### Dielectric Properties Of Crystal Quartz At High Frequencies

Dielectric measurements can in general be divided into three methods suitable for various frequency ranges—bridge methods for frequencies up to a few me./s., resonant circuits using lumped elements for frequencies between about 10 and 100 mc./s., and for higher frequencies resonant circuits with distributed constants.

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Dielectric properties may be conveniently represented by a complex dielectric constant given by ∊ = ∊′−j∊″. The two properties to be considered here are the normal dielectric constant or permittivity which is given by ∊′ and the loss factor tan δ which = ∊″/∊′. For low-loss dielectrics at high frequencies ∊′ << ∊′ and tan δ becomes equal to the power factor cos φ. Experimentally, ∊′ is found from effects due to a change in capacity and tan δ by a change in the losses of the circuit in which the dielectric is placed.

Measurements of these properties of crystal quartz are dominated by the facts that quartz is a piezo-electric material, has low losses of the same order as the best dielectrics known, and that only physically small specimens are conveniently available.

A quartz crystal consists roughly of a hexagonal prism with a hexagonal pyramid at either end. The line joining the vertices of the pyramids is a trigonal axis of symmetry—the principal or optic axis (Z) which is a direction of zero piezo-electric effect. Three digonal axes of symmetry also exist in a plane at right angles to this axis and these are parallel to the faces of the hexagonal prism. These digonal or electric axes (X) are directions of maximum piezo-electric effect and the three directions co-planar with, and perpendicular to, the electric axes are called the mechanical axes (Y). Measurements have been taken with specimens cut so that the electric field is parallel to these various axes.

The low loss of quartz necessitates the use of resonant sections with the lowest possible attenuation and this is best given by some type of resonant cavity where the supports of the central conductor (if any) and of the specimen form part of the cavity itself.

For the 100–300 mc./s. region a convenient method allowing for the size of resonant sections and of the quartz, is a re-entrant cavity with the quartz in the gap. By using correct dimensions this can be treated as a shorted coaxial transmission line with lumped capacity at one end.

For the micro-wave region suitable methods, again allowing for the size of resonator and specimen and for ease of calculation, are a shorted coaxial line and a cavity resonant in the E_{010} mode.

*For the coaxial line* the fields and placing of the specimen are as shown:

*For the E _{010} resonant cavity*: The only factor controlling the resonant frequency is the radius: λ = 2.6125a. The length is immaterial, but the greater the length, i.e., the greater the volume to surface ratio, the greater is the Q factor.

At about 10 cm. the E_{010} resonant cavity is the simpler of the two; for longer wavelengths the coaxial section is more suitable, and for shorter wavelengths (say 3 cm.) another type, a cavity resonant in the H_{01n} mode is preferable.

A paper by Horner and others in the *J.I.E.E.*, Vol. 93, Part 3, January, 1946, gives formulae derived from Maxwell's equations, which are suitable for use with all three methods. Before this paper was available, solutions from transmission line equations were obtained for a variable frequency method with a shorted coaxial line section by using a similar mathematical treatment to that in a paper by Gent of the Standard Telephone Company, which considered an open-ended section used with a variable length method. The two treatments for the coaxial line section give the same solutions for the permittivity, but the evaluation of the loss factor is approached slightly differently.

* _{010} resonator for micro-wave region*: For the E

_{010}resonant cavity the formulae which have been used in calculations are taken from the paper by Horner and others.

∊′ is evaluated from the resonant frequency and the dimensions of the cavity and specimen. Tan δ is found from a formula which contains the dimensions and ∊′, and the difference of the reciprocals of the Q factors of the cavity when (1) the dielectric is present (say Q) or (2) the dielectric is replaced by a loss-free specimen of the same permittivity and dimensions (say Q′)

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i.e. tan δ ∝ [1/Q - 1/Q′]

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This value of Q′ cannot be determined experimentally and cannot be used with the experimental value Q since Q factors obtained in practice never approach theoretical values. The difference is attributed to a departure of the depth of current penetration from the theoretical value given by d = 1/√πμfσ

where μ = permeability

σ = conductivity.

The Q of an air-filled resonator is found experimentally (say Q_{a}) and used in the theoretical formula

Q_{T} = a/d (1 + a/1_{r})

where a = radius of resonator

1_{r} = length of resonator

to give an effective current depth d′ according to the equation

Q_{a} = a/d′ (1 + a/1_{r})

This value d′ after allowance for any frequency difference, is now used in the theoretical formula for Q′ to give an equivalent experimental value which can be used in the equation

tan δ ∝ (1/Q - 1/Q′)

The method has the disadvantage that each cavity can be used for only one frequency with a particular specimen; but the construction of them is relatively simple. The variable-frequency method only can be used for the determination of Q factors, i.e. Q = f_{0}/Δf where fo is the resonant frequency and Δf is the width at half-power points.

Suitable sizes of specimens and cavities were shown. The diameters here are 3 011 in. and 2.510 in. and the length is 1 in., which is much less than the value at which oscillation in other modes commences. The specimens are half an inch in diameter. Two cavities must be used, since the introduction of the quartz changes the resonant frequency to a value outside the range of the oscillators available, e.g., the smaller changes from approximately 3,600 to 2,875 or 2.845 mc./s., depending on which cut of quartz is used.

The Q values are quite low. For the air-filled cavity the theoretical value at 10 cm. is approximately 6800. This, of course, is never approached owing to the difference in current penetration from the theoretical, but it is also lowered appreciably by the coupling loops, the holes in the curved surfaces, and the poor finish of the surface. The best experimental value used in calculations is 2900, but this has recently been raised by decreasing the coupling considerably. Plating and better finishing of the surface would allow these Q values to be raised appreciably when further cavities are constructed.

Experimentally a signal from a klystron oscillator is fed into the cavity by a small loop, resonance being indicated by a crystal detector and mirror galvanometer, which has been calibrated and gives a square-law response. The signal is heterodyned with another from a fixed klystron oscillator using a crystal mixer and the beat frequency determined on a calibrated HRO receiver. The frequency of the fixed klystron is obtained using a coaxial line wave-meter. An alternative method is to measure the beat frequency on a discriminator with a calibrated response characteristic, but this does not have the flexibility of the receiver which covers a wide range of frequencies with varying degrees of band spreading.

*Results* so far obtained give for specimens cut parallel to the × and Y axes

∊′ = 4.42 at 2.875 mc./s. tan δ = 0.0002

and for quartz cut parallel to Z axis ∊′=4.60 at 2,845 mc./s. tan δ = 0.0003

*Re-entrant cavity for 100–300 mc./s. region*.—From transmission line theory when all losses are small, the condition for resonance of a shorted coaxial section with capacity at one end is
Z_{0} tan β 1 = 1/ωC

where Z_{0} = characteristic impedance of line.

With dielectric in the condenser and a resonant frequency f_{1}
Z_{0} tan β_{1} 1 = 1/ω_{1}∊′C

With air as dielectric and resonant frequency f_{2}
Z_{0} tan β_{2} 1 = 1/ω_{2} C
∊′ = f_{2}/f_{1} · tan β_{2} 1/tan β_{1} 1

To obtain the loss factor the equivalent series resistances must be considered.

The equivalent circuit with dielectric present is:
tan δ = ω ∊′ C R_{C}
Q_{1} = ω L/R_{L} + R_{C} = 1/R_{L} + R_{C} · 1/ω ∊′ C

Remove the dielectric and retune so the capacity is again ∊′ C, i.e. same resonant frequency
Q_{2} = 1/R_{L} ω ∊′ C
tan δ = 1/Q_{1} − 1/Q_{2}

Several corrections have to be made to these simple formulae—allowance must be made for the edge or fringing capacity at the gap—this is of the same order as the direct capacity and cannot be eliminated from the equations. A small change in line length occurs on retuning the line after the dielectric is removed and for constructional purposes it is more convenient not to have the gap at one end of the cavity. Control of the resonant frequency lies in the capacity, and to work at any particular frequency it is in general necessary to have different values for the gap separation and the thickness of the specimen. It is also desirable to be able to use various diameters of specimens.

Solutions have been obtained covering all the above cases.

In general, where dimensions are as shown:

C_{0} = edge capacity

C_{1} = capacity of air condenser of same size as dielectric

C_{2} = capacity of air condenser, radius b, separation (t–d)

C_{2} = capacity of ring condenser, radii a and b, and separation t

∊′ = 1/t/d [f_{1}/f_{2} · (tan β_{1} 1_{A} + tan β_{1} 1_{B})/tan β_{2} 1_{A} + tan β_{2} 1_{B}) · A—3] + 1

where A = 1 − (C_{2} + C_{0}) ω_{2} Z_{0} [tan β_{2} 1_{A} + tan β_{2} 1_{B}]/1 − (C_{2} + C_{0}) ω_{1} Z_{0} [tan β_{1} 1_{A} + tan β_{1} 1_{B}]
tan δ = [1/Q_{1} − 1/Q_{2} · tan β_{1} 1_{A} + tan β_{1}. 1_{B} ± ∧ 1_{B}/tan β_{1} 1_{A} + tan β_{1} 1_{B}]
x (∊′ C_{1} + C_{2}) [∊′ C_{1} C_{2} + (C_{2} + C_{0}) (∊′ C_{1} + C_{2})]/∊′ C_{1} C_{22}

The dimensions of the cavity were chosen so that the length is greater than the diameter, and the radius much less than a quarter wave length in order to satisfy the resonance conditions. The diameter of the outer conductor is 6 in. and the inner 1.5 in, giving a ratio approximately equal to that for an optimum Q value.

The length can be changed by inserting additional sections giving overall lengths of 9 to 12 in, in order to give a greater frequency coverage. The top shorting plate can be removed for insertion of the specimen and the upper portion of the central conductor controlled by a micrometer moves through spring finger contacts to give an adjustable gap of known dimensions. The cavity is fed through a small coupling loop near the base and a similar output loop feeds to a crystal detector as a resonance indicator.

Frequencies are measured by heterodyning a variable oscillator with the fundamental or harmonics of a second oscillator which is in turn heterodyned with a standard signal generator to give a final beat frequency which can be measured on the HRO receiver.

Curves of resonant frequency are of the form shown. For the 9 in. cavity the values are approximately 300 mc./s. at 0.3 in. and 195 mc./s. at 0.05 in. For the 12 in. cavity, 230 mc./s. at 0.3 in. and 150 mc./s. at 0.05 in. The frequency is, of course, lowered when the dielectric is inserted.

Experimental values of the edge capacity are obtained by determining the total capacity at the gap from the equation and subtracting the calculated geometrical capacity.

It varies from about 1.20 pf. at a separation of 0.1 in. to 1.05 pf. at a separation of 0.2 in. At these separations the calculated direct capacity which has a curve of the form shown has the approximate values 3.9 pf. and 2.0 pf. respectively.

The Q factor of the air filled cavity varies approximately as shown.

Accurate results for the permittivity and loss factor of quartz have not. yet been obtained, since at present there is a slight variation from parallelism in the gap faces due to difficulties in construction. A difference of 1/1000 in. introduces an appreciable error. The results do show, however, that plates cut perpendicularly to the × and Y axes give similar values which are lower than those for plates cut perpendicularly to the Z axis (same as in microwave case).

Values obtained for various diameters and thicknesses of plates range from 4.25 to 4.45 for the former and 4.4 to 4.7 for the latter over frequencies between 200 and 250 mc./s., but, in view of the known inaccuracies, these should be taken only as an indication of the true values.

*The permittivity of quartz* as given by different authors in available literature shows appreciable variations over the lower frequencies up to 30 mc./s. and in tables a most probable value is generally quoted. Piezo-electric effects are responsible for some of the discrepancies as shown by variations for plates of slightly different thicknesses at the same frequencies.

The results obtained so far by the above two methods are in the same range as at lower frequencies and show a similar greater value of the permittivity and loss factor for specimens with the optic axis parallel to the direction of the-electric field than for those in which the axis is perpendicular to the field.