on the quadrants or on the needle, as the case may be, is increased, the period gradually lengthens, and a value is finally reached at which the vane becomes unstable at a certain point of the scale. This behavior is due to the occurrence of nonlinear terms in the expression for the capacity between the vane and the quadrants. By careful adjustment the importance of these terms can be diminished but never eliminated.

In setting up the Dolezalek electrometer, the vane should not be too close to either the top or the bottom of the box, since small variations due to changes in temperature and so forth will change the characteristics. Also, irregularities in the vane may make important nonlinear terms in the capacity between the vane and the quadrants. Although the maximum charge sensitivity is not affected by this distance, the optimum voltage and the voltage sensitivity are affected.

The instrument is leveled until the piece holding the vane is in the center of the circular hole in the top of the box. A grounding switch, which may be manually or magnetically operated, must be provided. Care should be exercised not to introduce variable thermal e.m.f.'s.

The torsion head should be adjusted so that each half of the vane lies as nearly as possible symmetrically between two quadrants. With the vane grounded, small values of

+ V and –V are applied to the quadrants. Adjustment is made to the torsion head in the appropriate direction, so that whether a potential is on the quadrants or not, no motion of the vane takes place. After the full values of ~ V are placed on the quadrants, slight adjustments can be made with the potentiometer as shown in Figs. 10(b) and 10(c). This procedure is generally known as bringing the mechanical and electrical zeros together, and must be done with all forms of electrometers.

In operation, the actual useful working value of the charge sensitivity will be about

Sq = 1.3 X 10⁴ div./statcoulomb

= 0.4 X 10¹⁴ div./coulomb

= 0.6 X 10 div./electron.

The corresponding voltage sensitivity will probably lie in the range

= l000 to 1500 div./volt,

while the optimum value of the voltage applied to the quadrants will probably be between 50 and 100 volts on each side of ground, or if the high potential is placed on the needle, it will usually be between 100 and 200 volts, depending, among other things, on the size of the torsion fiber. It is assumed in the above that a scale with 1-mm divisions is used at the customary distance of 1 m.

The Compton electrometer⁸ was introduced in 1919 by the two Compton brothers.⁹ It is of the quadrant type but is so arranged that one quadrant can be raised or lowered with respect to the other three. Further dissymmetry is introduced by giving the vane an initial tilt. By proper adjustment of this movable quadrant the time consumed by the needle in returning to its initial position after a deflection may be lengthened (negative control) or shortened (positive control). The design of the instrument is shown in Fig. 11. The dissymmetry introduces additional nonlinear terms into the change of capacity as the needle moves, and an electrostatic torque is introduced which either opposes the torque of the suspension (negative control) or aids it (positive control). In the extreme case the action of the suspension can be more than completely neutralized, so that an unstable instrument results. This means that the voltage sensitivity can be made extremely high. Fig. 12(a) illustrates the relationship between the voltage sensitivity and the voltage on the needle for different degrees of positive control, while Fig. 12(b) shows the same relationship for various degrees of negative control. In this latter case a stiffer suspension was used. The circles on curves 6, 7, 8, 9 represent the highest sensitivity at which the zero of the instrument is sufficiently stable to allow satisfactory measurements to be made. The small figures above the curves of Figs. 12(a) and 12(b) represent the time required in each case for the needle to return to within 1 mm of the rest position after a deflection of 50 mm. Because of the small restoring torque and high air damping, the motion of the suspended system is aperiodic.

Where extremely small potentials are to be measured and where the demand on charge sensitivity is not too great, the Compton electrometer is very suitable. However, the same voltage sensitivity could be achieved with the usual quadrant electrometer by putting in a suspension fine enough to give the same time of return to zero, provided that the moving system were equally as light as that in the Compton. In fact, it will be noted from the curves that with neither positive nor negative control, shown by the straight line of Fig. 12(a), but with a fine fiber, voltage sensitivities can be obtained equal to those with a stiff fiber and large negative control. This high voltage sensitivity is not always useful when measuring electric charges, which is the main purpose of electrometers, for not only does the instrument become very sluggish, but drifts become bad. Also, for a given time of return from a given deflection the charge sensitivity has a maximum value.¹⁰ Wolf¹¹ states that the maximum usable charge sensitivity of the Compton electrometer is div./coulomb, which occurs at a voltage sensitivity of 5000 div./volt.

The Hoffmann electrometer¹² combines the highest charge sensitivity of any commercial instrument with stability, that is, lack of drift, and ease of working. Great care has been exercised to eliminate contact potentials, thermal e.m.f.'s, and air currents. To achieve the elimination of alr currents, heav~round the movable system to insure that thermal gradients are kept at a minimum, A decided advantag;e is gained also by evacuating the case to a few milliIneters of mercury, thus making the instrument "deadbeat."

The instrument operates upon essentially the same principle as the quadrant electrometer. The chief difference is that a half vane is used for the movable system, so that, instead of quadrants, only two conductors, or binants, are necessary. Fig. 13 represents the relationships of the essential parts. The platinum needle and mirror together weigh approximately 5 mg, and the suspension is a 3 microns (0.0003-cm) Wollaston wire.

To achieve a sensitivity independent of temperature, it is necessary to keep the vane or needle at the same distance from the binants. This is accomplished by inserting into the supports of the upper part of the electrometer case, which in turn supports the suspensicn, a metal of such coefficient of thermal expansion that the over-all expansion completely neutralizes the change of length of the torsion fiber with temperature.

Contact and thermal electromotive forces are kept at a minimum by making everything from or plating it with platinum. Also, insulation is protected by metal, so that possible spurious charges cannot affect the system.

Since the Hoffmann electrometer combines so many desirable features, it may be well to list some of them. These characteristics must be combined in any other instrument with which it is intended to push the charge sensitivity to that limit set by Brownian motion, and still have freedom from drift and a reasonable working period.

1. The moment of inertia of the moving system must be small. (See Eq. 13.)

2. The suspension must be made of material which has a small coefficient of internal friction; that is, the needle must return to zero after a deflection.

3. Air currents must be kept at a minimum. This means that the moving system must be surrounded with heavy copper pieces. The suspension should be closely surrounded by metal pieces as well.

4. The case must be evacuated to keep the working time within a reasonable limit.

5. Temperature compensation is needed if the distance between the vane and the stationary parts is to remain constant.

6. Thermal and contact electromotive forces musb be eliminated.

In addition to the above, it is usually desirable to have the scale approximately linear.

Two additional features of the Hoffmann electrometer are (1) an electromagnetic grounding switch and (2) an induction ring for inducing a charge on the movable system.

To facilitate making the necessary electrical connections, a control mechanism is supplied with the instrument when it is purchased. Although not absolutely necessary, the control mechanism is a great aid, since the proper connections are made and broken at the right time by only one operation.

The latest model of the Hoffmann electrometer¹³ combines all the desirable features of the earlier models but permits greater accessibility to the essential parts. Also the adjustments are much more easily made; for example, in the older types the instrument had to be exhausted after adjusting~the binants, while in the new design th~s adjustment is made through a sylphon from the outside.

String electrometers are divided into two main divisions: (1) those with a fiber supported only at one end and (2) those in which the fiber is kept taut by a fine spring. The latter are the most common and, as far as is known, are the only ones on the market.

Electrometers of the first class are easily made and are often very satisfactory when high sensitivity is not needed. The two plates can be flat and the fiber hung down between them as shown in Fig. 14. There should be an adjustment either on the plates or on the fiber or on both to bring the mechanical and electrical zeros together. Some adjustment may be made by tipping the instrument in the appropriate direction. A microscope must be provided to read the deflection. With the plates 1 cm apart the fiber should be about 25 microns in diameter and 4 cm long if the potential on the plates is not to exceed 100 volts.

Of the second type, that designed by Wülf¹⁴ is, perhaps, typical. It is shown diagrammatically in Fig. 15. The fiber is usuallyaWollaston wire 2 microns in diameter, kept taut by a quartz-fiber bow. Screw adjustments permit movement of the plates with respect to the fiber as well as change in the tension of the fiber.

With all string electrometers the deflection is not a linear function of the applied charge or voltage at high sensitivity. It frequently happens that at high sensitivities the fiber leaves the field of view of the microscope as it reaches a position where instability occurs.

The chief advantages of string electrometers are (1) portability, (2) ease of adjustment, and (3) short working time.

The Perucca electrometer¹⁵ is similar to the string electrometer, except that the part between the plates consists of two conducting quartz fibers supported on a torsion fiber as shown in Fig. 16. The two movable fibers are brought together at one end, and a small index, which is viewed with a microscope, is provided. The charge and voltage sensitivities are each greater than can be obtained with a string electrometer. Not only is it more sensitive, but it also combines all the advantages of the latter instrument.

The Lindemann electrometer¹⁶ was developed primarily for use with photoelectric cells mounted on telescopes in measuring light from stars. Such use requires that the sensitivity and position of the moving system be independent of tilt. The first is accomplished by making all parts very rigid and the second by using for the moving system a light needle mounted on a stretched torsion fiber. Since a mirror and scale would be cumbersome for such uses, the deflection of the needle is read with a microscope with a micrometer ocular. The whole electrometer weighs but 80 g. The quadrants and needle mountings are represented in Fig. 17.

The principle on which the instrument works is similar to 11 the quadrant electrometer. The quadrants are 1.5 cm broad and 1 cm high, with a slot 2 mm wide, into which the needle may pass, cut in each. These plates are mounted 5 mm apart on quartz rods. The torsion-fiber mounting and needle are placed between the plates so that the junction of the needle and torsion fiber is symmetrically located with respect to the four plates. The whole is mounted in an aluminum box with suitable connections. Through a glass window in one side of the box the motion of the end of the needle is observed with a microscope. A window directly opposite on the other side of the case permits light to enter.

The needle may be balanced so that a rotation of the instrument through 90 degrees makes less than 0.06 mm motion of the end. The needle is usually about 1 cm long, and the torsion fiber is 6 microns in diameter. The needle and torsion fiber are covered with a conducting coating of metal, and a suitable connection,is made l~o the outside of the case. The quartz frame for holding the torsion fiber serves as insulation as well.

Electrical connections are the same as for any quadrant electrometer. Instability occurs at a potential of about 100 volts. At 3 volts below this unstable value the deflection will reach 99 per cent of its final value in 1 second. The voltage sensitivity under these conditions is 0.76 mm/volt motion of the end of the needle. With a suitable microscope a workable sensitivity of 500 div./volt can be obtained. The electrostatic capacity is about 2 cm.

The instrument may conveniently be used with a leak to measure currents of from 10 to 10 ampere. When not too great demands are to be met, this quite inexpensive electrometer will meet many needs, especially where portability is a requirement.

For a discussion of circuits, sensitivities, and limitations of the vacuum-tube electrometer, which uses specially constructed vacuum tubes, see Chapter X.

Some practical considerations in the use of electrometers and electroscopes. Useful sensitivity in X-ray work. Electrometers are frequently used with an ion chamber in X-ray work. As is well known, ions are formed not only by the X-ray beam but by (1) cosmic rays, (2) local radiation from radioactive matter in the surroundings, and (3) radioactive contamination on the inner walls af the ion chamber. Of these, (3) can be reduced to a small value in comparison with the others by two effective means. The inner walls can be painted with a mixture of collodion and lampblack, each of which is quite free from radioactive materials. The thickness should be about 0.05 mm to stop all the particles. The other method is to maintain a fine wire grid at a suitable potential to drive the ions formed by the particles back into the walls. Since in general (3) is due to particles which will have a range of less than 5 cm at normal air pressure, the range can be kept within the grid by a gas of high molecular weight or increased pressure, or both. Even rubbing Carborundum paper on the walls of the chamber will often help considerably in lowering the emission. As for (1), cosmic rays could be reduced to an extremely low value by going into a mine 100 to 200 feet below the surface of the ground, while (2) could be made negligible with 4 inches of surrounding lead. However, since it is not practical to go to such trouble, in most cases it is necessary to make the best of the situation.

Since the error of a result which depends on the difference or sum of two readings is¹⁷

(16)

where and are the errors of the two readings, it is hopeless to try to push the sensitivity of the measuring device beyond a certain point, and the only hope of increasing accuracy is by a longer period of observation. It is easily seen that the optimum useful value of the sensitivity is reached when the deflection due to background only and the deflection due to the X-ray beam only are equal for the same time of observation.

If the ratio of background to beam readings is to be made as small as possible, it is obvious that the volume also should be made as small as possible, since the background reading goes up with the volume.

Example. Assume an ion chamber of 1000 cm³. Cosmic rays will contribute about 3 ions/cm³/sec./atmosphere of air at sea level. Local radiation will be from 3 to 5 I (ions cm sec. atmosphere of air), while the background may vary over wide limits; from 0.1 to 10 ions cm sec. atmosphere of air will probably include the extreme cases. Since the particle paths will usually end in the gas, an increase of pressure will not change the number of ions formed by the particles, but the ionization due to local radiation and cosmic rays will go up as the pressure increases. Let us assume 7I as due to electrons and 5I as due to particles. The average path length of the electrons is about 10 cm, and at 60 ions/cm of path this corresponds to about 12 electrons/sec. crossing the chamber. In order to have a mean relative error of , then according to the laws of probability, particles must cross the chamber. Let be 0.03, or 3 per cent; then 10³ particles must be counted. This will take 80 seconds on the above assumptions. These 10³ electrons will form 6 X 10⁵ ions: Hence if we assume the same fluctuations in the ions from the beam, then according to Eq. 16, to have an average error of 4 per cent in a reading, we must time for at least a minute, and the sensitivity of the electrometer need not be greater than 10 div./ion if we estimate to 0.1 div.

As for the particles from the walls, their effect may be considered as follows: Supposing they amount to 5I, which is not an uncommon value, then there will be 5000 ions/sec. formed. Now an particle will form, on the average, about 10,000 ions in the gas. Hence there is 0.5 particle/sec. emitted by the walls. Now if is the mean absolute error in a given reading and in another reading, the mean relative error in the sum will be

(17)

where is the deflection of the instrument If is the number of particles per second of one kind of particle and the number of ions formed per particle, then it can be shown from Eq. 17 that the mean relative error of the sum is

under the above assumptions if the subscript 1 refers to the electrons and 2 refers to the particles. It is then necessary to count for 400 seconds to gain an accuracy of 4 per cent. In this time 5 X 10⁵ ions will have been collected. In order to read this to 4 per cent we need a sensitivity no greater than 10 div./ion.

The above calculations ha~re been made to show (1) the importance of eliminating particles as much as possible and (2) that when this is done completely, the sensitivity of the electrometer has a limit beyond which there is no gain.

If charges are to be collected where there is very little background, such as in photoelectric work, then there is no reason why the sensitivity cannot be pushed to the maximum. In all cases, if possible, an electrometer or an electroscope suitable to the accuracy required should be chosen.

Useful sensitivity in cosmic-ray work. In case the instrument is subject only to cosmic rays and no shielding is used, the ionization is due to random electrons and "X" particles, which ionize the gas the same as electrons. When this is so, the mean relative error is , where N is the total number of particles, the effects of which are measured. If there are n high-energy particles/cm²/sec., and if the mean relative error is for one reading, then we must observe for a time

where

for a spherical ionization chamber, and if a is the specific ionization, since the average path length is , the total number of ions collected in the time will be

and this must give a deflection which can be read with no larger relative error than . If = 60 ions cm, R = 10 cm, and = 0.01, then v = 8 X 10⁶ ions, and we need a sensitivity of about 3 X 10 div./ion. Now n 0.02 electron/cm²/sec. at sea level. Hence the minimum time of observation should be 30 minutes, and for each observation the mean error will be 1 per cent. This calculation, of course, neglects the error introduced by the background radiation.

Frequently, however, the ionization chamber is surrounded with shields made of iron or lead. These do two things: (1) In general, they lessen the intensity of the radiation, and (2) they introduce new radiations. All the particles passing through the ionization chamber are no longer randomly distributed in time, for, in addition, there now exist showers, consisting of from two to several hundred electrons, which come all at the same time from some region of the shield. These introduce larger fluctuations than would otherwise exist, and the time of observation for the size of ion chamber assumed above may be from two to four times as long for the same error, and a correspondingly less sensitivity of the measuring instrument will serve the purpose.

Steady deflection measurements. In some cases it may be desired to use the constant deflection instead of the drift method. This may be done by using the electrometer to measure the drop in potential across a fixed resistance as shown in Fig. 18. Assume that it is desired to measure a constant ion source I. Let the capacity to ground of the external system be cl and of the electrometer , and let the drop in potential be measured across . Then

The equation for the potential across the electrometer is then

Solving and putting in the boundary condition that when t = 0, V = 0,

Thus the potential across the electrometer rises exponentially. If we say arbitrarily that we shall wait until the deflection is 99 per cent of the ultimate deflection, then we must wait a time t = 4.6 , where . The deflection will be approximately after this time. Had we measured I by the drift method, we should have the same deflection in a time , the difference, of course, being due to the fact that in the second case the drift is constant, while in the first case the drift begins at the same rate, that is, as if , but gradually slows down, becoming very slow toward the last.

It is therefore much more satisfactory to use the drift method for measuring feeble currents, while larger currents are conveniently measured by the steady deflection method. The drift method can be used in measuring large currents also by inserting a capacity of the appropriate value to lengthen the time of drift.

Limitations of various types of instruments. Limitation on the charge sensitivity of electroscopes and electrometers has already been pointed out. For the former the maximum charge sensitivity is

and for the latter

The capacity of an electroscope which has no external lead will depend on the particular design, but for the Wulf or torsion type it will lie between 0.4 and 1 cm. That of an electrometer with its added external capacity will probably be between 20 and 100 cm. The restoring constant, k, of the suspension can be reduced in each to a point where the sluggishness of the motion makes the instrument tedious to work with, or in the case of most electroscopes, where the collecting potential becomes too small to collect most of the ions. Since an electrometer case can be evacuated, it is possible to adjust the pressure until the motion of the vane or needle becomes critically damped.

If the electrometer case is not evacuated, the working period may become excessively long when high sensitivities are desired. Much can be achieved by making the needle or vane small and light, as is done in the Lindemann and Perucca electrometers, and as is inherently the case with string electrometers.

Limitations imposed by drift. The amount of drift during a reading is often the limiting factor in electrometers. This frequently becomes bothersome long before the maximum sensitivity has been reached. One of the chief reasons for the drift is that the mechanical and electrical zeros gradually drift apart. The deflection caused by the zeros being different may be many times the actual amount they are apart. Drift, among other causes, is due tq (1) fluctuations in battery voltage and (2) nonelastic changes of strain in the suspension. If the drift were constant, proper allowances could be made, but there are so many factors which depend in a different way upon changes of voltages, temperature, humidity, and so forth, that it is often very difficult, if not impossible, to eliminate completely or take account of the drift. This is especially true with vacuum-tube electrometers, even though balanced circuits are used.

Limitations on the amount of useful magnification. Two methods are in general use for determining the amount of deflection in an electroscope or electrometer: (1) Microscope with micrometer ocular and (2) mirror and scale. For the Lindemann electrometer and most electroscopes the microscope is used. The limitation as far as magnification is concerned amounts to a limitation Qf resolution. Magnification can continue until the position of a diffraction band cannot be located to within 0.1 div. in the eyepiece. Beyond this, nothing is gained. With a numerical aperture of 1 and an image distance of 20 cm the shortest useful focal length is about 3 mm with a 100-div. scale in the eyepiece 1 cm long.

If a mirror and scale are used, there is a certain minimum mirror size which will allow sufficient resolution. With a 1 mm div. scale at the customary distance of 1 m, it is necessary to have a mirror at least 2 mm in diameter to read to 0.1 div. on the scale.

In all cases, whether in resolution, amount of drift, fluctuations, or the like, it should always be possible to estimate to 0.1 of the smallest division on the scale, and in general it is useless to push the sensitivity of any instrument beyond the point where 0.1 div. Ioses its significance.

Limitations imposed by Brownian motion. It is part of the classical theory of the equipartition of energy that all bodies have a mean thermal energy of for each degree of freedom, where K is Boltzmann's nonstant and T the absolute temperature. This Brownian motion of the instrument is evidenced by random fluctuations about the point of equilibrium. It is evident that before a superimposed steady deflection can be detected, it must be at least as large as this mean Brownian deflection.

The mechanical energy of a moving system with a restoring force proportional to the displacement is , where k is the restoring force (or torque) per unit of displacement. If is the mean Brownian deflection, then the k corresponding to this is giveen by

or

Now with the electroscope the maximum charge sensitivity is reached when . Consequently, the corresponding charge sensitivity is

if the deflection is equal to the mean Brownian deflection. At room temperature the maximum charge sensitivity is thus limited for electroscopes to

div./electron,

where c is in centimeters. For electrometers the expression becomes

It is obvious that the electrostatic capacity of the instrument should be as small as possible if it is intended to push the charge sensitivity to the limit. Inherently the capacity of the electroscope is much less than that of the electrometer. This not only makes it possible to have a higher charge sensitivity for the same torsion constant but allows it to be used.

It is interesting to compare the above limit with that obtainable with a Geiger counter. In some applications the number of counts and the number of unit charges collected are comparable. The mean error with a Geiger counter in a single count of N particles distributed at random is , so that if it is desired to have a mean relative error of 1 per cent, it is necessary to count 1/(0.01)² or 10⁴ particles. With an electroscope having a capacity of 0.5 cm, it is necessary to collect 8 X 10⁴ electrons to have the same mean error if the deflection can be read to 0.1 div. This is, of course, disregarding the backgrounds in each case.

A comparison of various types of instruments. Probably the most sensitive electrometer on the market is the Hoffmann. The maximum sensitivity which can be reached with this instrument is approximately 5 X 10¹⁵ div./coulomb. Drift has been eliminated to such an extent that sufficient time can elapse to detect an average of 1 electron/sec. For ease of working, however, it is advisable to keep the charge sensitivity in the neighborhood of 1 X 10¹⁵ div./coulomb. Much is gained in the Hoffmann by evacuating the case, thereby not only shortening the working time but greatly eliminating the effects of convection currents.

The vacuum-tube electrometer has gained much favor in the past few years. It has the advantage that it can be used in places where it would be inconvenient or impossible to use the conventional type of electrometer. The sensitivity can be made comparable to that of the Hoffmann, although it is very much inferior as far as drifts are concerned. Ordinary precautions consist in having large storage batteries for plate and filament supply which are kept at as constant a temperature as possible, with all leads well shielded. Resistances must also be kept constant. A1though with the proper circuit and circuit constants the effects of voltage fluctuations are reduced to a minimum, it is still not possible to eliminate the drift, and it is usually necessary to wait several hours after the connections are made for conditions to become only approximately steady.

When possible, an instrument should be chosen for the problem at hand. Frequently it is desirable to use an electroscope in place of the electrometer. The advantages to be gained may be listed as follows: (1) Freedom from external changes of temperature and humidity, (2) freedom from changes in battery potentials and resistances, (3) freedom from drifts, (4) need for only one potential, (5) ease of setting up and operating, (6) portability, and (7) low cost. The disadvantages are that (1) except with the torsion type the sensitivity is not as high as with the ordinary electrometer, (2) the sensitivity is not readily varied, and (3) it is not convenient to use a null method of reading.

In Table I are listed the approximate characteristics of some instruments. The values of charge sensitivities listed are not the maximum attainable but represent those that can be reached and worked without great difficulty. The values of the voltage sensitivities are those which correspond to these values of the charge sensitivities. In some cases the voltage sensitivity can be made much higher, in particular with the Compton, with which it is possible to reach 50,000 div./volt. The working period represents approximately the time for the deflection to become zero after the net charge is removed.

The above operations should be carried out in a room in which the motion of the air is at a minimum. It is often advisable to wear a mask or deflector over the nose to avoid blowing the leaf about.

Preparation of Wollaston wire.¹⁸ The Hoffmann and many string electrometers use a fine platinum suspension known as Wollaston wire. It may be obtained in various sizes from 1.5 to 5 microns. To produce such a fine wire of uniform size, the following process of manufacture is used. Upon a much larger platinum wire is electroplated a uniform layer of silver. The combination is then drawn down until the fine thread of platinum in the middle is of the proper size. The silver is etched off with acid. Since the resulting platinum wire is quite delicate, special care must be used in the etching as well as in subsequent handling.

In order to avoid small bubbles collecting on the wire and interfering with the etching, or in some cases breaking the fine wire, a special solution of chemically pure nitric acid in distilled water at a density of 1.10 g cm is used. To insure uniform etching, the wire should be thoroughly cleaned before it is immersed in the acid. As an aid to handling after etching, a bead two to three times the diameter of the silver wire is formed on one end with a small oxygen flame before the silver is etched off. A section of wire is then cut off, perhaps an inch longer than the necessary suspension. The solution is placed in a tall vessel, such as a graduate, and the straightened silver wire is supported in it vertically. The suspension should be left in for a longer rather than a shorter time, since the platinum is not damaged by the solution.

It is necessary that all the silver be etched off, or the suspension may be ruined in the annealing process. The small bead marks the lower end as the suspension is drawn from the solution. Before soldering it into place, it may be desirable to mount the suspension on a "wishbone," the distance between the two prongs being somewhat greater than the length of the mounted suspension. If quartz fibers 20 to 30 microns in diameter are mounted in the tips of the prongs and the Wollaston wire is fastened to these with a hard wax, there will be much less chance of breakage.

Either before or after mounting, the suspension should be placed in a horizontal position and annealed with a small gas flame. In still air the flame is passed beneath at such a distance that the platinum is heated to a bright red color. If all the silver has been etched off, the suspension will appear a uniform brightness throughout its length. The annealing is necessary, if the wire is to be used in an electrometer, to relieve the strains which resulted from the drawing.

In soldering the suspension in place, a c.p. solution of zinc chloride is a good flux. The heat is best applied with a small soldering iron, not directly at the point at which the suspension touches the solder but at a short distance away, relying upon the conductivity of the metal support. It is best to work under a magnifying glass or, better still, a binocular microscope. The joint should be rigidly inspected to see that the platinum is actually embedded in the solder and not just held by the solidified flux.

Insulators used in electrometer and electroscope work. The insulator ordinarily used in electrometers is amber. The amber now on the market is usually a manufactured product which has as good insulation properties as the natural amber and has the advantage of being obtainable in a variety of sizes. Amber has a high volume resistivity, and the surface resistance of clean amber is also high. If the surface is contaminated, the best remedy is to remove some of the amber with a clean tool by turning it in a lathe. If this is not convenient, the amber may be covered with a thin coat of ceresin, as will be described later.

The best insulator known is clean, dry, fused quartz. By clean quartz is meant quartz which has not touched anything since being heated to the softening pQint, and by dry quartz is meant quartz either in a good vacuum or in a gas dried by phosphorus pentoxide. Fused quartz is also superior to other insulators in that the soak-in is far less. Under comparable conditions amber has at least ten times the soak-in possessed by quartz.

Ceresin is a natural wax which has remarkable electrical insulation properties.¹⁹ It is about the same hardness as ordinary paraffin, each at 20 degrees C. However, it has a somewhat higher melting point than either paraffin or the artificial ceresin, being liquid at 65 degrees C. Its insulation properties have been measured by Curtiss²⁰ of the Bureau of Standards. He gives the surface resistivity as greater than 101' ohm cm even at 90 per cent humidity. One of its main uses in the laboratory is to~ improve the surface resistance of other insulators. If the solid insulator and the ceresin are each heated to around 100 degrees C. and a light coating of ceresin applied, the surface leakage will usually be found greatly reduced, sometimes by a factor of 100.

¹ This treatment follows, in general, that given by Hoffmann, G., in Handbuch der Exp. Physik, X, 42 (1928). Wein, W., and Harms, F., editors; Leipzig.

² Jeans, J. H., Mathematical Theory of Electricity and Magnetism, Fifth Edition, page 95. New York: The Macmillan Company

³ Wilson, C. T. R., Cambridge Phil. Soc., Proc., 12, 135 (1903).

⁴ Kaye, G. W. C., Phys. Soc., Proc., 23, 209 (1911).This instrument is made by Cambridge Scientific Instrument Company, Ltd., Cambridge, England.

⁵ This instrument can be obtained from E. Leybold's Nachfolger A. G. Köln-Bayental, Bonner Strasse 500, Germany.

⁶ See "The Use of Fused Silica."

⁷ This instrument is made by many firms, including the Cambridge Scientific Instrument Company, Ltd., Cambridge, England, the Cambridge and Paul Instrument Company, Ltd., and E. Leybold's Nachfolger A. G. Köln-Bayental, Bonner Strasse 500, Germany.

⁸ This instrument is made by the Rubicon Company, 29 North Sixth Street, Philadelphia, and by the Cambridge Scientific Instrument Company, Ltd., Cambridge, England.

²⁰ Curtiss, L. F., Bulletin of the Bureau of Standards, 1915.