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Modeling[edit]

Electrical model[edit]

Electronic symbol for a piezoelectric crystal resonator 

 A quartz crystal can be modelled as an electrical network with a low impedance (series)

and a high impedance (parallel) resonance point spaced closely together. athematically

(using the !aplace transform) the impedance of this network can be written as"

or#

where s is the comple$ frequency ( )# is the series resonant angular 

frequency and is the parallel resonant angular frequency.

%chematic symbol and equi&alent circuit for a quartz crystal in an oscillator 

 Adding additional capacitance across a crystal will cause the parallel

resonance to shift downward. Adding additional inductance across a crystal will

cause the resonance to shift upward. 'his can be used to adust the frequency

at which a crystal oscillates. rystal manufacturers normally cut and trim their

crystals to ha&e a specified resonance frequency with a known *load*

capacitance added to the crystal. +or e$ample# a crystal intended for a , p+

load has its specified parallel resonance frequency when a ,.- p+ capacitor  is

placed across it. ithout this capacitance# the resonance frequency is higher.

Resonance modes[edit]

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Temperature effects[edit]

 A crystal*s frequency characteristic depends on the shape or *cut* of the crystal.

 A tuning fork crystal is usually cut such that its frequency o&er temperature is a

parabolic cur&e centered around /0 1. 'his means that a tuning fork crystal

oscillator will resonate close to its target frequency at room temperature# but

will slow down when the temperature either increases or decreases from room

temperature. A common parabolic coefficient for a 2/ k3z tuning fork crystal is

4-.-5 ppm617.

8n a real application# this means that a clock built using a regular 2/ k3z

tuning fork crystal will keep good time at room temperature# lose / minutes

per year at 9- degrees elsius abo&e (or below) room temperature and

lose : minutes per year at /- degrees elsius abo&e (or below) room

temperature due to the quartz crystal.

Electrical oscillators[edit]

 A crystal used in hobby radio controlequipment to select frequency.

'he crystal oscillator circuit sustains oscillation by taking a &oltage signal

from the quartz resonator # amplifying it# and feeding it back to the

resonator. 'he rate of e$pansion and contraction of the quartz is

the resonant frequency# and is determined by the cut and size of the

crystal. hen the energy of the generated output frequencies matches thelosses in the circuit# an oscillation can be sustained.

 An oscillator crystal has two electrically conducti&e plates# with a slice or

tuning fork of quartz crystal sandwiched between them. ;uring startup# the

controlling circuit places the crystal into an unstable equilibrium# and due

to the positi&e feedback in the system# any tiny fraction of noise will start to

get amplified# ramping up the oscillation. 'he crystal resonator can also be

seen as a highly frequency<selecti&e filter in this system" it will only pass a

&ery narrow subband of frequencies around the resonant one# attenuating

e&erything else. E&entually# only the resonant frequency will be acti&e. As

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the oscillator amplifies the signals coming out of the crystal# the signals in

the crystal*s frequency band will become stronger# e&entually dominating

the output of the oscillator. 'he narrow resonance band of the quartz

crystal filters out all the unwanted frequencies.

'he output frequency of a quartz oscillator can be either that of the

fundamental resonance or of a multiple of that resonance# called

aharmonic frequency. 3armonics are an e$act integer multiple of the

fundamental frequency. =ut# like many other mechanical resonators#

crystals e$hibit se&eral modes of oscillation# usually at appro$imately odd

integer multiples of the fundamental frequency. 'hese are termed

>o&ertone modes># and oscillator circuits can be designed to e$cite them.

'he o&ertone modes are at frequencies which are appro$imate# but not

e$act odd integer multiples of that of the fundamental mode# and o&ertone

frequencies are therefore not e$act harmonics of the fundamental.

 A maor reason for the wide use of crystal oscillators is their high ? factor .

 A typical Q &alue for a quartz oscillator ranges from 9-5 to 9-,# compared

to perhaps 9-/ for an ! oscillator . 'he ma$imum Q for a high stability

quartz oscillator can be estimated as Q @ 9., 9-B6f # where f  is the

resonance frequency in megahertz.

Cne of the most important traits of quartz crystal oscillators is that they can

e$hibit &ery low phase noise. 8n many oscillators# any spectral energy at

the resonant frequency will be amplified by the oscillator# resulting in a

collection of tones at different phases. 8n a crystal oscillator# the crystal

mostly &ibrates in one a$is# therefore only one phase is dominant. 'his

property of low phase noise makes them particularly useful in

telecommunications where stable signals are needed# and in scientific

equipment where &ery precise time references are needed.

3igh frequency crystals are often designed to operate at third# fifth# or

se&enth o&ertones. anufacturers ha&e difficulty producing crystals thin

enough to produce fundamental frequencies o&er 2- 3z. 'o produce

higher frequencies# manufacturers make o&ertone crystals tuned to put the

2rd# 0th# or Bth o&ertone at the desired frequency# because they are thicker 

and therefore easier to manufacture than a fundamental crystal that would

produce the same frequencyDalthough e$citing the desired o&ertone

frequency requires a slightly more complicated oscillator circuit. [9/][92][95][90]

[9,] A fundamental crystal oscillator circuit is simpler and more efficient and

has more pullability than a third o&ertone circuit. ;epending on the

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manufacturer# the highest a&ailable fundamental frequency may be

/0 3z to ,, 3z.[9B][9:]

En&ironmental changes of temperature# humidity# pressure# and &ibrationcan change the resonant frequency of a quartz crystal# but there are

se&eral designs that reduce these en&ironmental effects. 'hese include the

'C# C# and CC (defined below). 'hese designs (particularly the

CC) often produce de&ices with e$cellent short<term stability. 'he

limitations in short<term stability are due mainly to noise from electronic

components in the oscillator circuits. !ong term stability is limited by aging

of the crystal.

;ue to aging and en&ironmental factors (such as temperature and

&ibration)# it is difficult to keep e&en the best quartz oscillators within one

part in 9-9- of their nominal frequency without constant adustment. +or this

reason# atomic oscillators are used for applications requiring better long<

term stability and accuracy.

Spurious frequencies[edit]

/0<3z crystal e$hibiting spurious response

+or crystals operated at series resonance or pulled away from the main

mode by the inclusion of a series inductor or capacitor# significant (and

temperature<dependent) spurious responses may be e$perienced. 'hough

most spurious modes are typically some tens of kilohertz abo&e the

wanted series resonance their temperature coefficient will be different from

the main mode and the spurious response may mo&e through the mainmode at certain temperatures. E&en if the series resistances at the

spurious resonances appear higher than the one at wanted frequency a

rapid change in the main mode series resistance can occur at specific

temperatures when the two frequencies are coincidental. A consequence

of these acti&ity dips is that the oscillator may lock at a spurious frequency

(at specific temperatures). 'his is generally minimized by ensuring that the

maintaining circuit has insufficient gain to acti&ate unwanted modes.

%purious frequencies are also generated by subecting the crystal to

&ibration. 'his modulates the resonance frequency to a small degree by

the frequency of the &ibrations. %<cut crystals are designed to minimize

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the frequency effect of mounting stress and they are therefore less

sensiti&e to &ibration. Acceleration effects including gra&ity are also

reduced with % cut crystals as is frequency change with time due to long

term mounting stress &ariation. 'here are disad&antages with % cut shear 

mode crystals# such as the need for the maintaining oscillator to

discriminate against other closely related unwanted modes and increased

frequency change due to temperature when subect to a full ambient

range. % cut crystals are most ad&antageous where temperature control

at their temperature of zero temperature coefficient (turno&er) is possible#

under these circumstances an o&erall stability performance from premium

units can approach the stability of Fubidium frequency standards.

Commonly used crystal frequencies[edit]

Main article: Crystal oscillator frequencies

rystals can be manufactured for oscillation o&er a wide range of

frequencies# from a few kilohertz up to se&eral hundred megahertz. any

applications call for a crystal oscillator frequency con&eniently related to

some other desired frequency# so hundreds of standard crystal frequencies

are made in large quantities and stocked by electronics distributors. +or

e$ample# many (non<tele&ision) applications use 2.0BG050 3z crystals

since they are made in large quantities for H'% color tele&ision recei&ers.

Ising frequency di&iders#frequency multipliers and phase lockedloop circuits# it is practical to deri&e a wide range of frequencies from one

reference frequency.

Crystal structures and materials[edit]

ommon package types for quartz crystal products

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luster of natural quartz crystals

 A synthetic quartz crystal grown by the hydrothermal synthesis# about9G cm long

and weighing about9/B grams

'uning fork crystal used in a modern quartz watch.

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%imple quartz crystal

8nside construction of a modern high performance 3<5G packagequartz crystal

+le$ural and thickness shear crystals

'he most common material for oscillator crystals is quartz. At the

beginning of the technology# natural quartz crystals were usedJ now

synthetic crystalline quartz grown by hydrothermal synthesis ispredominant due to higher purity# lower cost# and more con&enient

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handling. Cne of the few remaining uses of natural crystals is for pressure

transducers in deep wells. ;uring orld ar 88 and for some time

afterwards# natural quartz was considered a strategic material by the I%A.

!arge crystals were imported from =razil. Faw >lascas># the source

material quartz for hydrothermal synthesis# are imported to I%A or mined

locally by oleman ?uartz. 'he a&erage &alue of as<grown synthetic

quartz in 9GG5 was ,- I%;6kg.[9G]

'wo types of quartz crystals e$ist" left<handed and right<handed# differing

in the optical rotation but identical in other physical properties. =oth left

and right<handed crystals can be used for oscillators# if the cut angle is

correct. 8n manufacture# right<handed quartz is generally used.[/-] 'he

%iC5 tetrahedrons form parallel helicesJ the direction of twist of the heli$

determines the left< or right<hand orientation. 'he heli$es are aligned along

the z<a$is and merged# sharing atoms. 'he mass of the heli$es forms a

mesh of small and large channels parallel to the z<a$isJ the large ones are

large enough to allow some mobility of smaller ions and molecules through

the crystal.[/9]

?uartz e$ists in se&eral phases. At 0B2 1 at 9 atmosphere (and at higher

temperatures and higher pressures) the K<quartz undergoesquartz

in&ersion# transforms re&ersibly to L<quartz. 'he re&erse process howe&er

is not entirely homogeneous and crystal twinning occurs. are has to be

taken during manufacture and processing to a&oid the phase

transformation. Cther phases# e.g. the higher<temperature

phases tridymite and cristobalite# are not significant for oscillators. All

quartz oscillator crystals are the K<quartz type.

8nfrared spectrophotometry is used as one of the methods for measuring

the quality of the grown crystals. 'he wa&enumbers 20:0# 20--# and

259- cm49 are commonly used. 'he measured &alue is based on

the absorption bands of the C3 radical and the infrared ? &alue is

calculated. 'he electronic grade crystals# grade # ha&e ? of 9.: million or

abo&eJ the premium grade = crystals ha&e ? of /./ million# and special

premium grade A crystals ha&e ? of 2.- million. 'he ? &alue is calculated

only for the z regionJ crystals containing other regions can be ad&ersely

affected. Another quality indicator is the etch channel densityJ when the

crystal is etched# tubular channels are created along linear defects. +or

processing in&ol&ing etching# e.g. the wristwatch tuning fork crystals# low

etch channel density is desirable. 'he etch channel density for swept

quartz is about 9-M9-- and significantly more for unswept quartz.

Nresence of etch channels and etch pits degrades the resonator*s ? andintroduces nonlinearities.[//]

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?uartz crystals can be grown for specific purposes.

rystals for A'<cut are the most common in mass production of oscillator

materialsJ the shape and dimensions are optimized for high yield of the

required wafers. 3igh<purity quartz crystals are grown with especially low

content of aluminium# alkali metal and other impurities and minimal

defectsJ the low amount of alkali metals pro&ides increased resistance to

ionizing radiation. rystals for wrist watches# for cutting the tuning fork

2/B,: 3z crystals# are grown with &ery low etch channel density.

rystals grow anisotropicallyJ the growth along the O a$is is up to 2 times

faster than along the a$is. 'he growth direction and rate also influences

the rate of uptake of impurities.[/2] P<bar crystals# or O<plate crystals with

long P a$is# ha&e four growth regions usually called Q# <# O# and %.

[/5] 'he distribution of impurities during growth is une&enJ different growth

areas contain different le&els of contaminants. 'he z regions are the

purest# the small occasionally present s regions are less pure# the Q$

region is yet less pure# and the <$ region has the highest le&el of impurities.

'he impurities ha&e a negati&e impact on radiation hardness# susceptibility

to twinning# filter loss# and long and short term stability of the crystals.

[/0] ;ifferent<cut seeds in different orientations may pro&ide other kinds of

growth regions.[/,] 'he growth speed of the <$ direction is slowest due to

the effect of adsorption of water molecules on the crystal surfaceJ

aluminium impurities suppress growth in two other directions. 'he content

of aluminium is lowest in z region# higher in Q$# yet higher in <$# and

highest in sJ the size of s regions also grows with increased amount of

aluminium present. 'he content of hydrogen is lowest in z region# higher in

Q$ region# yet higher in s region# and highest in <$.[/B] Aluminium inclusions

transform into color centers with gamma ray irradiation# causing a

darkening of the crystal proportional to the dose and le&el of impuritiesJ the

presence of regions with different darkness re&eals the different growth

regions.

rystals for %A de&ices are grown as flat# with large <size seed with low

etch channel density.

%pecial high<? crystals# for use in highly stable oscillators# are grown at

constant slow speed and ha&e constant low infrared absorption along the

entire O a$is. rystals can be grown as P<bar# with a seed crystal in bar

shape and elongated along the P a$is# or as O<plate# grown from a plateseed with P<a$is direction length and <a$is width. [/-] 'he region around

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the seed crystal contains a large number of crystal defects and should not

be used for the wafers.