Magnetic systems for refrigeration & thermometry * Paramagnetic salts * * Hyperfine enhanced systems * * Nuclear spins * Magnetic moments can be aligned by external magnetic field Thermal disorder counteracts the perfect alignment of spins => degree of order (magnetization) depends on the ratio B/T Measure M in known B and get T OR Keep M constant, so that change in B results in change in T 1
Thermodynamics Balance between magnetic energy: thermal energy: μb kbt Equidistant Zeeman states for spin J give the partition function: J Z =[ m x / J nn A e ] m= J with nn 2 J 1 x x = [sinh /sinh ] 2J 2J A B x= k BT 2
Basic relations... Entropy: T ln Z S = kb T Heat capacity: CB = T Magnetization: ln Z M = k BT B T S T B 3
... worked out exactly 2 J 1 x 2 J 1 x S x x = coth coth nr 2 J 2J 2J 2J 2 J 1 x x ln [sinh / sinh ] 2J 2J 2 2 CB 2 J 1 x x2 x 2 2 2 J 1 x = sinh sinh 2 2 nr 4 J 2J 2J 4J 2 J 1 2 J 1 x 1 M M x = = B J x = coth coth nn A M sat 2J 2J 2J 2J 4
High-T approximations Often x << 1 (kbt >> μb) so that it is safe to simplify S J 1 2 ln 2 J 1 x nr 6J CB J 1 2 x nr 3J M J 1 x M sat 3J 5
Entropy All relations above, including molar spin entropy Sn = S/n, depend just on the ratio of magnetic field and temperature: Sn = Sn (B/T) The limiting value at high temperatures: Sn Sn, max = R ln(2j+1) T In adiabatic processes entropy remains constant => T can be made to change in proportion to B. BUT entropy must differ notably from its maximum value to begin with. 6
Other related quantities Susceptibility χ: where 0 M J 1 0 M sat = x= VB 3J VB T J 1 nr 0 2 = 3 J V k 2B is the Curie constant, [λ] = K Now, for example 2 J 1 V B CB nrx 2 = 3J 0 T 7
Heat capacity Any magnetic system, regardless of the origin and magnitude of the magnetic moment, has maximum in the heat capacity, that depends ONLY on the value of the spin J! The maximum occurs at x B = 1... 2 J Jk B T 8
Internal magnetic field The spins do not respond to external magnetic field only, they also feel the weak fields of each other. This is represented by the internal field b. Roughly speaking b ~ kbtc/μ, where Tc is the magnetic ordering temperature of the material This adds quadratically to the external magnetic field: B tot B b 2 2 To have Btot ~ B, one must have rather small b => we are interested on materials with low Tc, i.e. weakly magnetic materials 9
Paramagnetic salts Weakest known conventional magnets (with atomic moments) are so called paramagnetic salts The magnetic element is buried in an ionic compound (salt) containing lots of crystal water The moment is due to 3d transition elements 4f rare earth metals Mn2+, Fe3+, Cr3+, Ce3+,... 10
Electronic moments The unpaired electron at the inner shell of these ions carry a magnetic moment = g B J with Landé factor g= 3 S S 1 L L 1 2 J J 1 and the Bohr magneton eℏ 24 B = 9.27 10 J /T 2 me 11
Useful compounds The strength of the magnetic interactions is controlled by the distance between the moments: Tc ~ μ2/r 3 compound MAS FAA CPA CMN Mn2+ SO4 (NH4)2SO4 6 H2O Fe3+2 (SO4)3 (NH4)2SO4 24 H2O Cr3+2 (SO4)3 K2SO4 24 H2O 2Ce3+ (NO3)3 3 Mg(NO3)2 24 H2O Tc/K 0.17 0.03 0.01 0.002 Cerium can be further diluted by nonmagnetic lantanum => LCMN 12
Entropy curves 13
Conventional magnetic thermometry Measure the magnetization V B M 0 T (M << Msat) or magnetic susceptibility 0 M = VB T (Curie's law) or dynamic magnetic susceptibility 0 dm = V db T In order to operate at mk-range, B must not be very large => internal field b is significant 14
Effective magnetic fields It is useful to distinguish three different contributions to Btot: external magnetic field Bext Weiss field BW due to spin-spin interactions demagnetization field Bd depending on the sample shape Btot = Bext + Bd + BW Demagnetization field Bd = Dμ0M/V can be written in terms of the demagnetization factor D which is for sphere: plane B: plane B: D = 1/3 D=1 D=0 15
Effective magnetic fields (cont.) Weiss field BW = (α + R)μ0M/V depends on the lattice symmetry and on the exchange interaction between the spins for cubic crystals α = 1/3 exchange parameter R can be positive or negative (ferro- or antiferromagnetic exchange) For a specific sample with Btot = Bext + Δ/λ μ0m/v Δ/λ = α + R D 16
Curie-Weiss law Spins adapt to total field: You measure 0 M spin = VB tot => V spin 0 M= B ext M 0 V => V spin B ext / 0 M= 1 spin / 0 dm spin = = V db ext 1 spin / = /T = 1 /T T 17
Inverse susceptibility vs. T 1/χ = T 1 slo positive Δ : ferromagnet negative Δ : antiferromagnet } offset Δ λ e p T 18
CMN (or LCMN) thermometers These salts are insulators => exchange is weak => R << 1 It is a good idea to make the sample nearly spherical For example cylinder with L ~ d => D ~ 1/3 Then α + R D ~ 0 => Curie behavior χ ~ λ/t Typically Δ ~ 0.1... 1 mk d L Limiting factor is often poor thermal contact to the sensor thin silver wires or foils are usually mixed in with the salt powdered salt can be used to measure T of helium liquids Chemical stability (dehydration) may be another problem has to be recalibrated in each cooldown 19
Some realizations Gradiometric arrangement improves sensitivity of the measurement 20
Measuring schemes Measuring field (B = 0.1... 10 mt) must have extremely good stability => superconducting shields & nonmagnetic construction relatively easy and quick method simple relationship with T needs a single point for calibration sensitivity improves towards low T 21
Combined with SC fixed point device Superconducting transition points provide calibration points CMN thermometer interpolates/extrapolates from these 22
Impurity paramagnetism Foreign magnetic atom (e.g. Fe) can polarize the paramagnetic host with large permeability (e.g. Pd) => giant moment ~ 10 μb Few ppm of Fe in Pd gives reasonable magnetization to measure Moment density is low & randomly located => no magnetic order Random moment distribution results in spin-glass transition at Tg ~ 0.1 mk xfe/ppm Above Tg : χ µ 1/T Below Tg : χ µ T 23
Magnetic refrigeration Initial condition with low entropy S = Si < Smax, (ΔS/Smax > 0.2) has to be prepared by high field Bi / low temperature Ti No other entropy in the system should be comparable to this The system must be made isolated (adiabatic); heat switch needed The magnetic field must be changed reversibly => corresponding change in temperature (B/T = const.) The weaker the magnetic system, the larger Bi/Ti is needed BUT the lower final temperature is achievable, as Bmin ~ b Cooling capacity depends on final field & moment strength When the cooling capacity is exhausted, the process has to be repeated again (single cycle cooling) 24
Cooling cycle Zeeman levels (J = ½) Precool to point A: Bext = 0, T = Ti Magnetize A B, remove heat QAB = TΔS Isolate the system, ΔS = 0 thereafter T T Demagnetize B C S Q = T Use the capacity C A, CA T dt = C B dt Tf B Tf QCA << QAB 25
Conventional magnetic refrigeration Uses paramagnetic salts (CMN, CPA, FAA,...) Easy initial conditions Bi = 1 T and Ti = 1 K are sufficient High cooling capacity remains also at Bext = 0 Temperature can be measured from inherent property: χ = λ/t Fast, demagnetization can be performed in few minutes BUT Poor thermal conductivity limits heat transfer Difficult to mount samples (except liquid 3He immersed in powdered CMN salt; 2 mk can be achieved) Single shot cooling Largely replaced by dilution refrigerators Space applications are still developed 26
Realizations More elaborate schemes with multiple stages producing continuous refrigeration from 4 K to mk-range have been demonstrated 27