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Temperature fluctuations in a solid-nitrogen cooled secondary frequency standard
Cryogenics 42 (2002) 45–48 www.elsevier.com/locate/cryogenics
Temperature fluctuations in a solid-nitrogen cooled secondary frequency standard J.G. Hartnett *, M.E. Tobar, E.N. Ivanov, P. Bilski Frequency Standards and Metrology Research Group, Department of Physics, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia Received 29 August 2001; accepted 3 December 2001
Abstract Liquid helium and liquid nitrogen are commonly used as coolants for highly frequency stable oscillators. In this paper, we present the use of solid nitrogen as an alternative coolant. Solid nitrogen solidifies near 52 K under vacuum and we show that the temperature fluctuations in the solid are an order of magnitude lower than in the liquid. This has the advantage of reducing the temperature control requirements necessary to limit temperature induced frequency fluctuations in cryogenic microwave resonatoroscillators. Ó 2002 Published by Elsevier Science Ltd. Keywords: Fractional temperature fluctuations
1. Introduction At the heart of modern radar, telecommunications, navigation, GPS and precise time keeping are low-noise highly stability resonator-oscillators. Generally acoustic modes of resonance in quartz resonators have been used to build such oscillators. Recently, electromagnetic modes of resonance in sapphire resonators have been utilized because of their very high quality factor. This ensures good short-term frequency stability resulting from a high sensitivity of the control electronics to frequency fluctuations. However, to improve the long-term frequency stability of the oscillator the temperature induced frequency fluctuations should be suppressed. At liquid helium temperatures, frequency-temperature compensation of monolithic sapphire resonators has been achieved by the residual paramagnetic impurities of the order of a few parts per million [1]. Intentionally doped sapphire with a larger concentration of Ti3þ ions was used to raise the temperature of compensation above 50 K [2]. Another technique achieved compensation at about 10 K in a composite ruby-sapphire structure [3]. A mechanical compensation technique [4] *
Corresponding author. Tel.: +61-8-9380-3443; fax: +61-8-93801235. E-mail address: [email protected] (J.G. Hartnett).
involves a coupled mode resonator to cancel the resonator’s temperature dependence above 77 K. Another technique utilizes composite dielectric structures consisting of more than one low loss monocrystal. For example, two very thin slices of monocrystalline rutile were clamped tightly against the ends of a cylinder of sapphire monocrystal [5–7]. The temperature coefficients of permittivity of sapphire and rutile are of opposite sign and thus the temperature coefficient of the composite resonator was annulled at temperatures above 50 K. The new generation of atomic fountain primary standards needs ever increasingly stable flywheel oscillators as a pump source. The maximum allowable frequency instability for a flywheel oscillator at present is about 1 part in 1014 at 1 s [8]. In order to take full advantage of the reduced sensitivity to temperature fluctuations in resonators as described above and to reach the required level of oscillator-frequency stability it is necessary to measure the temperature fluctuations in the resonator under operating conditions. The temperature fluctuations of both a liquid and a solid nitrogen bath, at 77 and 52 K, respectively, were directly measured. Using a free running loop microwave oscillator at 12.03 GHz, based on a temperature compensated resonator, the temperature fluctuations in the resonator were also measured. The results indicate that the temperature fluctuations in solid nitrogen at 52 K were an order of magnitude lower than in liquid nitrogen at 77 K.
0011-2275/02/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. PII: S 0 0 1 1 - 2 2 7 5 ( 0 1 ) 0 0 1 7 8 - 3
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J.G. Hartnett et al. / Cryogenics 42 (2002) 45–48
2. Thermometers Any temperature control system for a highly stable oscillator or any temperature measurement system will have a temperature sensitive element at its heart. In order to determine which one would be the most suitable for our application at temperatures accessible with liquid or solid nitrogen, it was necessary to characterize the temperature dependence of the commercially available resistance-thermometers. The sensitivity of a resistancethermometer is defined as hðT Þ ¼
T oRTh ; RTh oT
ð1Þ
where T is the temperature (in Kelvin) and RTh is the resistance (in Ohms) across the thermometer at temperature T. A number of thermometers were investigated and their dimensionless coefficient or sensitivity compared (see Fig. 1). Over the desired temperature range (50–77 K), the platinum resistance thermometer (PRT) (curve 1) had the highest sensitivity, and, as a result of this experiment, was chosen as the best choice.
3. AC-bridge read-out The measurement system implemented was a bridge with ac excitation (see Fig. 2). The ac-bridge circuit was part of a custom-made temperature controller. The bridge driving voltage (u0 ) was supplied by an on-board (150 mV, 238 Hz) signal oscillator. The bridge was balanced using the variable resistor (RB ). The 2 kX and 10 X resistors were high-precision low-temperature coefficient type resistors. Only the PRT and its leads were in the cryogenic environment. The bridge error voltage was amplified by an ac-coupled amplifier, then demodulated and read out on a DVM and recorded by computer.
Fig. 1. Dimensionless coefficients of five resistance-thermometers measured over the range 50–77 K. Curve 1 was a standard platinum strip (PRT), curve 2 a carbon-glass Lakeshore CGR-1-2000 type, curve 3 an Alan Bradley resistor, curve 4 a Neocera CryoHybrid, and curve 5 was a Scientific Instruments (USA) ruthenium oxide RO105 type thermometer.
Fig. 2. The ac bridge used to measure voltage fluctuations in a platinum resistance thermometer (labelled RPRT ).
The bridge error voltage fluctuations at the output of the mixer in Fig. 2 (dumix ) are related to the temperature fluctuations of the PRT by dumix RPRT
ou dT hðT0 Þ ; oRPRT T0
ð2Þ
where ou=oRPRT is a parameter that characterizes the measurement system sensitivity, dT are the ambient temperature fluctuations and RPRT is the resistance of the PRT at some mean temperature T0 . If the bridge stays balanced, the noise from the source is rejected. (There was some systematic drift from this balance, which was subtracted from the raw data.) After taking the statistical Allan variance [9] of fractional fluctuations of voltage, the square root Allan variance or Allan deviation of fractional temperature fluctuations (rT ) was calculated. From (2), the Allan deviation of fractional temperature fluctuations (rT ) is related to the Allan deviation of fractional voltage fluctuations (ru ), by ru : ð3Þ rT ðou=oRPRT ÞRPRT ðT0 ÞhðT0 Þ
4. Temperature fluctuations of the nitrogen bath Using the circuit of Fig. 2, the voltage fluctuations out of the bridge were recorded with the PRT positioned deep in the cryogenic fluid. The nitrogen was solidified by pumping constantly on the fluid at a pressure below 101 Torr. The temperature was maintained at around 52 K. The measurement system noise floor (curve 3) was measured with a temperature-insensitive resistor. The solid nitrogen data were very close to the noise floor for integration times less than 10 s. Using (3), the voltage data were converted into Allan deviation of fractional temperature fluctuations ðrT Þ (see Fig. 3). A comparison is made between room temperature (curve 3) and liquid nitrogen (curve 1) and solid nitrogen (curve 2).
5. Loop oscillator A thermal design was implemented for a cryogenic sapphire-loaded-cavity (SLC) resonator attached to a
J.G. Hartnett et al. / Cryogenics 42 (2002) 45–48
Fig. 3. The Allan deviation of fractional temperature fluctuations of a liquid nitrogen bath (curve 1) and a solid nitrogen bath (curve 2) measured with the ac-bridge. Curve 3 are the measured room temperature fractional fluctuations. Curve 4 are the measured fractional temperature fluctuations in the cavity determined by measuring the oscillator frequency fluctuations cooled by a liquid nitrogen bath (77 K). Curve 5 are the measured fractional temperature fluctuations in the cavity determined by measuring the oscillator frequency fluctuations cooled to 58 K by a solid nitrogen bath and a foil heater and temperature controller. Curve 6 are the inferred fractional temperature fluctuations in the cavity passively cooled by a solid nitrogen bath (52 K).
copper supporting post. The copper post was connected to the liquid or solid nitrogen bath by a stainless-steel post. This was all housed within a vacuum can immersed in the bath. When the vacuum was sustained, the stainless-steel post provided good thermal impedance and hence low pass thermal filtering of the bath temperature fluctuations. The temperature fluctuations at the copper cavity could not be measured using the acbridge because the measurement system noise floor was insufficiently low enough. For this reason, a free running loop oscillator (Fig. 4) was implemented to get an estimate of the effects of temperature fluctuations on the oscillator frequency stability. The loop oscillator incorporated a 11–14 GHz amplifier, a custom-made band-pass filter, a mechanical
Fig. 4. Schematic of loop oscillator used.
47
phase shifter and a mechanical attenuator in die-cast aluminum box anchored to a 16-mm thick aluminum plate with stainless screws. The amplifier was thermally grounded to the base-plate. The resonator coupling on both probes was approximately 0.005, set previously to measure the Q-factor as a function of temperature. This resulted in a resonator insertion loss of 44 dB. As a result, no Pound frequency discriminator and electronicnoise-reduction servo was implemented. The oscillator frequency was down-converted by beating it with a HP8673G synthesizer locked to an Oscilloquartz 5 MHz reference, and counted on a HP53131A frequency counter. The Allan deviation of fractional frequency fluctuations (ry ) was calculated, for the free running oscillator at both 58 and 77 K. To determine the stability of the HP synthesizer, we used a similar synthesizer with both referenced by their own internal quartz 10 MHz quartz oscillators, and recorded the beat frequency of the two 12.03 GHz signals separated by about 65 kHz. The Allan deviation of fractional frequency fluctuations for the synthesizer was calculated to be 1:1 1012 at 1 s rising to 2:5 1012 at 100 s. The Oscilloquartz reference oscillator has been measured to have 3 1013 stability, and as a result the HP8673G synthesizer, referenced to this oscillator, would have this stability too.
6. Temperature fluctuations in resonator Provided temperature fluctuations in the resonator are the dominant noise mechanism, the fractional frequency fluctuations of an oscillator may be related to the fractional temperature fluctuations of the resonator by df T0 of dT ¼ : ð4Þ f0 f0 oT T0 After calculating the Allan deviation of both frequency and temperature fluctuations, after re-arranging, (4) becomes T0 of rT ¼ ry : ð5Þ f0 oT The dimensionless temperature coefficient of frequency (TCF) ðT0 =f0 Þdf =dT for the sapphire resonator was calculated from measured frequency-temperature data of the chosen mode of resonance, where the bath was allowed to warm slowly from solid to liquid at 1 atm. Knowing the oscillator’s fractional frequency stability (ry ), the fractional temperature stability (rT ) can be calculated from (5). From the measurements of the freerunning oscillator frequency, at 77 K, rT was calculated. It has been assumed here that the frequency stability is solely due to temperature fluctuations. The amplifier
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J.G. Hartnett et al. / Cryogenics 42 (2002) 45–48
phase noise contribution to the oscillator frequency instability was measured at the 1:7 1012 level, a factor of 4 below the temperature induced contribution. Therefore, by operating at a temperature where df =dT is large, we determined the temperature stability at the cavity to be about 1 lK at 1 s rising to 10 lK at 100 s (curve 4 in Fig. 3). However this was not the case for the data taken at 58 K. The frequency stability data here could not be assumed to be solely due to temperature fluctuations, because no Pound servo scheme was implemented and the data here were dominated by the electronic noise resulting from frequency shifts in the loop oscillator, particularly by the modulation of the Teflon in the coaxial lines leading to the cryogenic resonator. In calculating the temperature fluctuations from these data, because df =dT was greatly reduced due to the self-compensated resonator design (with a turning point at 54 K), the result (curve 5 in Fig. 3) was greater in the former. This was clearly not driven by temperature fluctuations and as a result should be disregarded. However, from a comparison of the liquid and solid nitrogen temperature fluctuations (curves 1 and 2 in Fig. 3), the fractional temperature fluctuations at the cavity in a solid nitrogen bath can be inferred to be an order of magnitude less than in a liquid nitrogen bath (curve 4 in Fig. 3). This means that temperature fluctuations of the SLC temperature at 52 K (curve 6 in Fig. 3) would be about 100 nK at 1 s rising to 500 nK at 100 s.
7. Discussion The fractional frequency of a frequency–temperature compensated resonator sufficiently close to the turning point (TTP ) can be expressed as a quadratic function in (T TTP ). When operating an oscillator with such a resonator (4), must be replaced by an equation in the second derivative, from which we can derive 1 o2 f ry ¼ TTP ð6Þ rT DT ; f0 oT 2 where DT ¼ jT TTP j, which represents the operating temperature offset from the turning point. The Allan deviation of an oscillator’s fractional frequency fluctuations due to temperature fluctuations may be calculated from (6) using our estimate of temperature fluctuations (rT ) for a solid–nitrogen-cooled resonator. The fractional curvature, 1=f0 ðo2 f =oT 2 Þ, was measured in a 12 GHz mode of resonance at the TTP ¼ 53:8 K to be 8 108 K2 [5]. By assuming that the turning point can be found within DT ¼ 1 mK results in ry ¼ 6:4 1018 at 1 s rising to 4 1017 at 100 s. Therefore the temperature control requirements are considerably relaxed, even increasing DT by 100 times the temperature
induced noise floor is still less than the target of 1 1014 at 1 s.
8. Conclusion The temperature fluctuations of both a liquid and a solid nitrogen bath, at 77 and 52 K, respectively, were directly measured. A solid nitrogen bath was found to have an order of magnitude lower temperature instability than liquid nitrogen. The temperature fluctuations in a resonator, filtered by a high thermal impedence, were also measured. This yielded a value of 100 nK at 1 s of averaging time. Calculations based on solid nitrogen bath fluctuations, passively filtered at the cavity, indicate that a flywheel oscillator with sub-1014 fractional frequency stability at 1 s, due to temperature control, is a modest expectation provided a temperature compensated resonator is used.
Acknowledgements The authors wish to acknowledge the contributions of the Australian Research Council (ARC) and Poseidon Scientific Instruments Pty Ltd.
References [1] Jones SK, Blair DG, Buckingham MJ. The effects of paramagnetic impurities on the frequency of sapphire loading superconducting resonators. Electron Lett 1988;24:346–7. [2] Hartnett JG, Tobar ME, Mann AG, Ivanov EN, Krupka J. Frequency–temperature compensation in Ti3þ and Ti4þ doped sapphire whispering gallery mode resonators. Proc IEEE Int Freq Cont Symp 1998:512–7. [3] Dick GJ, Wang RT. Cryo-cooled sapphire oscillator for the Cassini Ka-band experiment. Proc IEEE Int Freq Cont Symp 1997: 1009–14. [4] Dick GJ, Santiago DG, Wang RT. Temperature compensated sapphire resonator for ultra-stable oscillator capability at temperatures above 77 Kelvin. Proc IEEE Freq Cont Symp 1994:421–32. [5] Hartnett JG, Tobar ME, Ivanov EN, Cros D. High-Q frequency– temperature compensated sapphire/rutile resonator. Electron Lett 2000;36:726–7. [6] Tobar ME, Krupka J, Hartnett JG, Ivanov EN, Woode RA. Sapphire-rutile frequency–temperature compensated whispering gallery microwave resonators. Proc IEEE Int Freq Cont Symp 1997:1000–8. [7] Tobar ME, Krupka J, Hartnett JG, Ivanov EN, Woode RA. HighQ sapphire-rutile frequency–temperature compensated microwave dielectric resonators. IEEE Trans UFFC 1998;45:830–6. [8] Santarelli G, Laurent P, Lemonde P, Caliron A, Mann AG, Chang S, Luiten AN, Salomon C. Quantum projection noise in an atomic fountain: a high stability cesium frequency standard. Phys Rev Lett 1999;82:4619–22. [9] Allan DW. Time and frequency (time domain) characterization, estimation, and prediction of precision clocks and oscillators. IEEE Trans Ultrasonics, Ferroelectrics Freq Cont 1987;34:647–54.
Temperature fluctuations in a solid-nitrogen cooled secondary frequency standard J.G. Hartnett *, M.E. Tobar, E.N. Ivanov, P. Bilski Frequency Standards and Metrology Research Group, Department of Physics, University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia Received 29 August 2001; accepted 3 December 2001
Abstract Liquid helium and liquid nitrogen are commonly used as coolants for highly frequency stable oscillators. In this paper, we present the use of solid nitrogen as an alternative coolant. Solid nitrogen solidifies near 52 K under vacuum and we show that the temperature fluctuations in the solid are an order of magnitude lower than in the liquid. This has the advantage of reducing the temperature control requirements necessary to limit temperature induced frequency fluctuations in cryogenic microwave resonatoroscillators. Ó 2002 Published by Elsevier Science Ltd. Keywords: Fractional temperature fluctuations
1. Introduction At the heart of modern radar, telecommunications, navigation, GPS and precise time keeping are low-noise highly stability resonator-oscillators. Generally acoustic modes of resonance in quartz resonators have been used to build such oscillators. Recently, electromagnetic modes of resonance in sapphire resonators have been utilized because of their very high quality factor. This ensures good short-term frequency stability resulting from a high sensitivity of the control electronics to frequency fluctuations. However, to improve the long-term frequency stability of the oscillator the temperature induced frequency fluctuations should be suppressed. At liquid helium temperatures, frequency-temperature compensation of monolithic sapphire resonators has been achieved by the residual paramagnetic impurities of the order of a few parts per million [1]. Intentionally doped sapphire with a larger concentration of Ti3þ ions was used to raise the temperature of compensation above 50 K [2]. Another technique achieved compensation at about 10 K in a composite ruby-sapphire structure [3]. A mechanical compensation technique [4] *
Corresponding author. Tel.: +61-8-9380-3443; fax: +61-8-93801235. E-mail address: [email protected] (J.G. Hartnett).
involves a coupled mode resonator to cancel the resonator’s temperature dependence above 77 K. Another technique utilizes composite dielectric structures consisting of more than one low loss monocrystal. For example, two very thin slices of monocrystalline rutile were clamped tightly against the ends of a cylinder of sapphire monocrystal [5–7]. The temperature coefficients of permittivity of sapphire and rutile are of opposite sign and thus the temperature coefficient of the composite resonator was annulled at temperatures above 50 K. The new generation of atomic fountain primary standards needs ever increasingly stable flywheel oscillators as a pump source. The maximum allowable frequency instability for a flywheel oscillator at present is about 1 part in 1014 at 1 s [8]. In order to take full advantage of the reduced sensitivity to temperature fluctuations in resonators as described above and to reach the required level of oscillator-frequency stability it is necessary to measure the temperature fluctuations in the resonator under operating conditions. The temperature fluctuations of both a liquid and a solid nitrogen bath, at 77 and 52 K, respectively, were directly measured. Using a free running loop microwave oscillator at 12.03 GHz, based on a temperature compensated resonator, the temperature fluctuations in the resonator were also measured. The results indicate that the temperature fluctuations in solid nitrogen at 52 K were an order of magnitude lower than in liquid nitrogen at 77 K.
0011-2275/02/$ - see front matter Ó 2002 Published by Elsevier Science Ltd. PII: S 0 0 1 1 - 2 2 7 5 ( 0 1 ) 0 0 1 7 8 - 3
46
J.G. Hartnett et al. / Cryogenics 42 (2002) 45–48
2. Thermometers Any temperature control system for a highly stable oscillator or any temperature measurement system will have a temperature sensitive element at its heart. In order to determine which one would be the most suitable for our application at temperatures accessible with liquid or solid nitrogen, it was necessary to characterize the temperature dependence of the commercially available resistance-thermometers. The sensitivity of a resistancethermometer is defined as hðT Þ ¼
T oRTh ; RTh oT
ð1Þ
where T is the temperature (in Kelvin) and RTh is the resistance (in Ohms) across the thermometer at temperature T. A number of thermometers were investigated and their dimensionless coefficient or sensitivity compared (see Fig. 1). Over the desired temperature range (50–77 K), the platinum resistance thermometer (PRT) (curve 1) had the highest sensitivity, and, as a result of this experiment, was chosen as the best choice.
3. AC-bridge read-out The measurement system implemented was a bridge with ac excitation (see Fig. 2). The ac-bridge circuit was part of a custom-made temperature controller. The bridge driving voltage (u0 ) was supplied by an on-board (150 mV, 238 Hz) signal oscillator. The bridge was balanced using the variable resistor (RB ). The 2 kX and 10 X resistors were high-precision low-temperature coefficient type resistors. Only the PRT and its leads were in the cryogenic environment. The bridge error voltage was amplified by an ac-coupled amplifier, then demodulated and read out on a DVM and recorded by computer.
Fig. 1. Dimensionless coefficients of five resistance-thermometers measured over the range 50–77 K. Curve 1 was a standard platinum strip (PRT), curve 2 a carbon-glass Lakeshore CGR-1-2000 type, curve 3 an Alan Bradley resistor, curve 4 a Neocera CryoHybrid, and curve 5 was a Scientific Instruments (USA) ruthenium oxide RO105 type thermometer.
Fig. 2. The ac bridge used to measure voltage fluctuations in a platinum resistance thermometer (labelled RPRT ).
The bridge error voltage fluctuations at the output of the mixer in Fig. 2 (dumix ) are related to the temperature fluctuations of the PRT by dumix RPRT
ou dT hðT0 Þ ; oRPRT T0
ð2Þ
where ou=oRPRT is a parameter that characterizes the measurement system sensitivity, dT are the ambient temperature fluctuations and RPRT is the resistance of the PRT at some mean temperature T0 . If the bridge stays balanced, the noise from the source is rejected. (There was some systematic drift from this balance, which was subtracted from the raw data.) After taking the statistical Allan variance [9] of fractional fluctuations of voltage, the square root Allan variance or Allan deviation of fractional temperature fluctuations (rT ) was calculated. From (2), the Allan deviation of fractional temperature fluctuations (rT ) is related to the Allan deviation of fractional voltage fluctuations (ru ), by ru : ð3Þ rT ðou=oRPRT ÞRPRT ðT0 ÞhðT0 Þ
4. Temperature fluctuations of the nitrogen bath Using the circuit of Fig. 2, the voltage fluctuations out of the bridge were recorded with the PRT positioned deep in the cryogenic fluid. The nitrogen was solidified by pumping constantly on the fluid at a pressure below 101 Torr. The temperature was maintained at around 52 K. The measurement system noise floor (curve 3) was measured with a temperature-insensitive resistor. The solid nitrogen data were very close to the noise floor for integration times less than 10 s. Using (3), the voltage data were converted into Allan deviation of fractional temperature fluctuations ðrT Þ (see Fig. 3). A comparison is made between room temperature (curve 3) and liquid nitrogen (curve 1) and solid nitrogen (curve 2).
5. Loop oscillator A thermal design was implemented for a cryogenic sapphire-loaded-cavity (SLC) resonator attached to a
J.G. Hartnett et al. / Cryogenics 42 (2002) 45–48
Fig. 3. The Allan deviation of fractional temperature fluctuations of a liquid nitrogen bath (curve 1) and a solid nitrogen bath (curve 2) measured with the ac-bridge. Curve 3 are the measured room temperature fractional fluctuations. Curve 4 are the measured fractional temperature fluctuations in the cavity determined by measuring the oscillator frequency fluctuations cooled by a liquid nitrogen bath (77 K). Curve 5 are the measured fractional temperature fluctuations in the cavity determined by measuring the oscillator frequency fluctuations cooled to 58 K by a solid nitrogen bath and a foil heater and temperature controller. Curve 6 are the inferred fractional temperature fluctuations in the cavity passively cooled by a solid nitrogen bath (52 K).
copper supporting post. The copper post was connected to the liquid or solid nitrogen bath by a stainless-steel post. This was all housed within a vacuum can immersed in the bath. When the vacuum was sustained, the stainless-steel post provided good thermal impedance and hence low pass thermal filtering of the bath temperature fluctuations. The temperature fluctuations at the copper cavity could not be measured using the acbridge because the measurement system noise floor was insufficiently low enough. For this reason, a free running loop oscillator (Fig. 4) was implemented to get an estimate of the effects of temperature fluctuations on the oscillator frequency stability. The loop oscillator incorporated a 11–14 GHz amplifier, a custom-made band-pass filter, a mechanical
Fig. 4. Schematic of loop oscillator used.
47
phase shifter and a mechanical attenuator in die-cast aluminum box anchored to a 16-mm thick aluminum plate with stainless screws. The amplifier was thermally grounded to the base-plate. The resonator coupling on both probes was approximately 0.005, set previously to measure the Q-factor as a function of temperature. This resulted in a resonator insertion loss of 44 dB. As a result, no Pound frequency discriminator and electronicnoise-reduction servo was implemented. The oscillator frequency was down-converted by beating it with a HP8673G synthesizer locked to an Oscilloquartz 5 MHz reference, and counted on a HP53131A frequency counter. The Allan deviation of fractional frequency fluctuations (ry ) was calculated, for the free running oscillator at both 58 and 77 K. To determine the stability of the HP synthesizer, we used a similar synthesizer with both referenced by their own internal quartz 10 MHz quartz oscillators, and recorded the beat frequency of the two 12.03 GHz signals separated by about 65 kHz. The Allan deviation of fractional frequency fluctuations for the synthesizer was calculated to be 1:1 1012 at 1 s rising to 2:5 1012 at 100 s. The Oscilloquartz reference oscillator has been measured to have 3 1013 stability, and as a result the HP8673G synthesizer, referenced to this oscillator, would have this stability too.
6. Temperature fluctuations in resonator Provided temperature fluctuations in the resonator are the dominant noise mechanism, the fractional frequency fluctuations of an oscillator may be related to the fractional temperature fluctuations of the resonator by df T0 of dT ¼ : ð4Þ f0 f0 oT T0 After calculating the Allan deviation of both frequency and temperature fluctuations, after re-arranging, (4) becomes T0 of rT ¼ ry : ð5Þ f0 oT The dimensionless temperature coefficient of frequency (TCF) ðT0 =f0 Þdf =dT for the sapphire resonator was calculated from measured frequency-temperature data of the chosen mode of resonance, where the bath was allowed to warm slowly from solid to liquid at 1 atm. Knowing the oscillator’s fractional frequency stability (ry ), the fractional temperature stability (rT ) can be calculated from (5). From the measurements of the freerunning oscillator frequency, at 77 K, rT was calculated. It has been assumed here that the frequency stability is solely due to temperature fluctuations. The amplifier
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J.G. Hartnett et al. / Cryogenics 42 (2002) 45–48
phase noise contribution to the oscillator frequency instability was measured at the 1:7 1012 level, a factor of 4 below the temperature induced contribution. Therefore, by operating at a temperature where df =dT is large, we determined the temperature stability at the cavity to be about 1 lK at 1 s rising to 10 lK at 100 s (curve 4 in Fig. 3). However this was not the case for the data taken at 58 K. The frequency stability data here could not be assumed to be solely due to temperature fluctuations, because no Pound servo scheme was implemented and the data here were dominated by the electronic noise resulting from frequency shifts in the loop oscillator, particularly by the modulation of the Teflon in the coaxial lines leading to the cryogenic resonator. In calculating the temperature fluctuations from these data, because df =dT was greatly reduced due to the self-compensated resonator design (with a turning point at 54 K), the result (curve 5 in Fig. 3) was greater in the former. This was clearly not driven by temperature fluctuations and as a result should be disregarded. However, from a comparison of the liquid and solid nitrogen temperature fluctuations (curves 1 and 2 in Fig. 3), the fractional temperature fluctuations at the cavity in a solid nitrogen bath can be inferred to be an order of magnitude less than in a liquid nitrogen bath (curve 4 in Fig. 3). This means that temperature fluctuations of the SLC temperature at 52 K (curve 6 in Fig. 3) would be about 100 nK at 1 s rising to 500 nK at 100 s.
7. Discussion The fractional frequency of a frequency–temperature compensated resonator sufficiently close to the turning point (TTP ) can be expressed as a quadratic function in (T TTP ). When operating an oscillator with such a resonator (4), must be replaced by an equation in the second derivative, from which we can derive 1 o2 f ry ¼ TTP ð6Þ rT DT ; f0 oT 2 where DT ¼ jT TTP j, which represents the operating temperature offset from the turning point. The Allan deviation of an oscillator’s fractional frequency fluctuations due to temperature fluctuations may be calculated from (6) using our estimate of temperature fluctuations (rT ) for a solid–nitrogen-cooled resonator. The fractional curvature, 1=f0 ðo2 f =oT 2 Þ, was measured in a 12 GHz mode of resonance at the TTP ¼ 53:8 K to be 8 108 K2 [5]. By assuming that the turning point can be found within DT ¼ 1 mK results in ry ¼ 6:4 1018 at 1 s rising to 4 1017 at 100 s. Therefore the temperature control requirements are considerably relaxed, even increasing DT by 100 times the temperature
induced noise floor is still less than the target of 1 1014 at 1 s.
8. Conclusion The temperature fluctuations of both a liquid and a solid nitrogen bath, at 77 and 52 K, respectively, were directly measured. A solid nitrogen bath was found to have an order of magnitude lower temperature instability than liquid nitrogen. The temperature fluctuations in a resonator, filtered by a high thermal impedence, were also measured. This yielded a value of 100 nK at 1 s of averaging time. Calculations based on solid nitrogen bath fluctuations, passively filtered at the cavity, indicate that a flywheel oscillator with sub-1014 fractional frequency stability at 1 s, due to temperature control, is a modest expectation provided a temperature compensated resonator is used.
Acknowledgements The authors wish to acknowledge the contributions of the Australian Research Council (ARC) and Poseidon Scientific Instruments Pty Ltd.
References [1] Jones SK, Blair DG, Buckingham MJ. The effects of paramagnetic impurities on the frequency of sapphire loading superconducting resonators. Electron Lett 1988;24:346–7. [2] Hartnett JG, Tobar ME, Mann AG, Ivanov EN, Krupka J. Frequency–temperature compensation in Ti3þ and Ti4þ doped sapphire whispering gallery mode resonators. Proc IEEE Int Freq Cont Symp 1998:512–7. [3] Dick GJ, Wang RT. Cryo-cooled sapphire oscillator for the Cassini Ka-band experiment. Proc IEEE Int Freq Cont Symp 1997: 1009–14. [4] Dick GJ, Santiago DG, Wang RT. Temperature compensated sapphire resonator for ultra-stable oscillator capability at temperatures above 77 Kelvin. Proc IEEE Freq Cont Symp 1994:421–32. [5] Hartnett JG, Tobar ME, Ivanov EN, Cros D. High-Q frequency– temperature compensated sapphire/rutile resonator. Electron Lett 2000;36:726–7. [6] Tobar ME, Krupka J, Hartnett JG, Ivanov EN, Woode RA. Sapphire-rutile frequency–temperature compensated whispering gallery microwave resonators. Proc IEEE Int Freq Cont Symp 1997:1000–8. [7] Tobar ME, Krupka J, Hartnett JG, Ivanov EN, Woode RA. HighQ sapphire-rutile frequency–temperature compensated microwave dielectric resonators. IEEE Trans UFFC 1998;45:830–6. [8] Santarelli G, Laurent P, Lemonde P, Caliron A, Mann AG, Chang S, Luiten AN, Salomon C. Quantum projection noise in an atomic fountain: a high stability cesium frequency standard. Phys Rev Lett 1999;82:4619–22. [9] Allan DW. Time and frequency (time domain) characterization, estimation, and prediction of precision clocks and oscillators. IEEE Trans Ultrasonics, Ferroelectrics Freq Cont 1987;34:647–54.