category is a section that treats the effects of instrumental and environmental noise on the precision and accuracy of spectroscopic absorption measurements. Also largely new is the development of general chromatographic theory from kinetic considerations. The chapter on optical instruments brings together information that was formerly found in chapters on infrared, ultraviolet and visible, and atomic absorption spectroscopy; flame and emission spectroscopy; and fluorescence and Raman spectroscopy. Since the publication of the first edition, a host of modifications of wellestablished instrumental methods have emerged and are now included. Among these new developments are Fourier transform nuclear magnetic resonance and infrared methods, infrared photometers for pollutant measurements, laser modifications of spectroscopic measurements, nuclear magnetic resonance measuremefices. Readout devices employ an output transducer to convert the amplified signal from the processor to a signal that can be read by the human eye. Readout devices take the form of meters, strip chart recorders, oscilloscopes, pointer scales, and digital devices.

-.d

Electronics Instrumentation

The growth of instrumental analysis has closely paralleled developmentsin the fieldof

electronics, because the generation, transduction, amplification,and display of a signal can be rapidly and conveniently accomplished with electronic circuitry. Numerous trans- . ducers have been developed for the conver- . sion of chemical signals to electrical fonn, and enormous amplification of the resulting electrical signals is possible.In turn, the eleo- . trical signals are readily presented on meters, on recorders, or in digital form. Owing to the large amount of electronic circuitry in the laboratory, the modem chemist is faced with the question of how much knowledge of electronics is needed to make most efficient use of the equipment available for analysis.We believethat it is not only desirable but feasible for the chemist to possess a qualitative knowledgeof how electronic circuits work. Thus, Chapters 2 and 3 are largely devoted to a presentation of this topic; subsequent chapters are concerned with the application of various instruments to the solution of chemical problems.

TABLE '-21 ...-. Photometer

SOME EXAMPLES OF INSTRUMENT SIpaIGeRenator Tungsten lamp. glass filter, sample Flame, monochromator, chopper, sample dc source, sample Sample X-Ray tube, sample Sunlight, sample A..tyticaI

COMPONENTST~

SignalAttenuated light beam UVor visible radiation Cell current Hydrogen ion activity Dilfra(.ted radiation Color

IIIpIIt Tnnsducer Photocell Photomultiplier tube Electrodes Glass-calomel electrodes Photographic film Eye

SipaI

SignalElectrical current Electrical potential Electrical current Electrical potential Latent image Optic nerve signal

~

ReadoutCurrent meter Chart recorder Chart recorder Digital unit Black images on film Visual color response

None Amplifier, demodulator Amplifier Amplifier, digitalizer Chemical developer Brain

s:... -"";';"

Electrical or mechanical

IIT'JI!

Atomic emission sp:trometer Coulometer pH meter X-Ray powder diffractometer Color comparator

_~_;.u_.~_,.:.=;.,

~.-/112.301

o

0

'I

Oi~ital un,t

Output transduceror readout

This chapter provides a brief and elementary review of the laws governing electrical currents, the properties of direct and alternating current circuits, and selected methods for measuring electrical quantities.!

An electrical current is the motion of a charge through a medium. In metallic conductors, only electrons are mobile; here, the current involves motion of negative charges only. In media such as ionic solutions and semiconductors, both negative and positive species are mobile and participate in the passage of electricity. '

Electrical UnitsThe unit of charge or quantity of electricity is the coulomb,: C, which is the charge required to convert O;-\....+ I/ IVL

VFIGURE 2-8 Behavior of (a) a series RC circuit and (b) a series RL circuit.

across the plates and prevents the continued flow of electrons. When the current ceases. the capacitor is said to be charged. If the switch now is moved from 1 to 2, electrons will flow from the negatively charged lower plate of the capacitor through the resistor R to the positive upper plate. Again, this movement constitutes a current which decays to zero as the potential between

the two plates disappears; here. the capacitor is said to be discharged. A useful property of a capacitor is its ability to store an electrical charge for a period of time and then to give up the stored charge as needed. Thus, if S in Figure 2-8a is first held at I until C is charged and is then moved to a position between I and 2. the capacitor will remain in a charged condition for

an extended period. Upon moving S to 2, discharge occurs in the same way as it would if the change from 1 to 2 had been rapid. The quantity of electricity, Q, required to charge a capacitor fully depends upon the area of the plates, their shape, the spacing between them, and the dielectric constant for the material that separates them. In addition, the charge, Q, is directly proportional to the applied voltage. That is,

Q = CV

(2-20)

as a counter-potential v when the current first begins, and tends to oppOse the flow of electrons. On the other hand, when the current in a cOnductor ceases, the magnetic field collapses; this process causes the development of a momentary potential which acts to continue the current. The magnitude of the potential v that develops during the increase or decrease in current is found to be directly proportional to the rate of current change di/dt. That is,

electric circuit tend to complement one another. Rate C..-ent Poteatial Cbaages in an RC Circuit. The rate at which a capacitor is charged or discharged is ~te. Consider, for example, the circuit shown in Figure 2-8a. From Kirchhoffs voltage law, we know that at any instant after the switch is moved to position I, the sum of the voltage across C and R (vc and VR) must equal the input voltage~. Thus,

or

Note that the product RC that appears in the last three equations has the units of time; since R = vR/i and C = q/vc, RC = volts coulombs/seconds x coulombs volts

= seconds The term RC is called the time constant for the circuit. The following example illustrates the use of the three equations that were just derived. EXAMPLE Values for the components in Figure 2-8a are ~ = 10 V, R = 1000 C = 1.00 pF or 1.00 x 10-6 F. Calculate: (a) the time constant for the circuit; and (b) i, vc, and VR after two-time constants (t = 2RC) have elapsed. (a) Time constant RC 1000 x 1.00 x 10-6 = 1.00 X 10-3 s or 1.00 ms (b) Substituting t = 2.00 ens in Equation 2-27 reveals

When V is the applied potential in volts and Q is the quantity of charge in coulombs, the proportionality constant C is the capacitance of a capacitor in farads, F. One farad, then, corresponds to one coulomb of charge per applied volt. Most of the capacitors used in electronic circuitry have capacitances in the microfarad (10-6 F) to picofarad (10-12 F) ranges. Capacitance is important in ac circuits, particularly because a voltage that varies with time gives rise to a time-varying charge-that is, a current. This behavior is seen by differentiating Equation 2-20 to give

di v = -Ldt

Y; = (2-23)

Vt + VR

(2-24)

dq dt

= Cdv

dt

(2-21)

By definition (Equation 2-1 ~ the current i is the rate of change of charge; that is, dq/dt = i. Thus,i=

C

dvdt

It is important to note that the current in a capacitor is zero when the voltage is time independent-that is, for a direct current. Because a direct current refers to a steady state, the initial transient current that charges the condenser is of no significance in considering the overall effect of a capacitor on a current that has a de component. Inductance. It is found experimentally that a magnetic field surrounds any conductor as it carries an electric current. The work required to establish this field manifests itself

Here, the negative sign indicates that the induced potential acts to oppose the change in current. The proportionality constant L in Equation 2-23 is termed the inductance of the conductor; it has the units of henrys, H. One henry of inductance produces a counterpotential of one volt when the rate of change of current is one ampere per second. Inductors employed in electronic circuits have inductances that range from a few pH (microhenry) to several H or more. Figure 2-8b shows a series RL circuit, which contains a battery ~,a resistor R, and an inductor L connected in series. The magnitude of L depends upon the number of turns in the wire coil. As was pointed out earlier, a capacitor stores energy as an electric field across a dielectric; an inductor, on the other hand, stores energy as a magnetic field surrounding a conductor. The electric field in the capacitor is proportional to the applied voltage; the magnetic field in the inductor is proportional to the current that exists in the inductor. The difference between a capacitor and an inductor can also be seen by comparing Equations 2-22 and 2-23. The former indicates that a change in potential across a capac' itor results in a current; the latter shows that a change in current through an inductor causes the development of a potential. Thus, the functions of the two components in an

Because Y; is constant; the increase in Vc that accompanies the charging of the capacitor must be exactly offset "y a decrease in VR Substitution of Equations 2-4 and 2-20 into this equation gives, upon rearrangement, Y;=~+iR Differentiating with ~pect C to time t yields

n.

=

=

0= dq/~t + R ~ .dt

(2-26)

.1=

10.0 -200/100 lOOOe . . or 1.35 mA

Here again, we have used lower case letters to represent instantaneous charge and current. As noted earlier, dq/dt = i. Substituting this expression into Equation 2-26 yields, upon rearrangement, di dt -= - RC

= 1.35 X 10-3

- We find from Equation 2-28 thatVR

= 10.0e-2.00/1.00 = 1.35 V

and by substituting into Equation 2-29V(" =

10.0 (I -

e-2.0011.00)

= 8.65 V

Integration between the limits of the initial current link and i gives i=linite-/Rc

(2-27)

In order to obtain a relationship between the instantaneous voltage across the resistor, Ohm's law is emplo)led to replace i and lini' in Equation 2-27. Th~s,VR

=

Y;e-/RC

(2-28)

Substitution of this expression into Equation 2-24 yields, upon rearrangement,Vc

= Y;(1 -

e-/RC)

(2-29)

The center illustration in figure 2-8a shows the changes in i, 11K, and Vc that occur during the charging cycle of an RC circuit. These plots were based upon the data given in the example just considered. Note that VR and i assume their maximum values the instant the switch in Figure 2-8a is moved to 1. At the same instant, on the other hand, the voltage across the capacitor incr~ rapidly from zero and ultimately approaches a constant value. For practical purpOses, a capacitor is considered to be fully charged after 5RCs have elapsed. At this point the current

will have decayed to less than 1 % of its initial value (e-5 = 0.0067 ~ 0.01). When the switch in Figure 2-8a is moved to position 2, the battery is removed from the circuit and the capacitor becomes a source of current. The flow of charge, however, will be in the opposite direction from what it was previously. Thus, dqldt = -i The initial potential battery. That is, will be that of the

It is useful to compare the behavior of RC and RL circuits during a variation in signaL The plots in Figure 2-8 illustrate that the two reactances are alike in the sense that both show a potential change which is out of phase with the current. Note again the complementary behavior of the two; for an inductor, the voltage leads the current, while for a capacitor, the reverse is the case. Response of Series RC and RL Circuits to Sinusoidal Inputs In the sections that follow, the response of series RC and RL circuits to a sinusoidal ac voltage signal will be considered. The input signal is described by Equation 2-17; that is,

VR

2OV""

v,

-+Vc

R 15kn

~~ 0 -0.5

"

\,\1,\

>

o,

C O.OO8I'F

a-

0.75 kHz"

~ "

E

VC=V; Employing these equations and proceeding as in the earlier derivation, we find that for the discharge cycle. 1= VR = Vc' -t/RC -lie

(2-30) (2-31)

v.

=

v" sin cot = V" sin 2nft

(2-35)

-

Vce-t/RC

andVc

=

Vce-t/RC

(2-32)

Voltage, Current, and Phue Relationship for Series RC Circuit. Application of Kirchholrs voltage law to the circuit shown in Figure 2-9a leads to the expressionv.

The bottom plot of Figure 2-8a shows how these variables change with time. It is important to note that in each cycle, the change in voltage across the capacitor is out of phDse with and lags behind that of the current and the potential across the resistor. Rate of Current and Potential Change Across an RL Circuit. Equation 2-23 and techniques similar to those in the previous section can be used to derive a set of expressions for the RL circuit shown in Figure 2-8b that are analogous to Equations 2-27 through 2-32. For example, when the switch is closed to position 1,VR =

=

Vc

+ VR

= V" sin cot FIGURE 2-9 Current, voltage, and phase relationships for an RC circuit: (a) Circuit, (b) currents at 0.75 and 7.5 kHz, (c) voltages at 0.75 kHz, and (d) voltages at 7.5 kHz.

Substitution of Equations 2-20 and 2-4 yields v" sin cot = ~ Differentiating gives

+ Ri

and replacing dqldt with i di C + Rdi

cov" cos cot =

The peak current occurs sin(wt + ep) is unity. Thus,

whenever

the

The solution of the differential equation can be shown to be5i=

JR

2

V;(1 -

e-R1L)

(2-33)

+

~

l/coe)

2

sin(cot + ep) (2-36)

I , - JR2

V

+ (1/coC)2

I'

(2-38)

cies and capacitance, on the other hand, ep becomes negligible and, for all practical purposes, the current and input voltage are in phase. Under these circumstances, (llwe) becomes negligible with respect to R and Equation 2-38 reverts to Ohm's law.

where the phase angle tan

ep

is given by coRC

1 ep=--

where LIR is the time constant for the circuit. These relationships for a typical RL circuit are shown in Figure 2-8b.

, J. J. Brophy, Basic Electronics/or Scientists. New York: McGraw-Hil~ 1966. pp. 57-59.

A comparison of Equations 2-36 and 2-35 reveals that the current in a series RC circuit is often not in phase with the input voltage signaL This condition is likely to prevail whenever the frequency (or the capacitance) is small; then, tan ep and thus ep (Equation 2-37) becomes significant with respect to cot in Equation 2-36. At sufficiently high frequen-

EXAMPLE AssuUJe the components in the RC circuit shown in Figure 2-9a have values of V" = 20V,R = 1.5 x 104n,and C= 0.OO80pF. (a) Calculate the peak current (1) if f = 0.15 kHz, (2) f = 75 kHz, and (3) with the

capacitor removed from the circuit. (b) Calculate the phase angles between the voltage and the current at the two frequencies.

and at 75 kHz arc tan

(a) (1)1.1 1 we = 2~jC = 2n x 0.75 x 103 x 8.0 = 2.65X X

q, =

arc tan

2.65 x W 1.5 x 104R v, L

= arc tan 0.0177 = 1.0 deg 109

Er::

104 n

Note that the quantity l/2njC has the dimensions of ohms. That is, 1 1 jC=s-ICV-I But the number of coulombs C is equal to the product of the current in amperes and the time in seconds. Thus,

_1_jC-s-IAsV-1-

1_!_nA-

Employing Equation 2-38, we write 1"=

20

J(1.5 x W)2 + (2.65 x 104)2

= 6.6 x 10-4 A

(2) 1= 2.65X 102

1X

2njC = 2n x 75 x 103 x 8.0

10-9

n

120 " - J(1.5 x W)Z + (2.65 x= 1.33 x 10-3 A (3) In the absence of C I" = 1.5

W?

20X

104 = 1.33 x 10

-3

Figure 2-9b, c, and d shows the instantaneous current, voltage, and phase relationships when the circuit characteristics are the same as those employed in the example. The time scales on the abscissas differ by a factor of 100' to accommodate the two frequencies. From Figure 2-9b and the example, it is evident that at the lower frequency, the current is significantly smalIer than at the higher because of the greater reactance of the capacitor at the lower frequency. This larger reactance is also reflected in the large phase difference between the current and the input voltage shown in Figure 2-9c (q, = 60.5 deg). At 75 kHz, this difference becomes considerably sm~ler (Figure 2-9d). Figures 2-9c and 2-9d show the relationship between the input voltages and the voltage drops across the capacitor and the resistor. At the lower frequency, the drop across the capacitor is large with respect to that across the resistor. In contrast, at 75 kHz, lie is small, indicating little reaction between the current and the capacitor. At frequencies greater than 1 MHz, essentialIy no interaction would occur, and the arrangement would behave as a purely resistive circuit. Note also the phase relationships among t: vc, and VR shown in Figure 2-9c. Voltage, Current, and Phase Relationships for Series RL Circuits. Applying Kirchhoff's voltage law to the circuit in Figure 2-10 gives v. =VR

20V'"

U

~ :>

>

-5>

f

0

~

~

FIG UR E 2-10 Current, voltage, and phase relationships in an RL circuit: (a) Circuit, (b) currents at 20 and 200 kHz, (c) voltages at 200 kHz, and (d) voltages at 20 kHz.

in an inductor opposes the current; thus the polarity of Ri and L di/dt must be the same. Integration of this equation yields6I

The peak current occurs where R sin (rot is equal to unity, or I _

+ q,)

.=

V" . A.) J R 2 + (roL) 2 SID(rot + 'I'

(2 - 39)

,,- JR

2

+ (roL)2

V"

A

+ VL

= V" sin rot tanq,=roLR

(b) Employing Equation 2-37, we find at 0.75 kHz. 2.65 x W tan q, = 1.5 X 104 = 1.77

SUb~tituting Equations. 2-4 and 2-23 yields V, sin rot = Ri

+ L di/dt

q, =

60.5 deg

The second term on the right-hand side of the equation is positive because the minus sign in Equation 2-23 means that the induced voltage

J. J. Brophy. Basic Eleclronicsfor McGraw-Hili. 1966. pp. 67-69.

Scientists. New York:

Figure 2-10 shows the relationships among currents, voltages, and phases for an RL circuit. Comparison with Figure 2-9 again reveals the complementary nature of the two types of reactive components. For inductors, reaction to the current occurs at high frequencies but tends to disappear at low. Here, the current lags the input potential rather than leading it.

Capacitative .nd Inductive Reactance; Impedance. An examination of Equations 2-41 and 2-38 shows a similarity to Ohm's law, with the denominator terms being an expression of the impedance exerted by the circuit to the flow of electricity. Note that at high frequencies, where l/wC ~ R, Equation 2-38 reverts to Ohm's law; similarly, when wL ~ R, Equation 2-41 behaves in the same way. The term l/wC in the denominator of Equation 2-38 is termed the capacitive reactance Xc, where

Similarly, the inductive reactance XL is defined as (see Equation 2-41) XL = wL = 27tjL (2-43)

cuits containing capacitive and inductive elements. A convenient way of visualizing these effects is by means of vector diagrams. Vector Diagrams for Reactive Circuits. Because the voltage lags the current by 90 deg in a pure capacitance, it is convenient to let q, equal -90 deg for a capacitive reactance. The phase angle for a pure inductive circuit is +90 deg .. For a pure resistive circuit, q, will be equal to 0 deg. The relationship ainong Xc, XL' and R can then be represented vectorially as shown in Figure 2-11. As indicated in Figure 2-11c, the impedance of a series circuit containing a resistor, an inductor, and a capacitor is determined by the relationship Z = JR2

XL

0z~ R Za .JR2+xL2XL ~=ilfCtanR

---:I I

+ (XL - Xc>2

(2-46)

zo:lIE

The impedance Z of the two circuits under discussion is a measure of the total effect of resistance and reactance and is given by the denominators of Equations 2-38 and 2-41. Thus, for the RC circuit

Because the outputs of the capacitor and inductor are 180 deg out of phase, their combined effect is determined from the difference of their reactances.EXAMPLE

.JR' +(XL -Xci' (X, - Xci arc tan -R--

FIGURE 2-11 Vector diagrams for series circuits: (a) RC circuit, (b) RL circuit, (c) series RLC circuit.

Z=JR2and for the RL circuit Z

+X~(2-45) 2-42 and 2-44

For the following circuit, calculate the peak current and voltage drops across the three components.

= JR2 + xl

Substitution of Equations into Equation 2-38 yields 1 = V"

, ZZ

A similar result is obtained when Equations 2-43 and 2-45 are substituted into Equation 2-41. In one sense, the capacitive and inductive reactances behave in a manner similar to that of a resistor in a circuit-that is, they tend to impede the flow of electrons. They differ in two major ways from resistance, however. First, they are frequency dependent; second, they cause the current and voltage to differ in phase. As a consequence of the latter, the phase angle must always be taken into account in considering the behavior of cir-

= J(SO)2

+ (40 - 2W

= 53.8 nA

1" = 10 V/53.8

n = 0.186 = 3.7 V

VR = 0.186 x SO = 9.3 V Vc = 0.186 x 20

VL = 0.186 x 40 = 7.4 V

Note that the sum of the three voltages (20.4) in the example exceeds the source voltage. This situation is possible because the voltages are out of phase, with peaks occur-

ring at different times. The sum .of the instantaneous voltages, however, would add up to 10V. High-Pass .nd Low-Pass Fikers. Series RC and RL circuits are often used as filters to attenuate high-frequency signals while passing low-frequency components (a low-pass filter) or, alternatively, to reduce lowfrequency components while passing the high (a high-pass filter). Figure 2-12 shows how series RC and RL circuits can be arranged to give high- and low-pass filters. In each case. the input and output are indicated as the volt~es (Vp)i and (Vp) . ; In order to employ an RC circuit as a high-pass filter, the' output voltage is taken across the resistor R. The peak current in this circuit is given by Equation 2-38. That is, I _I' -

Since the voltage drop across the resistor is in phase with the current,

1 = (VI')., R The ratio of the peak output to the peak input voltage is obtained by dividing the first equation by the second and rearranging. Thus,

JR

(Vp);2

+ (l/wC)2

A plot of this ratio as a function of frequency is shown in Figure 2-13a; here, a resistance of 1.0 x 10 n and a capacitance of 0.10 pF was employed. Note that frequencies below 50 Hz have been largely removed from the input signal. It should also be noted that because the input and output voltages are out of

resistor -behavior just opposite of that of the RC circuit. Curves similar to Figure 2-13 are obtaincd for RL filters. Low- and high-pass filters are of great importance in the design of electronic cirCUits. Resonant Circuits A resonant or RLC circuit consists of a resistor, a capacitor, and an inductor arranged in series or parallel, as shown in Figure 2-14. Series Resonant Filters. The impedance of the resonant circuit shown in Figure 2-14a is given by Equation 2-46. Note that the impedance of a series inductor and capacitor is the difference between their reactances and that the net reactance of the combination is zero when the two are identical. That is, the FIGURE 2-12 Filter circuits: (a) Highpass RC filter, (b) low-pass RC filter, (c) highpass RL filter, and (d) low-pass RL filter.

~-----~

energy stored in the magnetic field of the inductor during the other. Thus, during a one-half cycle, the energy of the capacitor forces the current through the inductor; in the other half cycle, the reverse is the case. In principle, a current induced in a closed resonant circuit would continue indefinitely except for the loss resulting from resistance in the leads and inductor wire. The resonant frequency I. can be readily calculated from Equations 2-42 and 2-43. That is, whenf=f., XL = Xc, and 1 2nf.C = 2nf.L Upon rearrangement, it is found that

1.= 21tJfC(bl

1

1.0 O.B

The follQwing example will show some of the properties of a series resonant circuit.

~ ~ ~phase, the peak voltages will occur at different times. For the low-pass filter shown in Figure 2-12b, we may write

0.6 0." 0.2 0.0 1 100 Frequency. Hz

FIGURE 2-14 Resonant circuits employed as filters: (a) series circuit and (b) parallel circuit.

iEXAMPLE Assume the following values for the components of the circuit in Figure 2-14a: (Vp)i = 15.0 V (peak voltage), L= 100 mH, R = 20 n, and C = 0200 JLF. (a) Calculate the peak current at the resonance frequency and at a frequency 75 Hz greater than the resonance frequency. (b) Calculate the peak potentials across the resistor, inductor, and capacitor at the resonant frequency. (a) We obtain the resonant frequency with Equation 2-49. Thus, 1

::

(Vp). = IpXcSubstituting Equation 2-42 gives

(al

(Vp). = Ip/(roC)Multiplying by Equation 2-38 and rearranging yields (Vp)., (VP)i 1 1

O.B

~

0.6 0.4

;0-

"i

=

roCJR2

+ (l/roC)2

= j(roCR)2 + 1(2-48)Frequency. Hz(b)

Figure 2-13b was obtained with this equation. As shown in Figure 2-12, RL circuits can also be employed as filters. Note, however, that the high-pass filter employs the potential across the reactive element while the low-pass filter employs the potential across the

only impedance in the circuit is that of the resistor or, in its absence, the resistance of the inductor coil and the other wiring. This behavior becomes understandable when it is recalled that in a series RC circuit, the potentials across the capacitor and the inductor are 180 deg out of phase (see the vector diagrams in Figure 2-11). Thus, if Xc and XL are alike, there is no net potential drop across the pair. Electricity continues to flow, however; therefore, the net reactance for the combination is zero. Also, when XL and Xc are not identical, their net reactance is simply the difference between the two. The condition for resonance isXL= Xc

1.;= 2nJl00= 1125 Hz

x 10

3

x 0.2 x 10

6

By defidition, XL = Xc at!.; therefore, Equation 2-46 becomes

FIGURE 2-13 Frequency response of (a) the high-pass filter shown in Figure 2-118 and (b) the low-pass filter shown in Figure 2-12b; R = 1.0 X 104 nand C = 0.10 JLF.

Here, the energy stored in the capacitor during one half cycle is exactly equal to the

To calculate I, at 1200 Hz, we substitute Equations 2-42 and 2-43 into Equation 2-46. Thus, Z = =

JR

2

+ [27tjL + [27t x + (754 -

(1/27tfC)j2 1200 x 100 x 10

J202

- 1/(27t x 1200 x 0.200 x lO-ti)j2 = J400 663)2 = 93 0

and I, = 15.0/93 = 0.16 A. (b) At f. = 1125 Hz, we have found that I,= 0.75 A. The potentials across the resistor, capacitor, and inductor (V,)Il' (V,)L, and (J-;,)c are

(V')ll

= 0.75= 0.75

x 20

= 15.0 VFrequency, Hz

1050

1100 . Frequenc:y. Hz

(V,)L = I,XLX

27t x 1125 x 100 x 10-3

= 530 V Since XL = Xc at resonance,f

FIGURE 2-15 Frequency response of a series resonant circuit; R = 20Q,L = l00mH, C 0.2 IlF, and (J-;,), 15.0 V.

=

=

FIGURE 2-16 Ratio of output to input voltage for the series resonant filter; L = 100 mH, C = 0.20 IlF, and (v,,)i = 15 V.

Part (b) of this example shows that, at resonance, the peak voltage across the resistor is the potential of the source; on the other hand, the potentials across the Inductor and capacitor are over 35 times greater than the input potential. It must be realized, however, that these peak potentials do not occur simultaneously. One lags the current by 90 deg and the other leads it by a similar amount. Thus, the instantaneous potentials across the capacitor and inductor cancel, and the potential drop across the resistor is then equal to the input potential. Clearly, in a circuit of; this type, the capacitor and inductor may have to withstand considerably larger potenti;als than the amplitude of the input voltage would seem to indicate. Figure 2-15, which was obtained for the circuit shown in the example, demonstrates that the output of a series resonant circuit is a narrow band of frequencies, the maximum of which depends upon the values chosen for L and C.

If the output of the circuit shown in Figure 2-14a is taken across the inductor or capacitor, a potential is obtained which is several times larger than the input potential (see part b of the example). Figure 2-16 shows a plot of the ratio of the peak voltage across the inductor to the peak input voltage as a function of frequency. A similar plot is obtained if the inductor potential is replaced by the capacitor potential. The upper curve in Figure 2-16 was obtained by substituting a 10-0 resistor for the 20-0 resistor employed in the example. Parallel Resonant Filters. Figure 2-14b shows a typical parallel resonant circuit. Again, the condition of resonance is that Xc = XL and the resonant frequency is given by Equation 2-49. The impedance of the parallel circuit is given by Z =

filter (Equation 2-46). Note that the impedance in the latter circuit is a minimum at resonance, when XL = Xc. In contrast, the impedance for the parallel circuit at resonance is a maximum and in principle is infinite. Consequently, a maximum voltage drop across (or a minimum current through) the parallel reactance is found at resonance. With both parallel and series circuits at resonance, a small initial signal causes resonance in which electricity is carried back and forth between the capacitor and the inductor. But current from the source through the reactance is minimal. The parallel circuit, sometimes called a tank circuit, is widely used as for tuning radio or television circuits. Tuning is ordinarily accomplished by adjusting a variable capacitor until resonance is achieved. Behavior of RC

ing upon the relationship between the widtp of the pulse and the time constant for the circuit. These effects are illustrated in Figure 2-17 where the input is a square wave having a pulse width of 1",' seconds. The second column shows the variation in capacitor potential as a function of time, while the third column shows the change in resistor potential at the same times. In the top set of plots (Figure 2-17a~ the time constant of the circuit is much greater than the input pulse width. Under these circumstances, the capacitor can become only partially charged during each pulse. It then discharges as the input potential returns to zero; a sawtooth output results. The output of the resistor under these circumstances rises instantaneously to a maximum value and then decreases essentially linearly during the pulse lifetime. The bottom set of graphs (Figure 2-17c) illustrates the two outputs when the time constant of the circuit is much shorter than the pulse width. Here, the charge on the capacitor rises rapidly and approaches full charge near the end of the pulse. As a consequence, the potential across the resistor rapidly decreases to zero after its initial rise. When V;goes to zero, the capacitor discharges immediately; the output across the resistor peaks in a negative direction and then quickly approaches zero. These various output wave forms find applications in electronic circuitry. The sharply peaked voltage output shown in Figure 2-17c is particularly important in timing and trigger circuits.

SIMPLE ELECTRICAL

MEASUREMENTS

JR2 + (XC-XLC XLX

)2

(2-50)

Circuits with Pulsed InputsWhen a pulsed input is applied to an RC circuit, the voltage outputs across the capacitor and resistor take various forms, depend-

It is of interest to compare this equation with the equation for the impedance of the series

This section describes selected methods for the measurement of current, voltage, and resistance. More sophisticated methods for measuring these and other electrical properties will be considered in Chapter 3 as well as in later chapters.

f1o-------AJV:J

The Ayrton Shunt. The Ayrton shunt, shown in Figure 2-19, is often employed to vary the range of a galvanometer. The example that follows demonstrates how the resistors for a shunt can be chosen.Shunt

Meter

'" :\,

'-

~R, 2 R. 3 R, 4

' ..!!..AftR.

(b)

RCO!!T.

(el RC~T.

1..

T+l .J LJnLJnL T+ !7l n JVi 0

1..

Vc 0

o

t

LJ U

:Tl iT1

_

L

T+~' 1.._V" 0

I

-,

I

-,

I

-

--Time-

0 --T~_

o

--Time---"

EXAMPLE Assume that the meter in Figure 2-19 registers full scale with a current of SO JlA; its internal resistance is 3000 n. Derive values for the four resistors such that the meter ranges will be 0.1, 1, 10, and 100 mA. For each setting of the contact, the circuit is equivalent to two resistances in parallel. One resistance is made up of the meter and the resistors to the left of the contact in Figure 2-19; the other consists of the remaining resistor to the right of the contact. The potential across parallel resistances is, of course, the same. Thus, IM(RM

t '.

FIGURE 2-19Ayrton shunt.

A current meter with an

We may also write, from Kirchholfs current law

FIGURE 2-17 Output signals VR and Vc for pulsed input signal J-l; (a) time constant ~ pulse width Tp; (b) time constant ~ pulse width; (c) time co~tant .-;""'"

Nonparallel radiation

Radiation ----Pumping

JZ:"

Active I~ing mediu'!'

.

=s --.transmitting mirror .

'X-i- ~of a typical

FIGURE 5-3 laser source.

Schematic representation

Lase,.The first laser was constructed in 1960.2 Since that time, chemists have found numerous useful applications for these sources in highresolution spectroscopy, kinetic studies of

processes with lifetimes in the range of 10-9 to 10-12 s, the detection and determination of extremely small concentrations of species in the atmosphere, and the induction of is0topically selective reactions.3 In addition, laser sources have become important in several routine analytical methods, including Raman spectroscopy (Chapter 91 emission spectroscopy (Chapter 12), and Fourier transform infrared spectroscopy (Chapter 8). The term laser is an acronym for light amplification by stimulated emission of radiation. As a consequence of their lightamplifying properties, lasers produce spatially narrow, extremely intense beams of radiation. The process of stimulated emission produces a beam of highly monochromatic (bandwidths of 0.01 nm or less) and remarkably coherent (p. 99) radiation. Because of these unique properties, lasers have become important sources for use in the ultraviolet, visible, and infrared regions of the spectrum. A limitation of early lasers was that the radiation from a given source was restricted to a relatively few discrete wavelengths or lines. Recently, however, dye lasers have become available; tuning of these sources provides a

For a more complete discussion or lasers. see: B. A. Lengye~ lAsers. New York: Wiley, 1971; S. R. Leone and C. B. Moore, Chemical allll Biological A.pplications of lAsers, eel. C. E. Moore. New York: Academic Press, 1974; and A.nalytical lAser Spectroscopy. eel. N. Omenetlo. New York: Wiley, 1979.1

For a review or some or Ihese applicalions. see: S. R. Leone, J. Chem. Educ.,53, 13 (1976~

narrow band of radiation at any chosen wavelength within the range of the source. Figure 5-3 is a schematic representation showing the components of a typical laser source. The heart of the device is a lasing medium. It may be a solid crystal, such as ruby, a semiconductor such as gallium arsenide. a solution of an organic dye, or a gas. The lasing material can be activated or pumped by radiation from an external source so that a few photons of proper energy will trigger the formation of a cascade of photons of the same energy. Pumping can also be carried out by an electrical current or by an electrical discharge. Thus, gas lasers usually do not have the external radiation source shown in Figure 5-3; instead, the power supply is connected to a pair of electrodes contained in a cell filled with the gas. A laser normally functions as an oscillator, in the sense that the radiation produced from the laser mechanism is caused to pass back and forth through the medium numerous times by means of a pair of mirrors as shown in Figure 5-3. Additional photons are generated with each passage, thus leading to enormous amplification. The repeated passage also produces a beam that is highly parallel because nonparallel radiation escapes from the sides of the medium after being reflected a few times (see Figure 5-3~ In order to obtain a usable laser beam, one of the mirrors is coated with a sufficiently

thin layer of reflecting material so that a fraction of the beam is trilnsmitted rather than reflected (see Figure 5-3). . Laser action can be understood by considering the four processes depicted in Figure 5-4, namely, (a) pumping, (b) spontaneous emission (fluorescence1 (c) stimulated emission, and (d) absorption. For purposes of illustration, we will focus on just .two of several electronic energy levels that wil( exist in the atoms, ions, or molecules making up the laser material; as shown in the figure, the two electronic levels have energies E, and Ex. Note that the higher electronic state is shown as having several slightly different vibrational energy levels, E" E~, E;, and so forth. We have not shown additional levels for the lower electronic state, although such often exist. Pumping. Pumping, which is necessary for laser action, is a process by which the active species of a laser is excited by means of an electrical discharge, passage of an electrical current, exposure to an intense radiant source, or interaction with a chemical species. During pumping, several of the higher electronic and vibrational energy levels of the active species will be populated. In diagram a-I of Figure 5-4, one atom or molecule is shown as being promoted to an energy state E;; the second is excited to the slightly higher vibrational level The lifetime of excited vibrational states is brief, however; after 10-13 to 10-14 S, relaxation to the lowest excited vibrational level (E, in diagram a-3) occurs with the production of an undetectable quantity of heat. Some excited electronic states of laser materials have lifetimes considerably longer (often I ms or more) than their excited vibrational counterparts; longlived states are sometimes termed metastable as a consequence. Spontaneous Emission. As was pointed QUt in the discussion of fluorescence, a species in an excited electronic state may lose all or part of its excess energy by spontaneous emission of radiation. This process is depicted in the

E;.

diagrams shown in Figure 5-4b. Note that the wavelength of the fluorescent radiation is directly related to the energy difference between the two electronic states, E, - Ex. It is also important to note that the instant at which the photon is emitted, and its direction, will vary from excited electron to excited electron. That is, spontaneous emission is a random process; thus, as shown in Figure 5-4, the fluorescent radiation produced by one of the particles in diagram b-I differs in direction and phase from that produced by the second particle (diagram b-2). Spon.taneous emission, therefore, yields incoherent radiation. Stimulated Emission. Stimulated emission, which is the basis of laser behavior, is depicted in Figure 5-4c. Here, the' excited laser particles are struck by externally produced photons having precisely the same energies (E,- Ex) as the photons produced by'spontaneous emission. Collisions; of this type cause the excited species to r~lax immediately to the lower energy statei and to emit simultaneously a photon of exactly the same energy as the photon that stimulated the process. More important, and more remarkable, the emitted photon is precisely in phase with the photon that triggered the event. That is, the stimulated emission is totally coherent with the incoming radiation. . Absorption. The absorption process, which competes with stimulated emission. is depicted in Figure 5-4d. Here, two photons with energies exactly equal to E, - Ex are absorbed to produce the metastable excited state shown in diagram d-3; note that the state shown in diagram d-3 is identical to that attained by pumping diagram a-3. Population Inversion a'" Light Amplification. In order to have light amplification in a laser, it is necessary that the number of~hotons produced by stimulated emission exceed the number lost by absorption. This condition will prevail only when the number of particles in the higher energy state exceeds the number in the lower; that is, a population inversion from the normal distribution of energy states must exist.

(1)

=+=~. .,- -rPumping

-_

EM E

:: ~

Heat

energy

I I

II

-+-

+E.

~~

-----

__=cvu-

----

FIGURE 5-4 Four processes important in laser action: (a) pumping (excitation by electrical. radiant. or chemical energy). (b) spontaneous emission. (c) stimulated emission. and (d) absorption.

Population inversions are brought about by pumping. Figure 5-5 contrasts the effect of incoming radiation on a noninverted population with an inverted one. 11Iree- Four-Level LIser Systems. Figure 5-6 shows simplified energy diagrams for the two common types of laser systems. In the three-level system, the transition responsible for laser radiation is between an excited state E, and the ground state Eo; in a four-level system, on the other hand, radiation is generated by a transition from E, to a state Ex that has a greater energy than the ground state. Furthermore, it is necessary that transitions between Ex and the ground state be rapid. The advantage of the four-level system is that the population inversions necessary for laser action are more readily achieved. To understand this, note that at room temperature a large majority of the laser particles will be in the ground-state energy level Eo'in both systems. Sufficient energy must thus be provided to convert more than 50% of the lasing species to the E, level of a three-level

system. In contrast, it is only necessary to pump sufficiently to make the number of particles in the E, energy level exceed the number in Ex of a four-level system. The lifetime of a particle in the Ex state is brief, however, because the transition to Eo is fast; thus, the number in the Ex state will generally be negligible with respect to Eo and also (with a modest input of pumping energy) with respect to E,. That is, the four-level laser usually achieves a population inversion with a smaller expenditure of pumping energy. Some Examples of Useful Lasers. The first successful laser, and one that still finds widespread use, was a three-level device in which a ruby crystal was the active medium. The ruby was machined into a rod about 4 em in length and 0.5 em in diameter. A ftash tube was coiled around the cylinder to produce intense ftashes of light. Because the pumping was discontinuous, a pulsed laser beam was produced. Continuous wave ruby sources are now available. A variety of gas lasers are sold commercially. An important example is the argon ion

E." ____ FISttransition nonradiative Ey

E.

:rE~ 1E,/--

Fast transition

FIGURE 6-6 Energy level diagrams for two types of laser systems.

AbIofption "',

Ey

rv\ri

rvvE, lal

I

V\rStimulated / emission

'UlrE,(b)

Ey

Light amplification

by stimulated emission

laser, which produces intense lines in the green (514.5 om) and blue (488.0 nm) regions. This laser is a four-level device in which argon ions are formed by an electrical or radio frequency discharge. The input energy is sufficient to excite the ions from their ground state, with a principal Quantum number of 3, to various 4p states. Laser activity then involves transitions to the 4s state. Dye lasers4 have become important radiation sources in chemistry because they are tunable over a range of 20 to 50 nm; that is, dye lasers provide bands of radiation of a chosen wavelength with widths of a few hundredths of a nanometer or less. The active materials in dye lasers are solutions of organic compounds capable.()fftuorescing in the ultraviolet. visible. or infrared regions. In contrast to ruby or gas lasers. however, the lower energy level for laser action (Ex in Figure 5-6) is not a single energy but a band of energies arising from the superposition of a large number of small vibrational and rotational energy states upon the base electronic energy state. Electrons in E). may then undergo transitions to any of these states, thus producing photons of slightly different energies.

Tuning of dye lasers can be readily accomplished by replacing the nontransmitting mirror shown in Figure 5-3 with a monochromator equipped with a reftection grating or a Littrow-type prism (p. 126) which will reftect only a narrow bandwidth of radiation into the laser medium; the peak wavelength can be varied by rotatioQ of the grating or prism. Emission is then stirllulated for only part of the fluorescent spec(rum, namely. the wavelength reflected from the monochromator.

WAVELENGTH SELECTION; MONOCHROMATORS With few minor exceptions, the analytical methods considered in the following seven chapters require dispersal of polychromatic radiation into bands that encompass a restricted wavelength region. The most common way of producing such bands is with a device called a monochromator.'

FIGURE 6-6 Passage of radiation through (a) a noninverted population and (b) an inverted population.

For rurther inrormation, see: R. B. Green, - Dye Laser Instrumentation," J. Chem. Educ. 54 (9) A365, (10) A407,(1977~

For a more complete d~ussion or monochromators, see: R. A. Bauman, AbJO,ption Spect,OM:OPY. ew York: N Wiley, 1962; F. A. Jenkins and H. E. White, FUMamentats of Optia. 3d ed. New York: McGraw-HiI~ 1957; E. J. Meehan, in T,etIliM 011 A""lytiClJI CMmUt'Y. cds. I. M. KolthofTand P. J. Elving. New York: Wiley, 1964. Part t, vol. 5. Chapter 55: and J. F. James and R. S. Sternberg, The Design of Optical Spectromners. London: Chapman and Hall. 1969,

j

... A

~~7Bsfit

Focal plane

A2/Exit/

grating, angular dispersion results. from diffraction, which occurs at the reflective surface. In both ~esigns, the dispersed radiation is focused on the focal plane AB where it appears as two "images of the entrance slit (one for each wavelength). Types of Instruments Employing Monochromators The method of detecting the radiation dispersed by a monochromator varies. In a spectroscope, detection is accomplished visually with a movable eyepiece located along the focal plane. The wavelength is then determined by measurement of the angle between the incident and dispersed beams. In a spectrograph, a photographic film or plate is mounted along the focal plane; the series of darkened images of the slit along the length of the developed film or plate is called a spectrogram. The monochromators shown in Figure 5-7 are also utilized for dispersing elements in spectr~ters. Such instruments have a fixed exit slit located in the focal plane. With a continuous source, the wavelength emitted from this slit can be varied continuously by rotation of the dispersing element. We need to define two other types of instruments to complete this discussion. A spectrophotometer is a spectrometer which contains a photoelectric device for determining the power of the radiation exiting from the slit. A photometer also employs a photoelectric detector but contains no monochromator; instead, filters are used to provide radiation bands encompassing limited wavelength regions. A photometer is incapable of providing a continuously variable band of radiation. Prism Monochromators Prisms can be used to disperse ultraviolet, visible, and infrared radiation. The material used for their construction will differ depending upon the wavelength region.

A

(bl

FIGURE 5-7 Two types of monochromators: (a) Bunsen prism monochromator, and (b) Czerney-Tumer grating monochromator. (In both instances, 11 > 12,)

Mooochromators for ultraviolet, visible, and infrared radiation are all similar in mechanical construction in the sense that they employ slits, lenses, mirrors, windows, and prisms or gratings. To be sure, the materials from which these components are fabricated will depend upon the wavelength region of intended use (see Figure 5-2b and c). Components of Monochromator All monochromators contain an entrance slit, a collimating lens or mirror to produce a parallel beam of radiation, a prism or grating as a dispersing element, 6 and a focusing element

Much less gencraIIy employed are interference wedges. These devices are described in the next section (p. 136).

which projects a series of rectangular images of the entrance slit upon a plane surface (the focal plane). In addition, most monochromators have entrance and exit windows, which are designed to protect the components from dust and corrosive laboratory fumes. Figure 5-7 shows the optical design of two typical monochromators, one employing a prism for dispersal of radiation and the other a grating. A source of radiation containing but two wavelengths, 11 and 12, is shown for purposes of illustration. This radiation enters the monochromators via a narrow rectangular opening or slit, is collimated, and then strikes the surface of the dispersing element at an angle. In the prism monochromator, refraction at the two faces results in angular dispersal of the radiation, as shown; for the

Construedoa ~terials. Clearly, the materials employed for fabricating the windows lenses, and prisms of a monochromator must transmit radiation in the frequency range of interest . ideally, the transmittance should approach tOO%, although it is sometimes necessary to use substances with transmittances as low as 20%. The refractive index of materials used in the construction of windows and prisms should be low in order to reduce reflective losses (see Equation 4-14, p. 103). However, for lenses which function by bending rays of radiation, a material with a high refractive index is desirable in order to reduce focal lengths. Ideal construction materials for both lenses and windows should exhibit little change in refractive index with frequency, thus reducing chromatic aberrations. In contrast, just the opposite property is sought for prisms, where1he extent of dispersion depends upon the rate of change of refractive index with frequency. In addition to the foregoing optical properties, it is desirable that monochromator components be resistant to mechanical abrasion and attack by atmospheric components and laboratory fumes. Needless to say, no single substance meets all of these criteria and trade-offs are thus required in monochromator design. The choices here depend strongly upon the wavelength region to be used. Table 5-1 lists the properties of the common substances employed for fabricating monochromator components. Clearly, no single material is suitable for the entire wavelength region. For the ultraviolet, visible, and near-infrared regions (up to about 3000 nm~ a quartz prism is often employed; it should be noted, however, that glass provides better resolution for the same size prism for wavelengths between 350 and 2000 nm. As shown in column 5 of Table 5-1, several prisms are required fo cover the entire infrared region. Types of Prisms and Prism Monochromators. Figure 5-8 shows the two most common types of prism configurations. The first is a 6O-deg design, which is ordinarily fabricated

FIGURE 5-8 Dispersion by a prism: (a) quartz Cornu type, and (b) Littrow type.

from a single block of material. When crystalline (but not fused) quartz is employed, however, the prism is usually formed by cementing two 3().deg prisms together, as shown in Figure 5-8a; one is constructed from right-handed quartz and the second from left-handed quartz. In this way, the opticaUyactive quartz causes no net polarization of the emitted radiation; this type of prism is

called a Cornu prism. Figure 5-7a shows a Bunsen monochromator, which employs a 6O-deg prism, likewise often made of quartz. Figure 5-8b depicts a Littrow prism, which is a 3().deg prism with a mirrored back. It is seen that refraction occurs twice at the same interface; the performance characteristics of the Littrow prism are thus similar to those of the 6O-deg prism. The Littrow prism permits

somewhat more compact monochromator designs. In addition, when quartz is employed, polarization is canceled by the reversal of the radiation path. A typical Littrow-type monochromator' is shown in Figure 7-10. Angular DispersioR of ~ The angular dispersion of a prism is the rate of change of the angle 8 in Figure 5-8 as a function of wavelength; that is, d8/dl. The spectral purity of the radiation exiting from a prism monochromator is dependent upon this quantity. The angular dispersion of a prism can be resolved into two parts d8 d)'

= dn . d)'

d8 dn

(5-1)

TABLE 11 OPTICS

CONSTRUCTION

MATERIAL

FOR SPECTROPHOTOMETER

MaterialFused silica Quartz Flint glass Calcium fluoride ~fluorite) Lit ium fluoride Sodium chloride Potassium bromide Cesium iodide KRS-5 (TlBr-TlI)

Tl'lIDSIIlittance Range, pm 0.18-3.3 0.20-3.3 0.35-2.2 0.12-12 0.12-6 0.3-17 0.3-29 0.3-70 1-40

RefractiveIndex 1.46 1.54 1.66 1.43 1.39 1.54 1.56 1.79 2.63

0.52 x 0.63 x 1.70 x 0.33 x

=w10-4 10-4 10-4 10-4 1.45 x 10-4

Uleful 1tuIe fer Prisms, IJI110.18-2.7 0.20-2.7 0.35-2 5-9.4 2.7-5.5 8-16 15-28 15-55 24-40

HarUelsDd CIIemicaI Resistance Excellent Excellent Excellent Good Poor Poor Poor Poor Good

where d8/dn represents the change in 8 as a function of the refractive index n of the prism material and dn/d)' expresses the variation of the refractive index with wavelength (that is, the dispersion of the substance from which the prism is fabricated). The magnitude of dO/dn is determined by the geometry of the prism and the angle of incidence i (Figure 5-Sa~ In order to avoid problems of astigmatism (double images), this angle should be such that the path of the beam through the prism is within a few degrees of being parallel to the base of the prism. Under these circumstances. dO/dll depends only upon the prism angle ClC (Figure 5-Sa) and increases rapidly with this quantity. Reflection losses, however. impose an upper limit of about 60 deg for IX. For a prism in which ClC = 60 deg, it can be shown that dO/dn = (I - n /4,-t2

rise in the refractive index for glass below 400 Dm corresponds to the sharp increase in absorption that prevents the use of this material below 350 Dm. in the region of 350 to 2000 DID, however, glass is greatly superior to quartz for prism fabrication because of its larger change in refractive index with wavelength (dll/d).). Focal-Plane DiSpersion of Prism Monochromators. The focal-plane dispersion of a monochromator refers to the variation in wavelength as a function of y, the linear distance along the line AB of the focal plane of the instrument (see Figure 5-7). That is, the focal-plane dispersion is given by dy/d)'. Figure 5-10 shows the focal-plane dispersion of two prism monochromators and one grating instrument. Note that the dispersion for the two prism monochromators is highly nonlinear, the longer wavelengths being bunched together over a relatively small distance. The two prism monochromators were Littrow types. cach having a height of 57 mm. Notc the much largcr dispersion exhibited by the monochromator with a glass prism, in the region of 350 to 800 nm.

K

~ .S ~~ .~

1.60

~OCk

0.29 x 10-4 0.94 x 10-4

(5-2)

----------salt .

Radians/ l'1li at 0.589 pm

The term dll/d)' in Equation 5-1 is related to the dispersion of the substance from which the prism is constructed. We have noted (p. 102) that the greatest dispersion for a given material is near its anomalous dispersion region, which in turn is close to a region of absorption. The dispersion of some substances employed for construction of prisms is shown in Figure 5-9. Note that the rapid

1.50

1.40

o

200

400

600

800nm

1000

1200

Wavelength.

FIGURE 5-9 materials.

Dispersion of several optical

Figure 5-10 illustrates one ofthe important advantages of grating mopochromatorslinear dispersion along the focal plane. Resolving Power of Prism Monocbromators. The resolving .power R of a prism gives the limit of its ability to separate adjacent images having slightly different wavelengths. Mathematically, this quantity is defined as

where dl represents the wavelength difference that can just be resolved and l is the average wavelength of the two images. It can be shown that the resolving power of a prism is directly proportional to the length of the prism base b (Figure 5-8) and the dispersion of its construction material. That is,

Gl'IIting Monochromatora for Wavelength Selection Dispersion of ultraviolet, visible, and infrared radiation can be brought about by passage of a beam through a transmission grating or by

reflection from a reflection grating. A transmission grating consists of a series of parallel and closely spaced grooves ruled on a piece of glass or other transparent material. A grating suitable for use in the ultraviolet and visible region has between 2000 and 6000 lines per millimeter. An infrared grating requires considerably fewer lines; thus, for the far infrared region, gratingS with 20 to 30 lines per millimeter may suffice. It is vital that these lines be equally spaced. throughout the several centimeters in length of the typical grating. Such gratings require elaborate apparatus for their production and are consequently expensive. Replica gratings are less costly. They are manufactured I;>y employing a master grating as a mold for' the production of numerous plastic replicas; the products of this process, while inferior in performance to an original grating, are adequate for many applications. When a transmission grating is illuminated from a slit, ~ch groove scatters radiation and thus effedtively becomes opaque. The nonruled portions then behave as a series of closely spaced slits, each of which acts as a new radiation source; interference among the multitude of beams results in diffraction of

Monochromatic

beam It incident ongtel .

. FIGURE 6-11 Schematic diagram illustrating the mechanism of diffraction from an echellette-type grating. (From R. P. Bauman, Absorption Spectroscopy. New York, Wiley, 1962, p. 65. With permission.)

200~. om

I

. 300

Gl'lting 500

I

IGlass prism

200X,om

350

400

450

500

600 800

A-,j\il9~;;; -.-,Quartz prism

I250 300

I350 400

I

I I II500600800

200 A,nm

IAI

I5.0

I

I

'!20.0

I

,

II8 25.0

0

10.0 15.0 Diltance y along focal plane, em

FIGURE 6-10 Dispersion for three types ofmonochromators. The points A and B on the scale correspond to the points shown in Figure 5-7.

the radiation as shown in Figure 4-00 (p. 98). The angle of diffraction, of course, depends upon the wavelength. Reflection gratings are used more extensively in instrument construction than their transmitting counterparts. Such gratings are made by ruling a polished metal surface or by evaporating a thin film of aluminum onto the surface of a replica grating. As shown in figure 5-11, the incident radiation is reflected from one of the faces of the groove, which then acts as a new radiation source. Interference results in radiation of differing wavelengths being reflected at different angles r. Diffraction by a Grating. figure 5-11 is a schematic representation of an echellette-type grating, which is grooved or blazed such that it has relatively broad faces from which reflection occurs and narrow unused faces. This geometry provides highly efficient diffraction of radiation. Each of the broad faces can be considered to be a point source of radiation; thus interference among the

reflected beams 1,2, and 3 can occur. In order for constructive interference to occur between adjacent beams, it is necessary that their path lengths differ by an integral multiple n of the wavelength l of the incident beam. In Figure 5-11, parallel beams of monochromatic radiation 1 and 2 are shown striking the grating at an incident angle i to the grating normal. Maximum constructive interference is shown as occurring at the reflected angle r. It is evident that beam 2 travels a greater distance than beam 1 and that this difference is equal to (CD - AB). For constructive interference to occur, this difference must equal n).. That is, M = (CD - AB) Note, however, that angle CAD is equal to angle i and that angle BDA is identical with angle r. Therefore, from simple trigonometry, we may write

where d is the spacing between the reflecting surfaces. It is also seen that AB= -dsin r

The minus sign, by convention, indicates that reflection has occurred. The angle r, then, is negative when it lies on the opposite side of the grating normal from the angle i (as in Figure 5-1I~ and is positive when it is on the same side. Substitution of the last two expressions into the first gives the condition for constructive interference. Thus,III =

and the wavelengths for the first-, secondo, and thUd-order reflections are 543, 271, and 181 nm, respectively. Similarly, when r = -10 deg (here r lies to the right of the grating n0"ttal) .t = SOO [sin 48 + sine-10)] = 284.7n n

RfIOIviDg Power of a Gntiac- It can be shown 7 that the resolving power R of a grating is given by the very simple expression R = .u = aN

For the grating, we employ Equation 5-7

.t -=aN .uFor the first-order spectrum (D = I) N 460.25 460 3' = 0.10 x I =. x 10 hnes

.t

(5-7)

Wavelength (nm) forr

where n is the diffraction order and N is the number of lines illuminated by the radiation from the entrance slit. Thus, as with a prism (Equation 5-4~ the resolving power depends upon the physical size of the dispersing element. EXAMPLE ~mpare the size of (I) a 6O-deg fused silica' prism; (2) a 6O-deg glass prism; and (3) a grating with 2000 lines/mm, that would be required to resolve two lithium emission lines at 460.20 and 460.30 nm. Average values for tlje dispersion (dn/d.t) of fused silica and glass ~n the region of interest are 1.3 x 10- 4 and 3.6 x 10-4 nm-I, respectively. For the two prisms, we employ Equation 5-4 R=-

I h 4.6 X 103 lines 0.1 em gra t 109 engt = 2000 Ii x -nes/mm mm 0.23 em Thus, in theory, the separation can be accomplished with a silica prism with a base of 3.5 em, a glass prism with a base of 1.3 em, or 0.23 em of grating. Gntinp versus Priims. Gratings offer several advantages over prisms as dispersing elements. One of these is that the dispersion is nearly constant with wavelength; this property makes the design of monochromators considerably more simple. A second advantage can be seen from the foregoing example, namely, that better dispersion can be expected for the same size of dispersing element. In addition, reflection gratings provide a means of dispersing radiation in the far ultraviolet and far infrared regions where absorption prevents the use prisms. Gratings have the disadvantage of producing somewhat greater amounts of stray radiation as well as higher-order spectra. These disadvantages are ordinarily not serious because the extraneous radiation can be minimized through the use of filters and by proper instrument design. For a number of years, high-quality gratings were more expensive than good prisms. This disadvantage has now largely disappeared.

d(sin i + sin r)

(5-5)

n=1 543 458 372 285

n=2 271 229 186 142

n=3 181 153 124 95

Equation 5-5 suggests that several values of.t exist for a given diffraction angle r. Thus, if a first-order line (D = I) of 800 nm is found at r, second-order (400 nm) and third-order (267 nm) lines also appear at this angle. Ordinarily, the first-order line is the most intense; indeed, it is possible to design gratings that concentrate as much as 90% of the incident intensity in this order. The higherorder lines can generally be removed by filters. For example, glass, which absorbs radiation below 350 nm, eliminates the higherorder spectra associated with first-order radiation in most of the visible region. The example which follows illustrates these points. EXAMPLE An echellette grating containing 2000 blazes per millimeter was irradiated with a polychromatic beam at an incident angle 48 deg to the grating normal. Calculate the wavelengths of radiation that would appear at an angle of reflection of +20, + 10,0, and -10 deg (angle r, Figure 5-11). To obtain d in Equation 5-5, we write Imm nm nm d=---x 106=500-2000 blazes mm blaze When r in Figure 5-11 equals + 20 deg , SOO (. 48 . 2) 542.6 A=sm +sm 0 =-D n

20 10 0 -10

A concave grating can be produced by ruling a spherical reflecting surface. Such a diffracting element serves also to focus the radiation on the exit slit and eliminates the need for a lens. Dispersion fJl GntiDp. The angular dispersion of a grating can be obtained by differentiating Equation 5-5 while holding i constant; thus, at any given angle of incidence dr D -=-d)' dcosr

.u =bd)'460.25

.t

dn

b-- x--

.t

I

.u

dn/d.t

------

460.30 - 460.20

x--

I

dn/d)'

or

(5-6)

For fused silica, b = 460.25 nm x I x 10-7 em 0.10 nm 1.3 x 10-4 nm-I om = 3.5 em and f