Weierstrass M-test

Phase diagram of liquid ³He–⁴He mixtures showing the phase separation.

Schematic diagram of a standard, or wet, dilution refrigerator

Schematic diagram of the low-temperature part of a dilution refrigerator.

The inside of a helium dilution refrigerator, with the vacuum cans removed.

Gas control system for a helium dilution refrigerator

Schematic diagram of a cryogen-free, or dry, dilution refrigerator precooled by a two-stage pulse tube refrigerator, indicated by the dotted rectangle.

A ³He/⁴He dilution refrigerator is a cryogenic device that provides continuous cooling to temperatures as low as 2 mK, with no moving parts in the low-temperature region. The cooling power is provided by the heat of mixing of the Helium-3 and Helium-4 isotopes. It is the only continuous refrigeration method for reaching temperatures below 0.3 K.

The dilution refrigerator was first proposed by Heinz London in the early 1950s, and was experimentally realized in 1964 in the Kamerlingh Onnes Laboratorium at Leiden University.

Theory of operation

The refrigeration process uses a mixture of two isotopes of helium: helium-3 and helium-4. When cooled below approximately 870 millikelvin, the mixture undergoes spontaneous phase separation to form a ³He-rich phase (the concentrated phase) and a ³He-poor phase (the dilute phase). As shown in the phase diagram, at very low temperatures the concentrated phase is essentially pure ³He, while the dilute phase contains about 6.6% ³He and 93.4% ⁴He. The working fluid is ³He, which is circulated by vacuum pumps at room temperature.

The ³He enters the cryostat at a pressure of a few hundred millibar. In the classic dilution refrigerator (known as a wet dilution refrigerator), the ³He is precooled and purified by liquid nitrogen at 77 K and a ⁴He bath at 4.2 K. Next, the ³He enters a vacuum chamber where it is further cooled to a temperature of 1.2–1.5 K by the "1 K bath", a pumped ⁴He bath (as decreasing the pressure of the helium reservoir depresses its boiling point). The 1 K bath liquifies the ³He gas and removes the heat of condensation. The ³He then enters the main impedance, a capillary with a large flow resistance. It is cooled by the still (described below) to a temperature 500–700 mK. Subsequently the ³He flows through a secondary impedance and one side of a set of counterflow heat exchangers where it is cooled by a cold flow of ³He. Finally, the pure ³He enters the mixing chamber, the coldest area of the device.

In the mixing chamber, two phases of the ³He–⁴He mixture, the concentrated phase (practically 100% ³He) and the dilute phase (about 6.6% ³He and 93.4% ⁴He), are in equilibrium and separated by a phase boundary. Inside the chamber, the ³He is diluted as it flows from the concentrated phase through the phase boundary into the dilute phase. The heat necessary for the dilution is the useful cooling power of the refrigerator, as the process of moving the ³He through the phase boundary is endothermic and removes heat from the mixing chamber environment. The ³He then leaves the mixing chamber in the dilute phase. On its way up, the cold, dilute ³He cools the downward flowing ³He via the heat exchangers until it enters the still. In the still, the ³He flows through superfluid ⁴He which is at rest. The pressure in the still is kept low (about 10 Pa) by the pumps at room temperature. The vapor in the still is practically pure ³He, which has a much higher partial pressure than ⁴He at 500–700 mK. The pump therefore creates an osmotic pressure difference, which drives more ³He from the concentrated to dilute phases in the mixing chamber, and then up from the mixing chamber to the still. Heat is supplied to the still to maintain a steady flow of ³He. The pumps compress the ³He to a pressure of a few hundred millibar and feed it back into the cryostat, completing the cycle.

Cryogen-free dilution refrigerators

Modern dilution refrigerators can precool the ³He with a cryocooler in place of liquid nitrogen, liquid helium, and a 1 K bath. No external supply of cryogenic liquids is needed in these "dry cryostats" and operation can be highly automatized. However, dry cryostats have high energy requirements and are subject to mechanical vibrations, such as those produced by pulse tube refrigerators. The first experimental machines were built in the 1990s, when (commercial) cryocoolers became available, capable of reaching a temperature lower than that of liquid helium and having sufficient cooling power (on the order of 1 watt at 4.2 K). Pulse tube coolers are commonly used cryocoolers in dry dilution refrigerators.

Dry dilution refrigerators generally follow one of two designs. One design incorporates an inner vacuum can, which is used to initially precool the machine from room temperature down to the base temperature of the pulse tube cooler (using heat-exchange gas). However, every time the refrigerator is cooled down, a vacuum seal that holds at cryogenic temperatures needs to be made, and low temperature vacuum feed-throughs must be used for the experimental wiring. The other design is more demanding to realize, requiring heat switches that are necessary for precooling, but no inner vacuum can is needed, greatly reducing the complexity of the experimental wiring. The latter type is used for the SCUBA-2 sub-millimetre camera on the James Clerk Maxwell Telescope.

Cooling power

At not too low temperatures the cooling power at the mixing chamber is approximately given by

${\dot {Q}}_{m}={\dot {n}}_{3}(12T_{i}^{2}-96T_{m}^{2})$

where ${\dot {n}}_{3}$ is the ³He molar circulation rate, T_m is the mixing-chamber temperature, and T_i the temperature of the ³He entering the mixing chamber. In the case of zero heat load there is a fixed ratio between the two temperatures

$T_{m}={\frac {T_{i}}{2.8}}.$

From this relation it is clear that a low T_m can only be reached if T_i is low. In dilution refrigerators T_i is reduced by using heat exchangers as shown in the schematic diagram of the low-temperature region above. However, at very low temperatures this becomes more and more difficult due to the so-called Kapitza resistance. This is a heat resistance at the surface between the helium liquids and the solid body of the heat exchanger. It is inversely proportional to T⁴ and the heat-exchanging surface area A. In other words: to get the same heat resistance one needs to increase the surface by a factor 10,000 if the temperature goes down by a factor 10. In order to get a low thermal resistance at low temperatures (below about 30 mK) a large surface area is needed. The lower the temperature, the larger the area. In practice one uses very fine silver powder.

Limitations

There is no fundamental limiting low temperature of dilution refrigerators. Yet the temperature range is limited to about 2 mK for practical reasons. At very low temperatures both the viscosity and the thermal conductivity of the circulating fluid become larger if the temperature is lowered. To reduce the viscous heating the diameters of the inlet and outlet tubes of the mixing chamber must go as T_m⁻³ and to get low heat flow the lengths of the tubes should go as T_m⁻⁸. That means that, to reduce the temperature by a factor 2, one needs to increase the diameter by a factor 8 and the length by a factor 256. Hence the volume should be increased by a factor 2¹⁴=16384. In other words: every cm³ at 2 mK would become 16.384 liter at 1 mK. The machines would become very big and very expensive. Fortunately there is a very powerful alternative for cooling below 2 mK and that is nuclear demagnetization.

See also

• Adiabatic demagnetization
• Magnetic refrigeration.
• Dry dilution refrigerator
• Helium-3 refrigerator
• Refrigerated transport Dewar
• Timeline of low-temperature technology

References

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
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• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

External links

• Low Temperature Physics: The What, the How, the Why - The National High Magnetic Field Laboratory
• 3He-4He Dilution Explanation
• Lancaster University, Ultra Low Temperature Physics - Description of dilution refrigeration.
• Harvard University, Marcus Lab - Hitchhiker's Guide to the Dilution Refrigerator.