Chapter 4
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NMR TRANSITIONFigure J.Four level state diagram (ms=±l/2;m.=±l/2) and the transitions of an ENDOR experiment. An allowed EMR transition is saturated, and the rf frequency is swept in order to detect NMR transitions.
The design of an ENDOR spectrometer system differs only slightly from the basic cw- or pulsed EMR spectrometer, and the subject has been exhaustively reviewed (Gothe, 1970; Kevan & Kispert, 1976; Box, 1977;
Leniart, 1979; Schweiger, 1987; Kurreck et al., 1988; Bender & Aisen, 1993; Thomann & Bernardo, 1993; Piekara-Sady & Kispert, 1994). But there is a gap in the ENDOR literature as concerns the so-called 'coil' that is used to generate the radiofrequency field that drives the NMR transitions, and this review is intended to meet that need. At issue is the need to irradiate the sample material by a secondary radiofrequency field (i.e. NMR transition), which entails the incorporation of a second 'resonator' without incurring serious deleterious effects on the primary microwave field (EMR transition). A loop or helical coil is used as a radiofrequency field generating element in a transmitter circuit that consists of a low power signal source or sweeper, a broadband power amplifier, and the coil network. The latter is typically two short sections of transmission line that are spanned by the coil and terminated by a power load (Figure 2). The manner in which the radiofrequency and microwave resonators are integrated affects both the method and the sensitivity of the spectroscopic measurement because one is combining two RLC structures in a distributed parameter network while trying to retain the behavior and properties of each device.
4. ENDORCOILS AND RELATED RADIOFREQUENCY CIRCUITS 157
AMPLIFIER SAMPLE TUBE
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Figure 2. The basic configuration for simultaneously irradiating a sample material with radio-and microwave fields for ENDOR.
Methods of generating magnetic fields in the radiofrequency domain of the electromagnetic spectrum are analogous to those used in DC electricity theory. A single linear conductor generates a magnetic field in a circular pattern that is orthogonal to the conductor's axial direction. The magnetic field polarity is determined by the direction of current flow through the conductor according to the right hand rule, and the field strength can therefore be enhanced by modifying the geometry of the conductor so that fields reinforce one another. Two common conductor shapes are the loop and its extension, the multi-turn coil. Both of these geometries are commonly used for ENDOR spectroscopy, and the term coil will be used in this review as a synonymous reference to both.
2. THE PRINCIPLES OF COIL IMPLEMENTATION
2.1 The Coil as a Field-Generating Element
At its simplest, an ENDOR coil may be constructed by forming a wire loop or helix around the sample holder (Figure 2). A wire loop or multi-tum coil constitutes a device known as an inductor, wherein a magnetic field is generated by the current flow through the conductor. For direct current circuits, the magnetic field strength at the center of a loop (or helix) is given as H
=
NoI /2 r (and B=
WH), whereN, I,r,andJ.l.are the number of turns in the loop, the magnitude of the current, the radius of the coil, and themagnetic permitivity of the medium (air and/or the sample material), respectively. The field is a vector quantity and has a direction that is determined by the direction of current flow. The loop therefore behaves as a magnetic dipole when alternating current is used. In the latter scenario, the energy stored in the magnetic field of the dipole is P=tL-I2, whereL is the inductance of the device in units of Henrys, and power is in Watt -sec (Jordan
& Balman, 1968).
The inductance of a conductor may be calculated by using its geometric parameters (ef Terman, 1955). Besides the circular loop and helix (Section 3.2), common inductors that find application in ENDOR probeheads are parallel pairs of wire (Section 3.4) and parallel rectangular bars (Section 3.5). For each of these commonly used ENDOR coils the inductance is related to geometry as follows:
Straight Wire: L
=
0.00508/ (In 4/ - 0.75) (1a) dParallel Wire Pair: L
=
0.01016/ (In2D - D - 8p) (1b)d /
Parallel Rectangular Bars:
D D b-v
c
L
=
0.01016/(In--+ 1.5 --+0.2235--) (I c)b+c / /
Circular Loop: L=0.01595D(1n8D
-2-8p) (I d)
d
Single Layer Helix: L
=
FN2D (Ie)where in each case L is expressed in microhenrys and all dimensions are expressed in inches. The geometric variable D designates the diameter (looplhelix) or the spacing between the conductors (parallel wire/bar); dis the diameter of the wire; band e are the width and breadth of the bar; and I is the length. The parameter F is a form factor (ef Terman, 1955) that depends on the diameter to length ratio of the helix, and is assigned the value 0.005854 for the familiar 1:4 ratio that is used in the cylindrical TMtJ o
cavity . The term OJi that appears in (lb) and (ld) is a numerical correction factor that is related to the skin depth and permeability of the conductor; it simply means that the numerical quantities within the parentheses are not as precise as would be inferred.
As an illustrative example, the multi-tum wire coil that is used in the commercial TM110ENDOR cavity will be analyzed as follows. The diameter of the coil is approximately 1.0 ern, or 0.39 inches, which means that L will scale as 0.002305 xN2• The estimated inductance of a 20-turn coil is therefore 9.22 x 10-7Henrys.
4. ENDORCOILS AND RELATED RADIOFREQUENCY CIRCUITS
2.2
The Effect of Reactance
159
Capacitors and inductors generate electric and magnetic fields that, in AC circuits, interact with the current and affect its time-dependent behavior by introducing a phase lag between the voltage and current. This so-called reactance is manifest as a frequency dependent complex valued term that in the case of the inductor is given by the formula i01L, where L is the value of the inductance. The AC current magnitude is related to the voltage in much the same way as the DC relation, except that the resistance is replaced by a complex-valued impedance, Z, which is equal to i01L for the inductor. Since the rf field power is proportional to the square of the current, that is, P
=
~L'I2, one desires to optimize the relation1= Y'V, where Y is the admittance ofthe circuit, or Z -I.
As depicted in Figure 2, the ENDOR coil circuit consists of a series pairing of the inductor/coil and a 50 n power resistor, often referred to as a load. The resistanceR is real-valued, and the inductance supplies a complex-valued impedancei01L, so that one writes the impedance of the network as Z
=
R + i01L. As such, V= Z.R, or 1= Y'V, and therefore the total power delivered to the network may be written as P=Z·/2 or P= y.V2. The admittanceY= (R +i01Lrl, and the power may be written:(2)
where the denominator represents the magnitude of the admittance. It follows from this equation that the term 012L2will limit the power delivered to the network since 01=21tf and varies with the rfcarrier frequency. For example, with R=50 nandL=9.22 X 10-7Henrys, as calculated in Section 2.1 for a 20-turn coil, parameters for the evaluation of
I
ZI
are :f
(MHz) cox 10-6 ro2L2ohms (R2+ro2L2) 112ohms1 6.28 33 50.3
5 31.4 839 57.8
10 62.8 3360 76.6
50 314 84000 294.1
100 630 335000 580.9
This simplified analysis ofthe ENDOR coil network, graphically depicted in Figure 3, illustrates why one seeks to minimize the inductance of the coil, which, as has been described in Section 2.1, can be controlled via the geometry of the device. For example, a pair of parallel rectangular strips(cf.
Section 3.5) of dimension 4 x 10 mm and separated by 4 mm (the approximate diameter of a standard sample tube)
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