Relation Between 1/f Noise and Q-Factor in Quartz Resonators at Room and Low Temperatures, First Theoretical Interpretation

Abstract

32, av. de 1'Observatoire -25000 Elesanqon -France and P. Handel Physics Department, University of Missouri, St Louis, MO 63121 Sumnary The ultimate s t a b i l i t y of q u a r t z c r y s t a l o s c i l l a t o r s is given by t h e frequency n o i s e of t h e quartz resonator, which corresponds i n t h e tine domain t o a f l o o r . The measurement of t h e resonance frequency fluctuations of the resonator itself can be made only i f the influence of t h e e l e c t r onics noise is reduced. By u s i n g a r( transmission network and a balanced phase bridge the driving source phase fluctuations can be rejected by 50 dB t o 60 dB. This measurement system w a s used a t r m temperature with a l a r g e number of p a i r s of resonat o r s , and a correlation between l/f frequency noise P l s and Q-factors was found, following a l / Q law. In order t o extend t h e measurements t o very l o w temperatures, a new measurement system was studied, which allows t o o p e r a t e w i t h p a i r s of non identical resonators. T h u s l/f frequency noise was measured a t 4K and a t 1K. A large decrease of t h e noise is observed corresponding t o t h e i n c r e a s e of the Q-factors.A theoretical explanation of t h e I/Q' law is attempted. It is based on three-phonon interaction processes, which a r e a t t h e o r i g i n of the acoustic attenuation and of variations of velocity. Both phenomena a r e r e l a t e d by means of the thermal phonon relaxation time.

Quartz resonator instabilities under cryogenic conditions

“…Three works give values of the flicker noise for resonators operating in cryostats [7], [10], [11]. note that all three works present values of the fractional frequency fluctuations S y ( f ) for just one value of the Fourier frequency f = 1 Hz, and not over a wide Fourier frequency range, as in Figs.…”

“…For 10-MHz resonators, this difference is about 500 Hz. This phenomenon is due to small differences in crystal blank orientations [10]. This difference cannot be covered with trimming capacitors as in the case of room-temperature measurements.…”

“…Furthermore, no studies were done concerning resonator nonlinearities, different mode comparison, or analysis of loss phenomena. some studies were partially related to quartz resonator proper phase noise at liquid helium temperatures [7], [10], [11], but their results are questionable and are discussed in this work. The nonsystematic approach of these studies is a consequence of the many difficulties associated with cryogenic experiments.…”

1∕f Performance Limits of Scanning Tunneling Microscopes

“…where C is the specific heat, T is the temperature, y is the Gruneisen constant, So is the speed of sound in the absence of dissipation, Q.o is the resonance frequency of the crystal, p is the crystal density, and x is the relaxation time [1]. To simplify, we let AE = CT <y^> = constant and also use the low frequency Young modulus Eo = p So^ = constant.…”

“…The relation between quartz quality (Q-factor), relaxation time and frequency given by [1], is Ex/AE~Q^ which fluctuations to be written as [1], is Ex/AE~Q^, which allows for the spectral density of fractional spatial…”