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{{Other uses}}
'''Superparamagnetism''' is a form of [[magnetism]], which appears in small [[ferromagnetic]] or [[ferrimagnetic]] [[nanoparticles]]. In sufficiently small nanoparticles, magnetization can randomly flip direction under the influence of temperature. The typical time between two flips is called the [[Néel relaxation theory|Néel relaxation time]]. In the absence of external magnetic field, when the time used to measure the magnetization of the nanoparticles is much longer than the ''Néel relaxation time'', their magnetization appears to be in average zero: they are said to be in the superparamagnetic state. In this state, an external magnetic field is able to magnetize the nanoparticles, similarly to a [[paramagnet]]. However, their [[magnetic susceptibility]] is much larger than the one of paramagnets.


In [[mathematics]], a '''sequence''' is an ordered list of objects (or events). Like a [[Set (mathematics)|set]], it contains [[Element (mathematics)|members]] (also called ''elements'', or ''terms''). The number of ordered elements (possibly infinite) is called the ''length'' of the sequence.  Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. A sequence is a [[Discrete mathematics|discrete]] [[function (mathematics)|function]].
== The Néel relaxation in the absence of magnetic field ==
{{Main|Néel relaxation theory}}
Normally, any ferromagnetic or ferrimagnetic material undergoes a transition to a paramagnetic state above its [[Curie temperature]]. Superparamagnetism is different from this standard transition since it occurs below the Curie temperature of the material.


For example, (M, A, R, Y) is a sequence of letters that differs from (A, R, M, Y), as the ordering matters, and (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''[[finite set|finite]]'', as in this example, or ''[[Infinite set|infinite]]'', such as the sequence of all [[even and odd numbers|even]] [[positive and negative numbers|positive]] [[integer]]s (2, 4, 6,...).  Finite sequences are sometimes known as ''strings'' or ''words'' and infinite sequences as ''streams''.  The empty sequence ( ) is included in most notions of sequence, but may be excluded depending on the context.
Superparamagnetism occurs in nanoparticles which are [[single domain (magnetic)|single-domain]], i.e. composed of a single [[magnetic domain]]. This is possible when their diameter is below 3–50 nm, depending on the materials. In this condition, it is considered that the magnetization of the nanoparticles is a single giant magnetic moment, sum of all the individual magnetic moments carried by the atoms of the nanoparticle. Those in the field of superparamagnetism call this “macro-spin approximation”.
[[Image:Cauchy sequence illustration2.svg|right|thumb|350px|An infinite sequence of real numbers (in blue). This sequence is neither increasing, nor decreasing, nor convergent, nor [[Cauchy sequence|Cauchy]]. It is, however, bounded.]]


== Examples and notation ==
Because of the nanoparticle’s [[magnetic anisotropy]], the magnetic moment has usually only two stable orientations antiparallel to each other, separated by an [[energy barrier]]. The stable orientations define the nanoparticle’s so called “easy axis”. At finite temperature, there is a finite probability for the magnetization to flip and reverse its direction. The mean time between two flips is called the Néel relaxation time <math>\tau_N</math> and is given by the following Néel-Arrhenius equation:<ref name=Neel49>{{cite article|first=L. |last=Néel|title=Théorie du traînage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites|journal=Ann. Géophys|volume=5|pages=99–136|year=1949}} (in French; an English translation is available in {{cite article|title=Selected Works of Louis Néel|editor-first=N.|editor-last=Kurti|publisher=Gordon and Breach|year=1988|isbn=2-88124-300-2|pages=407–427}}).</ref>
There are various and quite different notions of sequences in mathematics, some of which (''e.g.'', [[exact sequence]]) are not covered by the notations introduced below.


In addition to identifying the elements of a sequence by their position, such as "the 3rd element", elements may be given names for convenient referencing.  For example a sequence might be written as (''a''<sub>1</sub>, ''a''<sub>2</sub>, ''a''<sub>3</sub>, … ), or (''b''<sub>0</sub>, ''b''<sub>1</sub>, ''b''<sub>2</sub>, … ), or (''c''<sub>0</sub>, ''c''<sub>2</sub>, ''c''<sub>4</sub>, … ), depending on what is useful in the application.
:<math>\tau_N = \tau_0 \exp \left(\frac{K V}{k_B T}\right)</math>,


===Finite and infinite===
where:
A more formal definition of a '''finite sequence''' with terms in a set ''S'' is a [[function (mathematics)|function]] from  {1, 2, ..., ''n''} to ''S'' for some ''n'' > 0. An '''infinite sequence''' in ''S'' is a function from {1, 2, ... } to ''S''. For example, the sequence of prime numbers (2,3,5,7,11, … ) is the function 1→'''2''', 2→'''3''', 3→'''5''', 4→'''7''', 5→'''11''', … .
*<math>\tau_N</math> is thus the average length of time that it takes for the nanoparticle’s magnetization to randomly flip as a result of [[thermal fluctuations]].
*<math>\tau_0</math> is a length of time, characteristic of the material, called the ''attempt time'' or ''attempt period'' (its reciprocal is called the ''attempt frequency''); its typical value is 10<sup>−9</sup>–10<sup>−10</sup> second.
*''K'' is the nanoparticle’s magnetic anisotropy energy density and ''V'' its volume. ''KV'' is therefore the [[energy barrier]] associated with the magnetization moving from its initial easy axis direction, through a “hard plane”, to the other easy axis direction.
*k<sub>B</sub> is the [[Boltzmann constant]].
*''T'' is the temperature.


A sequence of a finite length ''n'' is also called an [[n-tuple|''n''-tuple]]. Finite sequences include the '''empty sequence''' ( ) that has no elements.
This length of time can be anywhere from a few nanoseconds to years or much longer. In particular, it can be seen that the Néel relaxation time is an exponential function of the grain volume, which explains why the flipping probability becomes rapidly negligible for bulk materials or large nanoparticles.


A function from ''all'' integers into a set is sometimes called a '''bi-infinite sequence''' or '''two-way infinite sequence'''.  An example is the bi-infinite sequence of all even integers ( … , -4, -2, 0, 2, 4, 6, 8… ).
=== Blocking temperature ===


===Multiplicative===
Let us imagine that the magnetization of a single superparamagnetic nanoparticle is measured and let us define <math>\tau_m</math> as the measurement time. If <math>\tau_m \gg \tau_N</math>, the nanoparticle magnetization will flip several times during the measurement, then the measured magnetization will average to zero. If <math>\tau_m \ll \tau_N</math>, the magnetization will not flip during the measurement, so the measured magnetization will be what the instantaneous magnetization was at the beginning of the measurement. In the former case, the nanoparticle will appear to be in the superparamagnetic state whereas in the latter case it will appear to be “blocked” in its initial state. '''The state of the nanoparticle (superparamagnetic or blocked) depends on the measurement time.''' A transition between superparamagnetism and blocked state occurs when <math>\tau_m = \tau_N</math>. In several experiments, the measurement time is kept constant but the temperature is varied, so the transition between superparamagnetism and blocked state is seen as a function of the temperature. The temperature for which <math>\tau_m = \tau_N</math> is called the '''blocking temperature''':
Let ''A'' = (a sequence defined by a function ''f'':{1, 2, 3, ...} → {1, 2, 3, ...}, such that ''a''<sub> ''i''</sub> = ''f''(''i'').


The sequence is '''multiplicative''' if ''f''(''xy'') = ''f''(''x'')''f''(''y'') for all ''x'',''y''  such that ''x'' and ''y'' are [[coprime]].<ref>{{cite book|title=Lectures on generating functions|last=Lando|first=Sergei K.|publisher=AMS|ISBN=0-8218-3481-9|chapter=7.4 Multiplicative sequences}}</ref>
:<math>T_B = \frac{K V}{k_B \ln \left(\frac{\tau_m}{\tau_0}\right)}</math>


==Types and properties==
For typical laboratory measurements, the value of the logarithm in the previous equation is in the order of 20–25.
A [[subsequence]] of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements.


If the terms of the sequence are a subset of an [[partially ordered set|ordered set]], then a ''monotonically increasing'' sequence is one for which each term is greater than or equal to the term before it; if each term is [[strict]]ly greater than the one preceding it, the sequence is called ''strictly monotonically increasing''. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the [[monotonic function|monotonicity]] property is called monotonic or ''monotone''. This is a special case of the more general notion of [[monotonic function]].
== Effect of a magnetic field ==


The terms ''nondecreasing'' and ''nonincreasing'' are used in order to avoid any possible confusion with strictly increasing and strictly decreasing, respectively.
[[Image:Langevin function.png|300px|thumb|right|Langevin function (red line), compared with <math>\tanh(x/3)</math> (blue line).]]


If the terms of a sequence are [[integer]]s, then the sequence is an  [[integer sequence]]. If the terms of a sequence are [[polynomial]]s, then the sequence is a [[polynomial sequence]].
When an external magnetic field is applied to an assembly of superparamagnetic nanoparticles, their magnetic moments tend to align along the applied field, leading to a net magnetization. The magnetization curve of the assembly, i.e. the magnetization as a function of the applied field, is a reversible S-shaped [[increasing function]]. This function is quite complicated but for some simple cases:
# If all the particles are identical (same energy barrier and same magnetic moment), their easy axes are all oriented parallel to the applied field and the temperature is low enough (''T<sub>B</sub> &lt; T ≲ KV/(10 k<sub>B</sub>)''), then the magnetization of the assembly is<br/><math>M(H) \approx n \mu \tanh\left(\frac{\mu_0 H \mu}{k_B T}\right)</math>.
# If all the particles are identical and the temperature is high enough (''T ≳ KV/k<sub>B</sub>''), then, irrespective of the orientations of the easy axes:<br/><math>M(H) \approx n \mu L\left(\frac{\mu_0 H \mu}{k_B T}\right)</math>
In the above equations:
* ''n'' in the density of nanoparticles in the sample
* <math>\mu_0</math> is the [[magnetic permeability]] of vacuum
* <math>\mu</math> is the magnetic moment of a nanoparticle
* <math>L(x) = 1/\tanh(x) - 1/x</math> is the [[Langevin function]]


If ''S'' is endowed with a [[topology]], then it becomes possible to consider ''convergence'' of an infinite sequence in ''S''. Such considerations involve the concept of the [[limit of a sequence]].
The initial slope of the <math>M(H)</math> function is the magnetic susceptibility of the sample <math>\chi</math>:
:<math>\chi = \frac{n \mu_0 \mu^2}{k_BT}</math> in the first case
:<math>\chi = \frac{n \mu_0 \mu^2}{3k_BT}</math> in the second case.
The later susceptibility is also valid for all temperatures <math>T > T_B</math> if the easy axes of the nanoparticles are randomly oriented.


If A is a set, the [[free monoid]] over A (denoted A<sup>*</sup>) is a [[monoid]] containing all the finite sequences (or strings) of zero or more elements drawn from A, with the binary operation of concatenation.  The [[free semigroup]] A<sup>+</sup> is the subsemigroup of A<sup>*</sup> containing all elements except the empty sequence.
It can be seen from these equations that large nanoparticles have a larger ''µ'' and so a larger susceptibility. This explains why superparamagnetic nanoparticles have a much larger susceptibility than standard paramagnets: they behave exactly as a paramagnet with a huge magnetic moment.


==Analysis==
=== Time dependence of the magnetization ===
In [[mathematical analysis|analysis]], when talking about sequences, one will generally consider sequences of the form
:<math>(x_1, x_2, x_3, \dots)\text{ or }(x_0, x_1, x_2, \dots)\,</math>
which is to say, infinite sequences of elements indexed by [[natural number]]s.


It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by  ''x<sub>n</sub>'' = 1/[[logarithm|log]](''n'') would be defined only for ''n'' ≥ 2.
There is no time-dependence of the magnetization when the nanoparticles are either completely blocked (<math>T \ll T_B</math>) or completely superparamagnetic (<math>T \gg T_B</math>). There is, however, a narrow window around <math>T_B</math> where the measurement time and the relaxation time have comparable magnitude. In this case, a frequency-dependence of the susceptibility can be observed. For a randomly oriented sample, the complex susceptibility<ref>{{cite journal |url=http://link.aps.org/doi/10.1103/PhysRevB.9.3891 |title=Superparamagnetism and relaxation effects in granular Ni-SiO<sub>2</sub> and Ni-Al<sub>2</sub>O<sub>3</sub> films  |first1=J. I. |last1=Gittleman |first2=B. |last2=Abeles |first3=S. |last3=Bozowski |journal=[[Physical Review B]] |volume=9 |pages=3891&ndash;3897 |year=1974 |doi=10.1103/PhysRevB.9.3891|bibcode = 1974PhRvB...9.3891G }}</ref> is:
When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices [[large enough]], that is, greater than some given ''N''.


The most elementary type of sequences are numerical ones, that is, sequences of real or [[complex number]]s.
:<math>\chi(\omega) = \frac{\chi_{sp} + i \omega \tau \chi_b}{1 + i \omega \tau}</math>
This type can be generalized to sequences of elements of some [[vector space]]. In analysis, the vector spaces considered are often [[function space]]s. Even more generally, one can study sequences with elements in some [[topological space]].


==Series==
where
{{main|Series (mathematics)}}
* <math>\tfrac{\omega}{2\pi}</math> is the frequency of the applied field
The sum of terms of a sequence is a [[series (mathematics)|series]]. More precisely, if (''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ...) is a sequence, one may consider the sequence of [[partial sum]]s (''S''<sub>1</sub>, ''S''<sub>2</sub>, ''S''<sub>3</sub>, ...), with
* <math>\chi_{sp} = \tfrac{n \mu_0 \mu^2}{3k_BT}</math> is the susceptibility in the superparamagnetic state
* <math>\chi_b = \tfrac{n \mu_0 \mu^2}{3KV}</math> is the susceptibility in the blocked state
* <math>\tau = \tfrac{\tau_N}{2}</math> is the relaxation time of the assembly


:<math>S_n=x_1+x_2+\cdots + x_n=\sum\limits_{i=1}^{n}x_i.</math>
From this frequency-dependent susceptibility, the time-dependence of the magnetization for low-fields can be derived:


Formally, this pair of sequences comprises the ''series'' with the terms ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ..., which is denoted as
:<math>\tau \frac{\mathrm{d}M}{\mathrm{d}t} + M = \tau \chi_b \frac{\mathrm{d}H}{\mathrm{d}t} + \chi_{sp} H</math>


:<math>\sum\limits_{i=1}^\infty x_i.</math>
==Measurements==
A superparamagnetic system can be measured with [[magnetic susceptibility|AC susceptibility]] measurements, where an applied magnetic field varies in time, and the magnetic response of the system is measured. A superparamagnetic system will show a characteristic frequency dependence: When the frequency is much higher than 1/τ<sub>N</sub>, there will be a different magnetic response than when the frequency is much lower than 1/τ<sub>N</sub>, since in the latter case, but not the former, the ferromagnetic clusters will have time to respond to the field by flipping their magnetization.<ref>{{cite web |url=http://www.qdusa.com/resources/pdf/1078-201.pdf |format=pdf |last1=Martien |first1=Dinesh |title=Introduction to: AC susceptibility |publisher=Quantum Design |accessdate=September 2011}}</ref> The precise dependence can be calculated from the Néel-Arrhenius equation, assuming that the neighboring clusters behave independently of one another (if clusters interact, their behavior becomes more complicated). It is also possible to perform magneto-optical AC susceptibility measurements with magneto-optically active superparamagnetic materials such as iron oxide nanoparticles in the visible wavelength range.<ref>{{cite journal |doi=10.1063/1.4801837|last1=Vandendriessche |first1=Stefaan et al. |title=Magneto-optical harmonic susceptometry of superparamagnetic materials|journal=Applied Physics Letters|volume=102|number=16|pages=161903-5}}</ref>


If the sequence of partial sums is convergent, one also uses the infinite sum notation for its limit. For more details, see [[series (mathematics)|series]].
==Effect on hard drives==
Superparamagnetism sets a limit on the storage density of [[hard disk drive]]s due to the minimum size of particles that can be used. This limit is known as the '''superparamagnetic limit'''.  


==Infinite sequences in theoretical computer science==
* Older hard disk technology uses [[longitudinal recording]]. It has an estimated limit of 100 to 200 Gbit/in²<ref>{{cite conference |last1=Kryder |first1=M. H. |title=Magnetic recording beyond the superparamagnetic limit |work=Magnetics Conference, 2000. INTERMAG 2000 Digest of Technical Papers. 2000 IEEE International |page=575 |doi=10.1109/INTMAG.2000.872350}}</ref>
Infinite sequences of [[numerical digit|digits]] (or [[character (computing)|characters]]) drawn from a [[finite set|finite]] [[alphabet (computer science)|alphabet]] are of particular interest in [[theoretical computer science]]. They are often referred to simply as ''sequences'' or ''streams'', as opposed to finite ''[[String (computer science)#Formal theory|strings]]''. Infinite binary sequences, for instance, are infinite sequences of [[bit]]s (characters drawn from the alphabet {0,1}). The set ''C'' = {0, 1}<sup>∞</sup> of all infinite, binary sequences is sometimes called the [[Cantor space]].
* Current hard disk technology uses [[perpendicular recording]]. {{As of|2010|8}} drives with densities of 667Gb/in<sup>2</sup> are available commercially. Perpendicular recording is predicted to allow information densities of up to around 1 Tbit/in² (1024 Gbit/in²).<ref>{{cite press release |title=Hitachi achieves nanotechnology milestone for quadrupling terabyte hard drive |publisher=[[Hitachi]] |date=October 15, 2007 |url= http://www.hitachi.com/New/cnews/071015a.html |format= |language= |trans_title= |accessdate=September 2011 |archiveurl= |archivedate= |quote= |ref= }}</ref>
* Future hard disk technologies currently in development include: [[heat-assisted magnetic recording]] (HAMR), which use materials that are stable at much smaller sizes. They require heating before the magnetic orientation of a bit can be changed; and [[bit-patterned recording]] (BPR).<ref>{{cite web |url=http://www.pcmag.com/article2/0,2817,2368023,00.asp |title=Will Toshiba's Bit-Patterned Drives Change the HDD Landscape? |last1=Murray |first1=Matthew |publisher=[[PC Magazine]] |date=2010-08-19 |accessdate=2010-08-21}}</ref>


An infinite binary sequence can represent a [[formal language]] (a set of strings) by setting the ''n''&thinsp;th bit of the sequence to 1 if and only if the ''n''&thinsp;th string (in [[shortlex order]]) is in the language. Therefore, the study of [[complexity class]]es, which are sets of languages, may be regarded as studying sets of infinite sequences.
==Applications of superparamagnetism==
===General Applications===
*[[Ferrofluid]]: tunable viscosity


An infinite sequence drawn from the alphabet {0, 1, ..., b&minus;1} may also represent a real number expressed in the base-''b'' [[positional number system]]. This equivalence is often used to bring the techniques of [[real analysis]] to bear on complexity classes.
===Biomedical applications===
 
* Imaging: [[Contrast agents]] in [[Magnetic Resonance Imaging]] (MRI)
== Vectors ==
* Magnetic separation: cell-, DNA-, protein- separation, RNA fishing
Sequences over a field may also be viewed as [[Vector (geometric)|vectors]] in a [[vector space]]. Specifically, the set of ''F''-valued sequences (where ''F'' is a [[field (mathematics)|field]]) is a [[function space]] (in fact, a [[product space]]) of ''F''-valued functions over the set of natural numbers.
* Treatments: [[targeted drug delivery]], [[magnetic hyperthermia]], [[magnetofection]]
 
In particular, the term ''[[sequence space]]'' usually refers to a [[linear subspace]] of  the set of all possible infinite sequences with elements in <math>\mathbb{C}</math>.
 
== {{anchor|Doubly infinite|Doubly infinite sequences|Doubly-infinite sequences}} Doubly infinite sequences ==
Normally, the term ''infinite sequence'' refers to a sequence which is infinite in one direction, and finite in the other—the sequence has a first element, but no final element (a ''singly infinite sequence''). A ''doubly infinite sequence'' is infinite in both directions—it has neither a first nor a final element. Singly infinite sequences are functions from the natural numbers ('''N''') to some set, whereas doubly infinite sequences are functions from the integers ('''Z''') to some set.
 
One can interpret singly infinite sequences as elements of the [[group ring|semigroup ring]] of the [[natural numbers]] <math>R[\N]</math>, and doubly infinite sequences as elements of the [[group ring]] of the [[integer]]s <math>R[\Z]</math>. This perspective is used in the [[Cauchy product]] of sequences.
 
==Ordinal-indexed sequence==
An [[Order_topology#Ordinal-indexed_sequences|ordinal-indexed sequence]] is a generalization of a sequence. If α is a [[limit ordinal]] and X is a set, an α-indexed sequence of elements of X is a function from α to X. In this terminology an ω-indexed sequence is an ordinary sequence.
 
== Sequences and automata ==
[[Automata theory|Automata]] or [[finite state machine]]s can typically be thought of as directed graphs, with edges labeled using some specific alphabet, Σ. Most familiar types of automata transition from state to state by reading input letters from Σ, following edges with matching labels; the ordered input for such an automaton forms a sequence called a ''word'' (or input word). The sequence of states encountered by the automaton when processing a word is called a ''run''. A nondeterministic automaton may have unlabeled or duplicate out-edges for any state, giving more than one successor for some input letter. This is typically thought of as producing multiple possible runs for a given word, each being a sequence of single states, rather than producing a single run that is a sequence of sets of states; however, 'run' is occasionally used to mean the latter.
 
===Types===
*[[±1-sequence]]
*[[Arithmetic progression]]
*[[Cauchy sequence]]
*[[Farey sequence]]
*[[Fibonacci number|Fibonacci sequence]]
*[[Geometric progression]]
*[[Look-and-say sequence]]
*[[Thue–Morse sequence]]
 
===Related concepts===
*[[List (computing)]]
*[[Order_topology#Ordinal-indexed_sequences|Ordinal-indexed sequence]]
*[[Recursion (computer science)]]
*[[Tuple]]
*[[Set theory]]
 
===Operations===
*[[Cauchy product]]
*[[Limit of a sequence]]


==See also==
==See also==
*[[Set (mathematics)]]
*[[Single-molecule magnet]]
*[[Net (topology)]] (a generalization of sequences)
*[[Iron oxide nanoparticles]]
*[[On-Line Encyclopedia of Integer Sequences]]
*[[Permutation]]
*[[Recurrence relation]]
*[[Sequence space]]


==References==
==References==
{{reflist}}
{{reflist}}
{{Refbegin}}
*{{cite journal |first1=L. |last1=Néel |url=http://hal.archives-ouvertes.fr/docs/00/23/42/17/PDF/ajp-jphysrad_1950_11_2_49_0.pdf |title=Théorie du traînage magnétique des ferromagnétiques en grains fins avec applications aux terres cuites |journal=Ann. Géophys. |volume=5 |pages=99&ndash;136 |year=1949 |language=French}} An English translation is available in {{cite book|title=Selected Works of Louis Néel |editor1-first=N. |editor1-last=Kurti |publisher=Gordon and Breach |location=New York |year=1988 |isbn=2-88124-300-2 |pages=407&ndash;427}}
*{{cite journal |first1=D. |last1=Weller |first2=A. |last2=Moser |url=http://dx.doi.org/10.1109/20.809134 |title=Thermal Effect Limits in Ultrahigh Density Magnetic Recording |journal= [[IEEE Transactions on Magnetics]]  |volume=35 |pages=4423&ndash;4439 |year=1999 |doi=10.1109/20.809134|bibcode = 1999ITM....35.4423W }}
{{Refend}}


==External links==
==External links==
*[http://oeis.org/ The On-Line Encyclopedia of Integer Sequences]
* [http://www.mff.cuni.cz/veda/konference/wds/contents/pdf05/WDS05_090_f3_Vejpravova.pdf Superparamagnetism of Co-Ferrite Nanoparticles]
*[http://www.cs.uwaterloo.ca/journals/JIS/index.html Journal of Integer Sequences] (free)
* [http://lmis1.epfl.ch/webdav/site/lmis1/shared/Files/Lectures/Nanotechnology%20for%20engineers/Archives/2004_05/Superparamagnetism.pdf Powerpoint presentation on Superparamagnetism in pdf]
*{{planetmath reference|id=397|title=Sequence}}
{{magnetic states}}
 
[[Category:Elementary mathematics]]
[[Category:Sequences and series|*]]


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Revision as of 03:59, 9 August 2014

Superparamagnetism is a form of magnetism, which appears in small ferromagnetic or ferrimagnetic nanoparticles. In sufficiently small nanoparticles, magnetization can randomly flip direction under the influence of temperature. The typical time between two flips is called the Néel relaxation time. In the absence of external magnetic field, when the time used to measure the magnetization of the nanoparticles is much longer than the Néel relaxation time, their magnetization appears to be in average zero: they are said to be in the superparamagnetic state. In this state, an external magnetic field is able to magnetize the nanoparticles, similarly to a paramagnet. However, their magnetic susceptibility is much larger than the one of paramagnets.

The Néel relaxation in the absence of magnetic field

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Normally, any ferromagnetic or ferrimagnetic material undergoes a transition to a paramagnetic state above its Curie temperature. Superparamagnetism is different from this standard transition since it occurs below the Curie temperature of the material.

Superparamagnetism occurs in nanoparticles which are single-domain, i.e. composed of a single magnetic domain. This is possible when their diameter is below 3–50 nm, depending on the materials. In this condition, it is considered that the magnetization of the nanoparticles is a single giant magnetic moment, sum of all the individual magnetic moments carried by the atoms of the nanoparticle. Those in the field of superparamagnetism call this “macro-spin approximation”.

Because of the nanoparticle’s magnetic anisotropy, the magnetic moment has usually only two stable orientations antiparallel to each other, separated by an energy barrier. The stable orientations define the nanoparticle’s so called “easy axis”. At finite temperature, there is a finite probability for the magnetization to flip and reverse its direction. The mean time between two flips is called the Néel relaxation time and is given by the following Néel-Arrhenius equation:[1]

,

where:

  • is thus the average length of time that it takes for the nanoparticle’s magnetization to randomly flip as a result of thermal fluctuations.
  • is a length of time, characteristic of the material, called the attempt time or attempt period (its reciprocal is called the attempt frequency); its typical value is 10−9–10−10 second.
  • K is the nanoparticle’s magnetic anisotropy energy density and V its volume. KV is therefore the energy barrier associated with the magnetization moving from its initial easy axis direction, through a “hard plane”, to the other easy axis direction.
  • kB is the Boltzmann constant.
  • T is the temperature.

This length of time can be anywhere from a few nanoseconds to years or much longer. In particular, it can be seen that the Néel relaxation time is an exponential function of the grain volume, which explains why the flipping probability becomes rapidly negligible for bulk materials or large nanoparticles.

Blocking temperature

Let us imagine that the magnetization of a single superparamagnetic nanoparticle is measured and let us define as the measurement time. If , the nanoparticle magnetization will flip several times during the measurement, then the measured magnetization will average to zero. If , the magnetization will not flip during the measurement, so the measured magnetization will be what the instantaneous magnetization was at the beginning of the measurement. In the former case, the nanoparticle will appear to be in the superparamagnetic state whereas in the latter case it will appear to be “blocked” in its initial state. The state of the nanoparticle (superparamagnetic or blocked) depends on the measurement time. A transition between superparamagnetism and blocked state occurs when . In several experiments, the measurement time is kept constant but the temperature is varied, so the transition between superparamagnetism and blocked state is seen as a function of the temperature. The temperature for which is called the blocking temperature:

For typical laboratory measurements, the value of the logarithm in the previous equation is in the order of 20–25.

Effect of a magnetic field

Langevin function (red line), compared with (blue line).

When an external magnetic field is applied to an assembly of superparamagnetic nanoparticles, their magnetic moments tend to align along the applied field, leading to a net magnetization. The magnetization curve of the assembly, i.e. the magnetization as a function of the applied field, is a reversible S-shaped increasing function. This function is quite complicated but for some simple cases:

  1. If all the particles are identical (same energy barrier and same magnetic moment), their easy axes are all oriented parallel to the applied field and the temperature is low enough (TB < T ≲ KV/(10 kB)), then the magnetization of the assembly is
    .
  2. If all the particles are identical and the temperature is high enough (T ≳ KV/kB), then, irrespective of the orientations of the easy axes:

In the above equations:

The initial slope of the function is the magnetic susceptibility of the sample :

in the first case
in the second case.

The later susceptibility is also valid for all temperatures if the easy axes of the nanoparticles are randomly oriented.

It can be seen from these equations that large nanoparticles have a larger µ and so a larger susceptibility. This explains why superparamagnetic nanoparticles have a much larger susceptibility than standard paramagnets: they behave exactly as a paramagnet with a huge magnetic moment.

Time dependence of the magnetization

There is no time-dependence of the magnetization when the nanoparticles are either completely blocked () or completely superparamagnetic (). There is, however, a narrow window around where the measurement time and the relaxation time have comparable magnitude. In this case, a frequency-dependence of the susceptibility can be observed. For a randomly oriented sample, the complex susceptibility[2] is:

where

From this frequency-dependent susceptibility, the time-dependence of the magnetization for low-fields can be derived:

Measurements

A superparamagnetic system can be measured with AC susceptibility measurements, where an applied magnetic field varies in time, and the magnetic response of the system is measured. A superparamagnetic system will show a characteristic frequency dependence: When the frequency is much higher than 1/τN, there will be a different magnetic response than when the frequency is much lower than 1/τN, since in the latter case, but not the former, the ferromagnetic clusters will have time to respond to the field by flipping their magnetization.[3] The precise dependence can be calculated from the Néel-Arrhenius equation, assuming that the neighboring clusters behave independently of one another (if clusters interact, their behavior becomes more complicated). It is also possible to perform magneto-optical AC susceptibility measurements with magneto-optically active superparamagnetic materials such as iron oxide nanoparticles in the visible wavelength range.[4]

Effect on hard drives

Superparamagnetism sets a limit on the storage density of hard disk drives due to the minimum size of particles that can be used. This limit is known as the superparamagnetic limit.

  • Older hard disk technology uses longitudinal recording. It has an estimated limit of 100 to 200 Gbit/in²[5]
  • Current hard disk technology uses perpendicular recording. Template:As of drives with densities of 667Gb/in2 are available commercially. Perpendicular recording is predicted to allow information densities of up to around 1 Tbit/in² (1024 Gbit/in²).[6]
  • Future hard disk technologies currently in development include: heat-assisted magnetic recording (HAMR), which use materials that are stable at much smaller sizes. They require heating before the magnetic orientation of a bit can be changed; and bit-patterned recording (BPR).[7]

Applications of superparamagnetism

General Applications

Biomedical applications

See also

References

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  1. ↑ Template:Cite article (in French; an English translation is available in Template:Cite article).
  2. ↑ 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
  3. ↑ Template:Cite web
  4. ↑ 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
  5. ↑ 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

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  6. ↑ Template:Cite press release
  7. ↑ Template:Cite web
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