Lecture XV. Jorge Quintanilla,

Magnetism and Superconductivity (PH752)

In the past couple of lectures we have discussed a different type of ordered magnetic states: anti-ferromagnetism.However, given that our description of such magnetic order has been approximate (based on mean-field the-ory) one wonders: how do we know that such order exists?

The case of ferromagnetism is pretty clear, as it leads to a spontaneous magnetization that can be detectedusing a magnetometer (see Lecture 1). However anti-ferromagnetism, having no net magnetization, cannotbe detected in this way.

In this lecture we will discuss two of the main techniques used to detect the microscopic arrangement ofmagnetic moments in a material: neutron scattering and muon spin rotation.

1 Magnetic neutron scattering

Neutrons have no charge, therefore they are very penetrating and can be used to probe the bulk of a magneticmaterial without being absorbed or reflected near the surface. At the same time, they do have a spin (withangular momentum quantum number s = 1/2) so they interact magnetically with the magnetic dipolemoments inside the sample. This is what makes them valuable in the study of magnetism.

They are not just useful for magnetism. In addition, they also bounce off the nuclei in the sample due to thestrong-force interaction with the other nuclei in them, so they can also be used to study the crystal structure.

Neutrons can be produced in two ways: through nuclear fission (in a fisision reactor) and through spalla-tion (using a proton accelerator).

In nuclear fission a heavy nucleus is hit by by a neutron from an earlier fission event and splits into twolighter nuclei, ejecting some excess neutrons in the process. Some of these neutrons go on to split othernuclei while the rest escape the reactor core and can be used to generate energy (by heating some water thatevaporates and moves a turbine) or, even better (!), to do neutron scattering!

In a spallation neutron source, a proton beam from a synchrotron or cyclotron is fired at a target made ofsome heavy element (e.g. Tungsten). The nuclei get hot and evaporate some of their neutrons.

In either case, the neutrons have to go through a moderator: a chunk of material that is kept at somesuitable temperature where the neutrons can thermalise. When they leave the moderator, the neutrons havea known, thermal (or quasi-thermal) energy distribution, corresponding to the moderator’s temperature, T .They have become thermal neutrons. The typical enegies of thermal neutrons are given by

kBT ∼p2

2mn, (1)

where p and mn are the linear momentum and mass of the neutron, respectively. The momentum is relatedto the neutron wavelength λ by means of the de Broglie wave-particle duality equation,

p =h

λ. (2)

Combining these two equations yields the thermal de Broglie wavelength of the neutron,

λ ∼h

√2mnkBT

. (3)

Substituting the mass of the neutron,

mn ≈ mp ≈ 1.7 10−27 Kg, (4)

we get

λ ∼1 nm�T/K

. (5)

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Thus for moderator temperatures T ∼ 100K we get neutron wavelengths λ ∼ 1 nm/10 = 1 Angstrom, whichof the order of the atomic spacings of solids. At these wavelengths interference effects result from featuresthat vary within the sample on the atomic scale. This is what makes neutron scattering useful for the studyof magnetism.

The choice of moderator will depend on what temperatures (i.e. what wavelengths) we wish to achieve,e.g. to study biological systems or long-wavelength helical magnetism (see Lecture XVI) we may wish forsomewhat longer wavelengths than to study, say, Neel order. Some common substances to use are water,solid hydrogen and solid methane.

Once the neutrons have been thermalised, they are guided towards our sample of magnetic material. Whilegoing through the sample, the neutrons will get deflected a certain angle 2θ from their incoming trajectory.

Each neutron is detected after being scattered and its exact energy is recorded. Recording the energy is ofcourse the same as recording the wavelength, since, using again two of the relations employed above,

E =p2

2mn(6)

=h2

2mnλ2. (7)

The energy of the neutron is determined as follows:

1. In a pulsed neutron source, the neutrons are produced in bunches, each lasting a very short time (ofthe order of a nanosecond). Since we know when each pulse of neutrons hits the sample, we can workout the energy of a neutron from the additional time t it takes it to hit the detector (time-of-flight).If the distance between the detector and the sample is l, then the neutron’s velocity is v = l/t and thusits momentum is p = mnv = mn (l/t) . The energy is of course E = p2/2mn = mnl

2/2t2.

Most spallation neutron sources are based on a proton synchrotron, where protons are acceleratedin compact bunches, and are thus pulsed sources.

2. In a continuous source, there is a constant flow of neutrons so the time-of-flight technique cannot beused. This is the case of most reactor-based neutron sources. In that case there are two potions (sic!):

(a) We can place a chopper between the source and the sample. The chopper is a rapidly-rotatingcylinder with two apertures which effectively chops the neutron beam into pulses.

3. Alternatively, and more commonly, we can use a monochromator, which is a crystal which scattersneutrons of different energies in different directions (using Bragg diffraction, which we explain below).By orienting the monochromator crystal in different directions we can select neutrons of differentwavelengths to reach our sample:

Here are the two arrangements:

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While reactors generate much higher fluxes of neutrons than spallation sources, in a pulsed source data fromall neutrons can be collected, while in a continuous one we have to throw away most of the neutrons. Theresult is that in the end both technologies have their advantages and disadvantages.

The scattering angle θ depends on the crystalline and magnetic structure of the system. It is very useful toregard a regular crystal lattice as a collection of sets of crystallographic planes. Each set is compoased ofmany parallel, equi-spaced planes containing an identical array of atoms. For a given crystal lattice, there aremany such sets of crystallographics planes, each of them with a given periodicity (the spacing between the

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planes) and direction (their orientation). For example, here we draw two of the many sets of crystallographicplanes that make up a square lattice:

When neutrons scatter off a crystal, trajectories corresponding to different scattering angles interfere. Thiswill lead to more scatering intensity for some values of θ than others. In particular, there will be intensescatering if the angle is such that trajectories correpsonding to reflections on parallel crystallographics planesinteract construcitvely. Fro this to happen two conditions are necessary:

1. The angle θ must correspond to specular reflection on a set of crystallographic planes existing in thecrystal.

2. The lengths of the “optical” paths corresponding to reflection on subsequent planes must differ by aninteger number of wavelengths, so as to ensure constructive interference.

The latter condition leads to the Bragg diffraction law:

2d sin θ = nλ, (8)

where d is the spacing between the relevant set of crystallographic planes and n = 0, 1, 2, 3, . . .

Throughout this discussion we are assuming elastic scattering, i.e. that the neutron does not exchangeenergy with the system. Most neutron scattering events are like this, so it is in general a good approximation

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to assume that this is the case always. (In inelastic experiments, on the other hand, neutrons that haveactually exchanged some energy are deliberately selected. We will discuss this later.) Under this assumptionthe magnitudes of the neutron momentum before and after it went through the sample are identical. Interms of the neutron’s wave vector k, which is related to the momentum via

p = �k, (9)

this translates to|k| =

��k�

�� . (10)

For elastic scattering, the scattering vectorQ = k� − k, (11)

that is, the change in the neutron’s wave vector on going through the sample, depends exclusively on therelative direction of k and k�, i.e. on the scattering angle θ. Its magnitude is given by

Q/2

k= sin θ ⇒ Q = 2k sin θ , (12)

as can be seen from the following geometric construction:

Evidently the scattering vector Q corresponding to a Bragg relfection will be perpendicular to the corre-sponding set of crystallographic planes.

Regarding its magnitude, it follows from the Bragg diffraction law (8). First we notice that k is related tothe wavelength of the neutrons via de de Broglie relation, Eq. (2):

�k =h

λ⇒ k =

2π

λ, (13)

thus

Q =4π

λsin θ. (14)

This allows us to re-write the Bragg law in the form 2d λ4πQ = nλ i.e.

Q = n2π

d. (15)

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The scattering vectors Q that are perpendicular to crystallogrpahic planes and whose magnitudes are torelated to the planes’ spacing by the above formula form the crystal’s reciprocal lattice.

A neutron diffraction experiment will show bright Bragg spots at the detectors corresponding to reciprocallattice vectors. When the system orders magnetically, the scattering of neutrons will become stronger.

The crucial reason why neutron scattering is so useful in the study of magnetism is that anti-ferromagneticcan alter the reciprocal lattice. Take the square lattice we looked at before. Assume we cool down the crystaland it goes through a Neel instability, so it develops anti-ferromagnetic order:

Since the neutrons couple to the magnetic moments, they are sensitive to the broken translational symmetry.This has been broken by the AF order, with the result that, for example, the set of crystallographic planesthat we colored above in green has its periodicity halved (since “spin-up” and “spin-down” planes are nolonger equivalent):

The corresponding reciprocal lattice vectors are twice as small:

QT>TN= n

2π

d→ QT<TN

= n2π

2d=

1

2QT>TN

. (16)

Giving values to n = 0, 1, 2, . . . we see that all the reciprocal lattice vectors that we had originally are stillthere, but now there are smaller ones that weren’t present in the paramagnetic crystal:

At T > TN

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At T < TN

As the temperature is lowered further, the magnetic peaks become more intense as the magnetic momentsbecome stronger:

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This is how antiferromagnetism was discovered and the main way to study it nowadays.

2 Muon spin rotation

Muons are sub-atomic particles. Like the electron, they are leptons, and they carry the same charge chargeas the electron, −e (negative muon, µ−) or the same charge of a positron, +e (positive muon, µ+). Also likethe electron and positron, muons have s = 1/2. Thus lke the neutron the muon can interact with magneticmoments - though the way we exploit that is quite different, as we shall see.

Unlike the electron, however, the muon has a farily large mass, mµ ≈ 200me, and a finite lifetime, τµ ≈ 2.2µs.

Muons consitute some of the most common forms of cosmic ray. We are regularly bombarded with muons.They are generated when high-energy particles hit nuclei in the ionisphere. Interestingly, at the velocitieswith which they reach the planet’s surface they wouldn’t be able to make the journey form the ionospherein the brief time spanned by their lifetimes, were it not for time dilation (resulting from their relativisticspeeds) which means that from their pespective time passes mucho more slowly than from the point of viewof our “stationary” frame of reference.

The muons we use in condensed matter research are not of the cosmic variety. They are generated here,on Earth, using proton beams. The muons are created by hitting a nucleus (any nucleus will do, but C orBe are commonly used) with a proton beam. When the proton hits the nucelus, it produces a pion whichquickly disintegrates, emitting a muon and the corresponding neutrino:

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Here there’s an extra bit of particle physics that helps us a lot: it turns out that the pion’s disintegrationinvolves the electro-weak interaciton, which violates parity conservation. As a result, the muon is producedwith its spin pointing anti-parallel to its momentum (the neutrino is also produced with its spin anti-parallelto its momentum; this way both spins add up to the pion’s s = 0). This means that the muon beam weproduce has a well-defined spin polarisaiton.

Usually for condensed matter experiments positive muons are employed. The are implanted in a sample andthen we wait for them to spontaneously disintegrate. Typically the muons will come into the sample in shortpulses (~100ns or so, which is much shorter than the muon lifetime, see above). When they go into thesample, they will very quickly come to rest in side it. When the muons disintegrate, they emit a positronand two neutrinos. Again the electro-weak interaction comes to help us here, because the positron is emittedwith higher probability in the direction in which the muon’s spin is pointing at the time of disintegration:

It is the positrons that we detect, but because of this coupling between their direction of emission and themuon’s spin direction, we gain information about where the muon’s spin was pointing inside the sample.

Because the spins of all the mouns were pointing in the same direction to begin with, if there are absolutely nomagnetic fields (either internal to the sample or externally-applied) then the moments would just continue topoint in that direction indefinitely and all the positrons will be detected in the same direction, too. However,if there is an applied magnetic field or, more interestingly, the sample has an intrinsic, internal magneticinduction (e.g. an effective exchange field Beff responsible for ferromagnetic or anti-ferromagnetic order)then the muon’sm magnetic moment will precess about this local field:

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This precssion will happen at the Larmor frequency of the muon. The derivation is entirely as what we didfor electrons early in this course, so we quote the result here:

ωLarmor = gµγµBlocal (17)

The gyromagnetic ratio and g-factor of the muon are

γµ =|e|

2mµ(18)

gµ ≈ 2M (19)

giving

ωLarmor ≈ 2π 135.5MHz ×Blocal

T. (20)

This precession can be seen as a rotation in the direction in which we detect positrons for about 10µs afterthe muon pulse has been implanted in the sample (after that time, most of the muons in the pulse havedied out, so we have to wait until the next pulse comes along before we can collect more data). Very often,what is measured is the “asymmetry” between the positron detections in two detectors pointing in differentdirections. This asymmetry oscillates as the mouns precess around the internal fields in the sample.

The following two animations illustrate much better than any words one could write how a muon rotationexperiment works:15

http://neutron.magnet.fsu.edu/images/uSR.gif

http://neutron.magnet.fsu.edu/images/uSR2.gif

Like neutron scattering, muon spin rotation alolows us to detect the magnetic moments that turn up belowTN in anti-ferromagnets. The key is that because the muon sits at a particular site within the crystallattice, it can see the magnetic moment of the nearest magnetic ion, without averaging the magnetizationfrom different ions on different magnetic sites. Muon spin rotation therefore allows us to determine thetemperature-dependence of the magnetic moment of each particular sub-lattice:

15 These videos come from “Muon spin relaxation”, Quantum Materials Group, Florida State University,http://neutron.magnet.fsu.edu/muon_relax.html (retrieved 23 November 2011).

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3 Neutron and muon facilities

Muon spin rotation and, specially, neutron scattering underpin all modern research in magnetism and thetechnologies that are based on it (such as magnetic data storage). Although individual experiments arecarried out by small groups of researchers, the neutron and muon sources themselves are huge enterprises(each of them has many different instruments on which different teams can be working with different samplesat the same time). The following video describing the ISIS facility at the Science and Technology FacilitiesCouncil’s Rutherford Appleton Laboratory, in Oxfordshire, gives an idea of the scale:16

http://www.isis.stfc.ac.uk/science/a-video-tour-of-the-isis-facility4469.html

16 ISIS Facility, http://www.isis.stfc.ac.uk (retrieved 10 November 2011).

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