Samarium–cobalt magnet

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Samarium%E2%80%93cobalt_magnet 

A samarium–cobalt (SmCo) magnet, a type of rare-earth magnet, is a strong permanent magnet made of two basic elements samarium and cobalt.

They were developed in the early 1960s based on work done by Karl Strnat at Wright-Patterson Air Force Base and Alden Ray at the University of Dayton. In particular, Strnat and Ray developed the first formulation of SmCo₅.

Samarium–cobalt magnets are generally ranked similarly in strength to neodymium magnets, but have higher temperature ratings and higher coercivity.

Attributes

Extremely resistant to demagnetization

Good temperature stability (maximum use temperatures between 250 °C (523 K) and 550 °C (823 K); Curie temperatures from 700 °C (973 K) to 800 °C (1,070 K)

Expensive and subject to price fluctuations (cobalt is market price sensitive)

Samarium–cobalt magnets have a strong resistance to corrosion and oxidation resistance, usually do not need to be coated and can be widely used in high temperature and poor working conditions.

They are brittle, and prone to cracking and chipping. Samarium–cobalt magnets have maximum energy products (BH_max) that range from 14 megagauss-oersteds (MG·Oe) to 33 MG·Oe, that is approx. 112 kJ/m³ to 264 kJ/m³; their theoretical limit is 34 MG·Oe, about 272 kJ/m³.

Sintered Samarium–cobalt magnets exhibit magnetic anisotropy, meaning they can only be magnetized in the axis of their magnetic orientation. This is done by aligning the crystal structure of the material during the manufacturing process.

Comparison of physical properties of sintered neodymium and Sm-Co magnets

Property (unit)	Neodymium	Sm-Co
Remanence (T)	1–1.5	0.8–1.16
Coercivity (MA/m)	0.875–2.79	0.493–2.79
Relative permeability (–)	1.05	1.05–1.1
Temperature coefficient of remanence (%/K)	–0.09..–0.12	−0.03..–0.05
Temperature coefficient of coercivity (%/K)	−0.40..–0.65	−0.15..–0.30
Curie temperature (°C)	310–370	700–850
Density (g/cm3)	7.3–7.7	8.2–8.5
CTE, magnetizing direction (1/K)	(3–4)×10−6	(5–9)×10−6
CTE, normal to magnetizing direction (1/K)	(1–3)×10−6	(10–13)×10−6
Flexural strength (N/mm2)	200–400	150–180
Compressive strength (N/mm2)	1000–1100	800–1000
Tensile strength (N/mm2)	80–90	35–40
Vickers hardness (HV)	500–650	400–650
Electrical resistivity (Ω·cm)	(110–170)×10−6	(50–90)×10−6

Series

Samarium–cobalt magnets are available in two "series", namely SmCo₅ magnets and Sm₂Co₁₇ magnets.

Series 1:5

These samarium–cobalt magnet alloys (generally written as SmCo₅, or SmCo Series 1:5) have one atom of rare-earth samarium per five atoms of cobalt. By weight this magnet alloy will typically contain 36% samarium with the balance cobalt. The energy products of these samarium–cobalt alloys range from 16 MG·Oe to 25 MG·Oe, that is, approx. 128–200 kJ/m³. These samarium–cobalt magnets generally have a reversible temperature coefficient of -0.05%/°C. Saturation magnetization can be achieved with a moderate magnetizing field. This series of magnet is easier to calibrate to a specific magnetic field than the SmCo 2:17 series magnets.

In the presence of a moderately strong magnetic field, unmagnetized magnets of this series will try to align their orientation axis to the magnetic field, thus becoming slightly magnetized. This can be an issue if postprocessing requires that the magnet be plated or coated. The slight field that the magnet picks up can attract debris during the plating or coating process, causing coating failure or a mechanically out-of-tolerance condition.

B_r drifts with temperature and it is one of the important characteristics of magnet performance. Some applications, such as inertial gyroscopes and travelling wave tubes (TWTs), need to have constant field over a wide temperature range. The reversible temperature coefficient (RTC) of B_r is defined as

(∆B_r/B_r) x (1/∆T) × 100%.

To address these requirements, temperature compensated magnets were developed in the late 1970s. For conventional SmCo magnets, B_r decreases as temperature increases. Conversely, for GdCo magnets, B_r increases as temperature increases within certain temperature ranges. By combining samarium and gadolinium in the alloy, the temperature coefficient can be reduced to nearly zero.

SmCo₅ magnets have a very high coercivity (coercive force); that is, they are not easily demagnetized. They are fabricated by packing wide-grain lone-domain magnetic powders. All of the magnetic domains are aligned with the easy axis direction. In this case, all of the domain walls are at 180 degrees. When there are no impurities, the reversal process of the bulk magnet is equivalent to lone-domain motes, where coherent rotation is the dominant mechanism. However, due to the imperfection of fabricating, impurities may be introduced in the magnets, which form nuclei. In this case, because the impurities may have lower anisotropy or misaligned easy axes, their directions of magnetization are easier to spin, which breaks the 180° domain wall configuration. In such materials, the coercivity is controlled by nucleation. To obtain much coercivity, impurity control is critical in the fabrication process.

Series 2:17

These alloys (written as Sm₂Co₁₇, or SmCo Series 2:17) are age-hardened with a composition of two atoms of rare-earth samarium per 13–17 atoms of transition metals (TM). The TM content is rich in cobalt, but contains other elements such as iron and copper. Other elements like zirconium, hafnium, and such may be added in small quantities to achieve better heat treatment response. By weight, the alloy will generally contain 25% of samarium. The maximum energy products of these alloys range from 20 to 32 MGOe, what is about 160-260 kJ/m³. These alloys have the best reversible temperature coefficient of all rare-earth alloys, typically being -0.03%/°C. The "second generation" materials can also be used at higher temperatures.

In Sm₂Co₁₇ magnets, the coercivity mechanism is based on domain wall pinning. Impurities inside the magnets impede the domain wall motion and thereby resist the magnetization reversal process. To increase the coercivity, impurities are intentionally added during the fabrication process.

Production

The alloys are typically machined in the unmagnetized state. Samarium–cobalt should be ground using a wet grinding process (water-based coolants) and a diamond grinding wheel. The same type of process is required if drilling holes or other features that are confined. The grinding waste produced must not be allowed to completely dry as samarium–cobalt has a low ignition point. A small spark, such as that produced with static electricity, can easily initiate combustion. The resulting fire produced can be extremely hot and difficult to control.

The reduction/melt method and reduction/diffusion method are used to manufacture samarium–cobalt magnets. The reduction/melt method will be described since it is used for both SmCo₅ and Sm₂Co₁₇ production. The raw materials are melted in an induction furnace filled with argon gas. The mixture is cast into a mold and cooled with water to form an ingot. The ingot is pulverized and the particles are further milled to further reduce the particle size. The resulting powder is pressed in a die of desired shape, in a magnetic field to orient the magnetic field of the particles. Sintering is applied at a temperature of 1100˚C–1250˚C, followed by solution treatment at 1100˚C–1200˚C and tempering is finally performed on the magnet at about 700˚C–900˚C. It then is ground and further magnetized to increase its magnetic properties. The finished product is tested, inspected and packed.^{[citation needed]}

Samarium can be substituted by a portion of other rare-earth elements including praseodymium, cerium, and gadolinium; the cobalt can be substituted by a portion of other transition metals including iron, copper, and zirconium.

Uses

1980s vintage headphones using Samarium Cobalt magnets

Fender used one of designer Bill Lawrence's Samarium Cobalt Noiseless series of electric guitar pickups in Fender's Vintage Hot Rod '57 Stratocaster. These pickups were used in American Deluxe Series Guitars and Basses from 2004 until early 2010.

In the mid-1980s some expensive headphones such as the Ross RE-278 used samarium–cobalt "Super Magnet" transducers.

Other uses include:

• High-end electric motors used in the more competitive classes in slotcar racing
• Turbomachinery
• Traveling-wave tube field magnets
• Applications that will require the system to function at cryogenic temperatures or very hot temperatures (over 180 °C)
• Applications in which performance is required to be consistent with temperature change
• Benchtop NMR spectrometers
• Rotary encoders where it performs the function of magnetic actuator

Rare-earth magnet

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Rare-earth_magnet 

 

Ferrofluid on glass, with a rare-earth magnet underneath

Rare-earth magnets are strong permanent magnets made from alloys of rare-earth elements. Developed in the 1970s and 1980s, rare-earth magnets are the strongest type of permanent magnets made, producing significantly stronger magnetic fields than other types such as ferrite or alnico magnets. The magnetic field typically produced by rare-earth magnets can exceed 1.4 teslas, whereas ferrite or ceramic magnets typically exhibit fields of 0.5 to 1 tesla.

There are two types: neodymium magnets and samarium–cobalt magnets. Rare-earth magnets are extremely brittle and also vulnerable to corrosion, so they are usually plated or coated to protect them from breaking, chipping, or crumbling into powder.

The development of rare-earth magnets began around 1966, when K. J. Strnat and G. Hoffer of the US Air Force Materials Laboratory discovered that an alloy of yttrium and cobalt, YCo₅, had by far the largest magnetic anisotropy constant of any material then known.

The term "rare earth" can be misleading, as some of these metals can be as abundant in the Earth's crust as tin or lead, but rare earth ores do not exist in seams (like coal or copper), so in any given cubic kilometre of crust they are "rare". The major source is currently China. Some countries classify rare earth metals as strategically important, and recent Chinese export restrictions on these materials have led some to initiate research programs to develop strong magnets that do not require rare earth metals.

Neodymium magnets (small cylinders) lifting steel balls. As shown here, rare-earth magnets can easily lift thousands of times their own weight.

Explanation of strength

The rare-earth (lanthanide) elements are metals that are ferromagnetic, meaning that like iron they can be magnetized to become permanent magnets, but their Curie temperatures (the temperature above which their ferromagnetism disappears) are below room temperature, so in pure form their magnetism only appears at low temperatures. However, they form compounds with the transition metals such as iron, nickel, and cobalt, and some of these compounds have Curie temperatures well above room temperature. Rare-earth magnets are made from these compounds.

The greater strength of rare-earth magnets is mostly due to two factors:

• First, their crystalline structures have very high magnetic anisotropy. This means that a crystal of the material preferentially magnetizes along a specific crystal axis but is very difficult to magnetize in other directions. Like other magnets, rare-earth magnets are composed of microcrystalline grains, which are aligned in a powerful magnetic field during manufacture, so their magnetic axes all point in the same direction. The resistance of the crystal lattice to turning its direction of magnetization gives these compounds a very high magnetic coercivity (resistance to being demagnetized), so that the strong demagnetizing field within the finished magnet does not reduce the material's magnetization.
• Second, atoms of rare-earth elements can have high magnetic moments. Their orbital electron structures contain many unpaired electrons; in other elements, almost all of the electrons exist in pairs with opposite spins, so their magnetic fields cancel out, but in rare-earths there is much less magnetic cancellation. This is a consequence of incomplete filling of the f-shell, which can contain up to 7 unpaired electrons. In a magnet it is the unpaired electrons, aligned so they spin in the same direction, which generate the magnetic field. This gives the materials high remanence (saturation magnetization J_s ). The maximal energy density B·H_max is proportional to J_s², so these materials have the potential for storing large amounts of magnetic energy. The magnetic energy product B·H_max of neodymium magnets is about 18 times greater than "ordinary" magnets by volume. This allows rare-earth magnets to be smaller than other magnets with the same field strength.

Magnetic properties

Some important properties used to compare permanent magnets are: remanence (B_r), which measures the strength of the magnetic field; coercivity (H_ci), the material's resistance to becoming demagnetized; energy product (B·H_max), the density of magnetic energy; and Curie temperature (T_C), the temperature at which the material loses its magnetism. Rare-earth magnets have higher remanence, much higher coercivity and energy product, but (for neodymium) lower Curie temperature than other types. The table below compares the magnetic performance of the two types of rare-earth magnets, neodymium (Nd₂Fe₁₄B) and samarium-cobalt (SmCo₅), with other types of permanent magnets.

Magnet	preparation	Br (T)	Hci (kA/m)	B·Hmax (kJ/m3)	TC (°C)
Nd2Fe14B	sintered	1.0–1.4	750–2000	200–440	310–400
Nd2Fe14B	bonded	0.6–0.7	600–1200	60–100	310–400
SmCo5	sintered	0.8–1.1	600–2000	120–200	720
Sm(Co,Fe,Cu,Zr)7	sintered	0.9–1.15	450–1300	150–240	800
Alnico	sintered	0.6–1.4	275	10–88	700–860
Sr-ferrite	sintered	0.2–0.4	100–300	10–40	450
Iron (Fe) bar magnet	annealed	?	800	?	770

Types

Samarium-cobalt

Main article: Samarium–cobalt magnet

Samarium–cobalt magnets (chemical formula: SmCo₅), the first family of rare-earth magnets invented, are less used than neodymium magnets because of their higher cost and lower magnetic field strength. However, samarium–cobalt has a higher Curie temperature, creating a niche for these magnets in applications where high field strength is needed at high operating temperatures. They are highly resistant to oxidation, but sintered samarium–cobalt magnets are brittle and prone to chipping and cracking and may fracture when subjected to thermal shock.

Neodymium

Neodymium magnet with nickel plating mostly removed

Neodymium magnets, invented in the 1980s, are the strongest and most affordable type of rare-earth magnet. They are made of an alloy of neodymium, iron, and boron (Nd₂Fe₁₄B), sometimes abbreviated as NIB. Neodymium magnets are used in numerous applications requiring strong, compact permanent magnets, such as electric motors for cordless tools, hard disk drives, magnetic holddowns, and jewelry clasps. They have the highest magnetic field strength and have a higher coercivity (which makes them magnetically stable), but they have a lower Curie temperature and are more vulnerable to oxidation than samarium–cobalt magnets.

Corrosion can cause unprotected magnets to spall off a surface layer or to crumble into a powder. Use of protective surface treatments such as gold, nickel, zinc, and tin plating and epoxy-resin coating can provide corrosion protection; the majority of neodymium magnets use nickel plating to provide a robust protection.

Originally, the high cost of these magnets limited their use to applications requiring compactness together with high field strength. Both the raw materials and the patent licenses were expensive. However, since the 1990s, NIB magnets have become steadily less expensive, and their lower cost has inspired new uses such as magnetic construction toys.

Hazards

The greater force exerted by rare-earth magnets creates hazards that are not seen with other types of magnet. Magnets larger than a few centimeters are strong enough to cause injuries to body parts pinched between two magnets or a magnet and a metal surface, even causing broken bones. Magnets allowed to get too near each other can strike each other with enough force to chip and shatter the brittle material, and the flying chips can cause injuries. Starting in 2005, powerful magnets breaking off toys or from magnetic construction sets started causing injuries and deaths. Young children who have swallowed several magnets have had a fold of the digestive tract pinched between the magnets, causing injury and in one case intestinal perforations, sepsis, and death.

A voluntary standard for toys, permanently fusing strong magnets to prevent swallowing, and capping unconnected magnet strength, was adopted in 2007. In 2009, a sudden growth in sales of magnetic desk toys for adults caused a surge in injuries, with emergency room visits estimated at 3,617 in 2012. In response, the U.S. Consumer Product Safety Commission passed a rule in 2012 restricting rare-earth magnet size in consumer products, but it was vacated by a US federal court decision in November 2016, in a case brought by the one remaining manufacturer. After the rule was nullified, the number of ingestion incidents in the country rose sharply, and is estimated to exceed 1,500 in 2019.

Applications

Since their prices became competitive in the 1990s, neodymium magnets have been replacing alnico and ferrite magnets in the many applications in modern technology requiring powerful magnets. Their greater strength allows smaller and lighter magnets to be used for a given application.

Common applications

Neodymium magnet balls

Common applications of rare-earth magnets include:

• computer hard disk drives
• wind turbine generators
• speakers / headphones
• bicycle dynamos
• MRI scanners
• fishing reel brakes
• permanent magnet motors in cordless tools
• high-performance AC servo motors
• traction motors and integrated starter-generators in hybrid and electric vehicles
• mechanically powered flashlights, employing rare earth magnets for generating electricity in a shaking motion or rotating (hand-crank-powered) motion
• industrial uses such as maintaining product purity, equipment protection, and quality control
• capture of fine metallic particles in lubricating oils (crankcases of internal combustion engines, also gearboxes and differentials), so as to keep said particles out of circulation, thereby rendering them unable to cause abrasive wear of moving machine parts

Other applications

Other applications of rare-earth magnets include:

• Linear motors (used in maglev trains, etc.)
• Stop motion animation: as tie-downs when the use of traditional screw and nut tie-downs is impractical.
• Diamagnetic levitation experimentation, the study of magnetic field dynamics and superconductor levitation.
• Electrodynamic bearings
• Launched roller coaster technology found on roller coaster and other thrill rides.
• LED Throwies, small LEDs attached to a button cell battery and a small rare earth magnet, used as a form of non-destructive graffiti and temporary public art.
• Neodymium magnet toys
• Electric guitar pickups
• Miniature figures, for which rare-earth magnets have gained popularity in the miniatures gaming community for their small size and relative strength assisting in basing and swapping weapons between models.

Rare-earth-free permanent magnets

The United States Department of Energy has identified a need to find substitutes for rare-earth metals in permanent-magnet technology and has begun funding such research. The Advanced Research Projects Agency-Energy (ARPA-E) has sponsored a Rare Earth Alternatives in Critical Technologies (REACT) program, to develop alternative materials. In 2011, ARPA-E awarded 31.6 million dollars to fund Rare-Earth Substitute projects.

Recycling efforts

The European Union's ETN-Demeter project (European Training Network for the Design and Recycling of Rare-Earth Permanent Magnet Motors and Generators in Hybrid and Full Electric Vehicles) is examining sustainable design of electric motors used in vehicles. They are, for example, designing electric motors in which the magnets can be easily removed for recycling the rare earth metals.

The European Union's European Research Council also awarded to Principal Investigator, Prof. Thomas Zemb, and co-Principal Investigator, Dr. Jean-Christophe P. Gabriel, an Advanced Research Grant for the project "Rare Earth Element reCYCling with Low harmful Emissions : REE-CYCLE", which aimed at finding new processes for the recycling of rare earth.

 

Paramagnetism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Paramagnetism 

 

When liquid oxygen is poured from a beaker into a strong magnet, the oxygen is temporarily contained between the magnetic poles owing to its paramagnetism.

 

 Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, diamagnetic materials are repelled by magnetic fields and form induced magnetic fields in the direction opposite to that of the applied magnetic field. Paramagnetic materials include most chemical elements and some compounds; they have a relative magnetic permeability slightly greater than 1 (i.e., a small positive magnetic susceptibility) and hence are attracted to magnetic fields. The magnetic moment induced by the applied field is linear in the field strength and rather weak. It typically requires a sensitive analytical balance to detect the effect and modern measurements on paramagnetic materials are often conducted with a SQUID magnetometer.

Paramagnetism is due to the presence of unpaired electrons in the material, so most atoms with incompletely filled atomic orbitals are paramagnetic, although exceptions such as copper exist. Due to their spin, unpaired electrons have a magnetic dipole moment and act like tiny magnets. An external magnetic field causes the electrons' spins to align parallel to the field, causing a net attraction. Paramagnetic materials include aluminium, oxygen, titanium, and iron oxide (FeO). Therefore, a simple rule of thumb is used in chemistry to determine whether a particle (atom, ion, or molecule) is paramagnetic or diamagnetic: if all electrons in the particle are paired, then the substance made of this particle is diamagnetic; if it has unpaired electrons, then the substance is paramagnetic.

Unlike ferromagnets, paramagnets do not retain any magnetization in the absence of an externally applied magnetic field because thermal motion randomizes the spin orientations. (Some paramagnetic materials retain spin disorder even at absolute zero, meaning they are paramagnetic in the ground state, i.e. in the absence of thermal motion.) Thus the total magnetization drops to zero when the applied field is removed. Even in the presence of the field there is only a small induced magnetization because only a small fraction of the spins will be oriented by the field. This fraction is proportional to the field strength and this explains the linear dependency. The attraction experienced by ferromagnetic materials is non-linear and much stronger, so that it is easily observed, for instance, in the attraction between a refrigerator magnet and the iron of the refrigerator itself.

Relation to electron spins

Constituent atoms or molecules of paramagnetic materials have permanent magnetic moments (dipoles), even in the absence of an applied field. The permanent moment generally is due to the spin of unpaired electrons in atomic or molecular electron orbitals. In pure paramagnetism, the dipoles do not interact with one another and are randomly oriented in the absence of an external field due to thermal agitation, resulting in zero net magnetic moment. When a magnetic field is applied, the dipoles will tend to align with the applied field, resulting in a net magnetic moment in the direction of the applied field. In the classical description, this alignment can be understood to occur due to a torque being provided on the magnetic moments by an applied field, which tries to align the dipoles parallel to the applied field. However, the true origins of the alignment can only be understood via the quantum-mechanical properties of spin and angular momentum.

If there is sufficient energy exchange between neighbouring dipoles, they will interact, and may spontaneously align or anti-align and form magnetic domains, resulting in ferromagnetism (permanent magnets) or antiferromagnetism, respectively. Paramagnetic behavior can also be observed in ferromagnetic materials that are above their Curie temperature, and in antiferromagnets above their Néel temperature. At these temperatures, the available thermal energy simply overcomes the interaction energy between the spins.

In general, paramagnetic effects are quite small: the magnetic susceptibility is of the order of 10⁻³ to 10⁻⁵ for most paramagnets, but may be as high as 10⁻¹ for synthetic paramagnets such as ferrofluids.

Delocalization

Selected Pauli-paramagnetic metals

Material	Magnetic susceptibility, ${\displaystyle \chi _{v}}$ [10−5] (SI units)
Tungsten	6.8
Caesium	5.1
Aluminium	2.2
Lithium	1.4
Magnesium	1.2
Sodium	0.72

In conductive materials, the electrons are delocalized, that is, they travel through the solid more or less as free electrons. Conductivity can be understood in a band structure picture as arising from the incomplete filling of energy bands. In an ordinary nonmagnetic conductor the conduction band is identical for both spin-up and spin-down electrons. When a magnetic field is applied, the conduction band splits apart into a spin-up and a spin-down band due to the difference in magnetic potential energy for spin-up and spin-down electrons. Since the Fermi level must be identical for both bands, this means that there will be a small surplus of the type of spin in the band that moved downwards. This effect is a weak form of paramagnetism known as Pauli paramagnetism.

The effect always competes with a diamagnetic response of opposite sign due to all the core electrons of the atoms. Stronger forms of magnetism usually require localized rather than itinerant electrons. However, in some cases a band structure can result in which there are two delocalized sub-bands with states of opposite spins that have different energies. If one subband is preferentially filled over the other, one can have itinerant ferromagnetic order. This situation usually only occurs in relatively narrow (d-)bands, which are poorly delocalized.

s and p electrons

Generally, strong delocalization in a solid due to large overlap with neighboring wave functions means that there will be a large Fermi velocity; this means that the number of electrons in a band is less sensitive to shifts in that band's energy, implying a weak magnetism. This is why s- and p-type metals are typically either Pauli-paramagnetic or as in the case of gold even diamagnetic. In the latter case the diamagnetic contribution from the closed shell inner electrons simply wins over the weak paramagnetic term of the almost free electrons.

d and f electrons

Stronger magnetic effects are typically only observed when d or f electrons are involved. Particularly the latter are usually strongly localized. Moreover, the size of the magnetic moment on a lanthanide atom can be quite large as it can carry up to 7 unpaired electrons in the case of gadolinium(III) (hence its use in MRI). The high magnetic moments associated with lanthanides is one reason why superstrong magnets are typically based on elements like neodymium or samarium.

Molecular localization

The above picture is a generalization as it pertains to materials with an extended lattice rather than a molecular structure. Molecular structure can also lead to localization of electrons. Although there are usually energetic reasons why a molecular structure results such that it does not exhibit partly filled orbitals (i.e. unpaired spins), some non-closed shell moieties do occur in nature. Molecular oxygen is a good example. Even in the frozen solid it contains di-radical molecules resulting in paramagnetic behavior. The unpaired spins reside in orbitals derived from oxygen p wave functions, but the overlap is limited to the one neighbor in the O₂ molecules. The distances to other oxygen atoms in the lattice remain too large to lead to delocalization and the magnetic moments remain unpaired.

Theory

The Bohr–Van Leeuwen theorem proves that there cannot be any diamagnetism or paramagnetism in a purely classical system. The paramagnetic response has then two possible quantum origins, either coming from permanent magnetic moments of the ions or from the spatial motion of the conduction electrons inside the material. Both description are given below.

Curie's law

For low levels of magnetization, the magnetization of paramagnets follows what is known as Curie's law, at least approximately. This law indicates that the susceptibility, ${\displaystyle \scriptstyle \chi }$, of paramagnetic materials is inversely proportional to their temperature, i.e. that materials become more magnetic at lower temperatures. The mathematical expression is:

${\displaystyle \mathbf {M} =\chi \mathbf {H} ={\frac {C}{T}}\mathbf {H} }$

where:

${\displaystyle \mathbf {M} }$ is the resulting magnetization, measured in amperes/meter (A/m),

${\displaystyle \chi }$ is the volume magnetic susceptibility (dimensionless),

${\displaystyle H}$ is the auxiliary magnetic field (A/m),

${\displaystyle T}$ is absolute temperature, measured in kelvins (K),

${\displaystyle C}$ is a material-specific Curie constant (K).

Curie's law is valid under the commonly encountered conditions of low magnetization (μ_BH ≲ k_BT), but does not apply in the high-field/low-temperature regime where saturation of magnetization occurs (μ_BH ≳ k_BT) and magnetic dipoles are all aligned with the applied field. When the dipoles are aligned, increasing the external field will not increase the total magnetization since there can be no further alignment.

For a paramagnetic ion with noninteracting magnetic moments with angular momentum J, the Curie constant is related the individual ions' magnetic moments,

${\displaystyle C={\frac {n}{3k_{\mathrm {B} }}}\mu _{\mathrm {eff} }^{2}{\text{ where }}\mu _{\mathrm {eff} }=g_{J}\mu _{\mathrm {B} }{\sqrt {J(J+1)}}.}$

where n is the number of atoms per unit volume. The parameter μ_eff is interpreted as the effective magnetic moment per paramagnetic ion. If one uses a classical treatment with molecular magnetic moments represented as discrete magnetic dipoles, μ, a Curie Law expression of the same form will emerge with μ appearing in place of μ_eff.

When orbital angular momentum contributions to the magnetic moment are small, as occurs for most organic radicals or for octahedral transition metal complexes with d³ or high-spin d⁵ configurations, the effective magnetic moment takes the form ( with g-factor g_e = 2.0023... ≈ 2),

${\displaystyle \mu _{\mathrm {eff} }\simeq 2{\sqrt {S(S+1)}}\mu _{\mathrm {B} }={\sqrt {N_{\rm {u}}(N_{\rm {u}}+2)}}\mu _{\mathrm {B} },}$

where N_u is the number of unpaired electrons. In other transition metal complexes this yields a useful, if somewhat cruder, estimate.

When Curie constant is null, second order effects that couple the ground state with the excited states can also lead to a paramagnetic susceptibility independent of the temperature, known as Van Vleck susceptibility.

Pauli paramagnetism

For some alkali metals and noble metals, conduction electrons are weakly interacting and delocalized in space forming a Fermi gas. For these materials one contribution to the magnetic response comes from the interaction between the electron spins and the magnetic field known as Pauli paramagnetism. For a small magnetic field ${\displaystyle \mathbf {H} }$, the additional energy per electron from the interaction between an electron spin and the magnetic field is given by:

${\displaystyle \Delta E=-\mu _{0}\mathbf {H} \cdot {\boldsymbol {\mu }}_{e}=-\mu _{0}\mathbf {H} \cdot \left(-g_{e}{\frac {\mu _{\mathrm {B} }}{\hbar }}\mathbf {S} \right)=\pm \mu _{0}\mu _{\mathrm {B} }H,}$

where ${\displaystyle \mu _{0}}$ is the vacuum permeability, ${\displaystyle {\boldsymbol {\mu }}_{e}}$ is the electron magnetic moment, ${\displaystyle \mu _{\rm {B}}}$ is the Bohr magneton, ${\displaystyle \hbar }$ is the reduced Planck constant, and the g-factor cancels with the spin ${\displaystyle \mathbf {S} =\pm \hbar /2}$. The ${\displaystyle \pm }$ indicates that the sign is positive (negative) when the electron spin component in the direction of ${\displaystyle \mathbf {H} }$ is parallel (antiparallel) to the magnetic field.

In a metal, the application of an external magnetic field increases the density of electrons with spins antiparallel with the field and lowers the density of the electrons with opposite spin. Note: The arrows in this picture indicate spin direction, not magnetic moment.

For low temperatures with respect to the Fermi temperature ${\displaystyle T_{\rm {F}}}$ (around 10⁴ kelvins for metals), the number density of electrons ${\displaystyle n_{\uparrow }}$ (${\displaystyle n_{\downarrow }}$) pointing parallel (antiparallel) to the magnetic field can be written as:

${\displaystyle n_{\uparrow }\approx {\frac {n_{e}}{2}}-{\frac {\mu _{0}\mu _{\mathrm {B} }}{2}}g(E_{\mathrm {F} })H\quad ;\quad \left(n_{\downarrow }\approx {\frac {n_{e}}{2}}+{\frac {\mu _{0}\mu _{\mathrm {B} }}{2}}g(E_{\mathrm {F} })H\right),}$

with ${\displaystyle n_{e}}$ the total free-electrons density and ${\displaystyle g(E_{\mathrm {F} })}$ the electronic density of states (number of states per energy per volume) at the Fermi energy ${\displaystyle E_{\mathrm {F} }}$.

In this approximation the magnetization is given as the magnetic moment of one electron times the difference in densities:

${\displaystyle M=\mu _{\mathrm {B} }(n_{\downarrow }-n_{\uparrow })=\mu _{0}\mu _{\mathrm {B} }^{2}g(E_{\mathrm {F} })H,}$

which yields a positive paramagnetic susceptibility independent of temperature:

${\displaystyle \chi _{\mathrm {P} }=\mu _{0}\mu _{\mathrm {B} }^{2}g(E_{\mathrm {F} }).}$

The Pauli paramagnetic susceptibility is a macroscopic effect and has to be contrasted with Landau diamagnetic susceptibility which is equal to minus one third of Pauli's and also comes from delocalized electrons. The Pauli susceptibility comes from the spin interaction with the magnetic field while the Landau susceptibility comes from the spatial motion of the electrons and it is independent of the spin. In doped semiconductors the ratio between Landau's and Pauli's susceptibilities changes as the effective mass of the charge carriers ${\displaystyle m^{*}}$ can differ from the electron mass ${\displaystyle m_{e}}$.

The magnetic response calculated for a gas of electrons is not the full picture as the magnetic susceptibility coming from the ions has to be included. Additionally, this formulas may break down for confined systems that differ from the bulk, like quantum dots, or for high fields, as demonstrated in the De Haas-Van Alphen effect.

Pauli paramagnetism is named after the physicist Wolfgang Pauli. Before Pauli's theory, the lack of a strong Curie paramagnetism in metals was an open problem as the leading model could not account for this contribution without the use of quantum statistics.

Examples of paramagnets

Materials that are called "paramagnets" are most often those that exhibit, at least over an appreciable temperature range, magnetic susceptibilities that adhere to the Curie or Curie–Weiss laws. In principle any system that contains atoms, ions, or molecules with unpaired spins can be called a paramagnet, but the interactions between them need to be carefully considered.

Systems with minimal interactions

The narrowest definition would be: a system with unpaired spins that do not interact with each other. In this narrowest sense, the only pure paramagnet is a dilute gas of monatomic hydrogen atoms. Each atom has one non-interacting unpaired electron.

A gas of lithium atoms already possess two paired core electrons that produce a diamagnetic response of opposite sign. Strictly speaking Li is a mixed system therefore, although admittedly the diamagnetic component is weak and often neglected. In the case of heavier elements the diamagnetic contribution becomes more important and in the case of metallic gold it dominates the properties. The element hydrogen is virtually never called 'paramagnetic' because the monatomic gas is stable only at extremely high temperature; H atoms combine to form molecular H₂ and in so doing, the magnetic moments are lost (quenched), because of the spins pair. Hydrogen is therefore diamagnetic and the same holds true for many other elements. Although the electronic configuration of the individual atoms (and ions) of most elements contain unpaired spins, they are not necessarily paramagnetic, because at ambient temperature quenching is very much the rule rather than the exception. The quenching tendency is weakest for f-electrons because f (especially 4f) orbitals are radially contracted and they overlap only weakly with orbitals on adjacent atoms. Consequently, the lanthanide elements with incompletely filled 4f-orbitals are paramagnetic or magnetically ordered.

μ_eff values for typical d³ and d⁵ transition metal complexes.

Material	μeff/μB
[Cr(NH3)6]Br3	3.77
K3[Cr(CN)6]	3.87
K3[MoCl6]	3.79
K4[V(CN)6]	3.78
[Mn(NH3)6]Cl2	5.92
(NH4)2[Mn(SO4)2]·6H2O	5.92
NH4[Fe(SO4)2]·12H2O	5.89

Thus, condensed phase paramagnets are only possible if the interactions of the spins that lead either to quenching or to ordering are kept at bay by structural isolation of the magnetic centers. There are two classes of materials for which this holds:

• Molecular materials with a (isolated) paramagnetic center.
  • Good examples are coordination complexes of d- or f-metals or proteins with such centers, e.g. myoglobin. In such materials the organic part of the molecule acts as an envelope shielding the spins from their neighbors.
  • Small molecules can be stable in radical form, oxygen O₂ is a good example. Such systems are quite rare because they tend to be rather reactive.
• Dilute systems.
  • Dissolving a paramagnetic species in a diamagnetic lattice at small concentrations, e.g. Nd³⁺ in CaCl₂ will separate the neodymium ions at large enough distances that they do not interact. Such systems are of prime importance for what can be considered the most sensitive method to study paramagnetic systems: EPR.

Systems with interactions

Idealized Curie–Weiss behavior; N.B. T_C=θ, but T_N is not θ. Paramagnetic regimes are denoted by solid lines. Close to T_N or T_C the behavior usually deviates from ideal.

As stated above, many materials that contain d- or f-elements do retain unquenched spins. Salts of such elements often show paramagnetic behavior but at low enough temperatures the magnetic moments may order. It is not uncommon to call such materials 'paramagnets', when referring to their paramagnetic behavior above their Curie or Néel-points, particularly if such temperatures are very low or have never been properly measured. Even for iron it is not uncommon to say that iron becomes a paramagnet above its relatively high Curie-point. In that case the Curie-point is seen as a phase transition between a ferromagnet and a 'paramagnet'. The word paramagnet now merely refers to the linear response of the system to an applied field, the temperature dependence of which requires an amended version of Curie's law, known as the Curie–Weiss law:

${\displaystyle \mathbf {M} ={\frac {C}{T-\theta }}\mathbf {H} }$

This amended law includes a term θ that describes the exchange interaction that is present albeit overcome by thermal motion. The sign of θ depends on whether ferro- or antiferromagnetic interactions dominate and it is seldom exactly zero, except in the dilute, isolated cases mentioned above.

Obviously, the paramagnetic Curie–Weiss description above T_N or T_C is a rather different interpretation of the word "paramagnet" as it does not imply the absence of interactions, but rather that the magnetic structure is random in the absence of an external field at these sufficiently high temperatures. Even if θ is close to zero this does not mean that there are no interactions, just that the aligning ferro- and the anti-aligning antiferromagnetic ones cancel. An additional complication is that the interactions are often different in different directions of the crystalline lattice (anisotropy), leading to complicated magnetic structures once ordered.

Randomness of the structure also applies to the many metals that show a net paramagnetic response over a broad temperature range. They do not follow a Curie type law as function of temperature however, often they are more or less temperature independent. This type of behavior is of an itinerant nature and better called Pauli-paramagnetism, but it is not unusual to see, for example, the metal aluminium called a "paramagnet", even though interactions are strong enough to give this element very good electrical conductivity.

Superparamagnets

Some materials show induced magnetic behavior that follows a Curie type law but with exceptionally large values for the Curie constants. These materials are known as superparamagnets. They are characterized by a strong ferromagnetic or ferrimagnetic type of coupling into domains of a limited size that behave independently from one another. The bulk properties of such a system resembles that of a paramagnet, but on a microscopic level they are ordered. The materials do show an ordering temperature above which the behavior reverts to ordinary paramagnetism (with interaction). Ferrofluids are a good example, but the phenomenon can also occur inside solids, e.g., when dilute paramagnetic centers are introduced in a strong itinerant medium of ferromagnetic coupling such as when Fe is substituted in TlCu₂Se₂ or the alloy AuFe. Such systems contain ferromagnetically coupled clusters that freeze out at lower temperatures. They are also called mictomagnets.