New Magnetocaloric Materials for Adiabatic Refrigeration Gadolinium-lithium tetrafluoride (Refrigerant) Pramosh Shrestha Master 2 internship report September 28, 2017 Supervised by Andreas J. Honecker Laboratoire de Physique Théorique et Modélisation (UMR 8089) Université de Cergy-Pontoise Saint-Martin site, Pontoise, France Acknowledgements This internship was my very first hand scholarly experience. I was delighted to recieve stipend(Which too was my firsthand experience in life), I forward my deep gratitude to the establishment of Université de Cergy-Pontoise, Saint-Martin site, Pontoise . I did not become anxious because I was aware I had two wise supervisor, I am again deeply grateful to them, namely Prof. Andreas J. Honecker([email protected]) and Mr. Mike Zhitomirsky([email protected]). I really received very condescending counselling, very pragmatic as well especially for a noob. i Contents 1 Introduction 1 2 Magnetic Refrigeration 2.1 Cryogenics and new method in it . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The magnetocaloric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 3 4 3 Gadolinium’s Compound 3.1 The compounds . . . . . 3.2 Electronic Configuration 3.3 g-factor . . . . . . . . . 3.3.1 Rudiments . . . 3.3.2 Landé g-factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 6 7 8 8 9 4 Demagnetization 4.1 Rudiments . . . . . . . . . . . . . . . . . . 4.2 Introductory explanation . . . . . . . . . 4.3 Mathematical background [33] . . . . . . . 4.3.1 Non-zero field . . . . . . . . . . . . ~ . . . . . . . . . . . 4.3.2 Solution for H 4.4 The demagnetising field inside a cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 10 10 11 11 12 13 . . . . 15 15 16 17 17 . . . . . . . . . of Gd(III) ions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Dipolar interaction energy 5.1 The energy . . . . . . . . 5.2 Hamiltonian matrix . . . 5.3 Geometrical shapes . . . . 5.4 The ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Preamble for energy evaluation 19 6.1 Prelude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.2 The convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6.3 Juxtaposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 7 Ground state configuration and energy 7.1 Approach . . . . . . . . . . . . . . . . . 7.2 Case of Terbium . . . . . . . . . . . . . 7.2.1 Data . . . . . . . . . . . . . . . . 7.2.2 Inference . . . . . . . . . . . . . 7.3 Case of Erbium . . . . . . . . . . . . . . 7.3.1 Data . . . . . . . . . . . . . . . . 7.3.2 Inference . . . . . . . . . . . . . 7.4 Case of Gadolinium . . . . . . . . . . . ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 24 25 25 26 26 26 27 27 CONTENTS CONTENTS 7.4.1 7.4.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Appendix A Program Coding A.1 Lattice summation series . A.1.1 Sphere . . . . . . . A.1.2 Ellipsiod . . . . . . A.1.3 Cylinder . . . . . . A.2 Hamiltonian Matrix . . . [a\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 31 31 32 33 34 Chapter 1 Introduction During the thesis project a magnetic refrigerant(Gd compound) was theoretically analyzed for it’s magnetic properties. A piece of atomic physics was revised. On the other hand quantum magnetism was not revised nonetheless it’s the quantum properties associated with electron that produces it’s magnetic phenomenon. Indeed the inter dipolar interaction energy is solely determined by electron quantum spin value ’S’. The compound Lithium Gadolinium Tetra Fluoride (LiGdF4 ) alongside with Gadolinium’s other alloys are considered as a a good refrigerant [23] for adiabatic magnetic refrigeration technique. LiGdF4 is a body-centered tarragons crystal with ion Gadolinium (Gd(III)) embedded in four lattice points. It’s physical properties, additionally energy calculations and it’s preludes are discussed in this report. The discovery of magnetic refrigeration is credited to a German physicist Warburg (1881) [32]. Subsequently by French physicist P. Weiss and Swiss physicist A. Piccard in 1917 [30]. The importance of this phenomenon is in cryogenics, superconductivity comes strikingly in mind. Low temperature, as low as in microkelvin can be achieved. It was first demonstrated experimentally by Giauque and his colleague D. P. MacDougall in 1933 for cryogenic purposes when they reached 0.25 K [13]. Between 1933 and 1997, advances in MCE cooling occurred [26]. In 1997, the first near room-temperature proof of concept magnetic refrigerator was demonstrated by Karl A. Gschneidner, Jr. by the Iowa State University at Ames Laboratory. This event attracted interest from scientists and companies worldwide who started developing new kinds of room temperature materials and magnetic refrigerator designs [2]. Recent research has focused on near room temperature as well [25]. A potential contemporary implications is in spacecraft. In 2014 giant anisotropic behavior of the magnetocaloric effect was found in HoM n2 O5 at 10 K. The anisotropy of the magnetic entropy change gives rise to a large rotating MCE offering the possibility to build simplified, compact, and efficient magnetic cooling systems by rotating it in a constant magnetic field [7]. In addition, in the wake of global environmental issues this refrigeration technique is conceived as greener, more efficient and feasible technology. Like every active research field this technique is also heavily emulated in industrial scale at least in contemporary prudentials. Here’re the quotations from [4], may help grasp the growing interest in it. “Magnetic refrigeration has the potential to reduce energy use by 30% and requires no refrigerant. Metkel Yebiyo and Graeme Maidment, of Sirach, describe the technology, its main applications, and the challenges facing firms trying to get the concept to market”. “There is no possibility of refrigerant leakage and no direct CO2 emissions, so magnetic refrigeration fully complies with all regulations”. Besides these practical topic the interest of this dissertation is on theoretical explanation of the phenomenon. The refrigerant in subject is a Gadolinium compound, it’s atomic structure as an element and an ion is introduced consecutively and then deep in- 1 Introduction [a\ sights about the inter-ionic dipolar magnetic interaction and magnetization phenomenon are confronted. The inter-ionic dipole interaction is an important acknowledgment ever since it was conceived as the factor contributing to the magnetocaloric effect. Such case means calculation of inter-ionic magnetic dipolar energy of specimen under study is necessary to progress. The interest remains in ground state configurations and it’s associated energy value. The ground state energy involved in this interaction can be minimized and is vehemently sought! This is a very tricky case. One can see how the summation series converges with the limit(number of ions). The low-temperature ordered states of LiRF4 (R is a rare Earth element, Gd is taken for this research) crystal is well discussed and published in 1977 [21]. By calculating the g tensor, in this case it pertains to the isotropy of lattice cell, one can make inference about the configuration in the specimen. One of the input in this research(low-temperature ordered states) was Luttinger-Tisza method [19, 20] which provides great deal of lattice(with ions) configuration. Furthermore the magnetic ordering of dipoles, which has the lowest, among the evaluated energy value(dipole interaction) will be the ground state of the compound. 2 Chapter 2 Magnetic Refrigeration 2.1 Cryogenics and new method in it In physics, cryogenics is the study of the production and behavior of materials at very low temperatures. Basically the laws of thermodynamics are to be complied in every sense. There stands no other governing foundation on it and probably never will be. It has good future promises. One of the attraction for researchers in cryogenics is superconductivity. The nullified resistance conductor has wide pragmatic applications in industries and modern engineering. Cryogenicists only have to deal with how to achieve it alongside they have to single out the optimized method and tamed inevitable effects. As aforementioned the governing laws are named laws of thermodynamics beside it is interesting to develop a working device compelling on it, the modification cryogenicist can to is how to comply on those laws. Up to now the chosen material for achieving low temperature were liquefied Helium, Nitrogen, Hydrogen and so forth. The process to cool them down is classical refrigeration, probably with certain updated engineering technique. These cryogens and the refrigeration equipments have their own cons to offer, likely as expensive, susceptible to explosions, feasibility, and so on. It is now that world is facing crisis in regards of environment, economy, safety regulations and so on, a relatively new or to be general a successor to these classical ways have been devised. The cryogen can be magnetic material and the low temperature production can be performed by magnetic phenomena. This method has already been coined into a term called Adiabatic magnetic refrigeration [10, 23, 29]. It is a cooling technology based on the magnetocaloric effect. This technique can be used to attain extremely low temperatures, as well as the ranges used in common refrigerators. Compared to traditional gas-compression refrigeration, magnetic refrigeration is safer, quieter, more compact, has a higher cooling efficiency, and is more environmentally friendly because it does not use harmful, ozone-depleting coolant gases [10, 16]. 2.2 The magnetocaloric effect The magnetocaloric effect (MCE, from magnet and calorie) is a magneto-thermodynamic phenomenon in which a temperature change of a suitable material is caused by exposing the material to a changing magnetic field. This is also known as adiabatic demagnetization since the cooling event occurs during isolation of system i.e. adiabatic process. The magnetocaloric effect can be represented with the equation below [25, 26], ! ! Z H1 T ∂M (T, H) ∆Tad = − dH (2.1) C(T, H) ∂T H0 H 3 H 2.3. THERMODYNAMICS Magnetic Refrigeration [a\ where T is the temperature, H is the applied magnetic field, C is the heat capacity of the working magnet(refrigerant), M is the magnetization of the refrigerant. From the equation we can see that magnetocaloric effect can be enhanced by: Applying a large field. Using a magnet with a small heat capacity. Using a magnet with a large change in magnetization vs. temperature, at a constant magnetic field. 2.3 Thermodynamics It is now the acknowledgment that thermodynamics is irreplaceable governing law for heat energy transport hereby Adiabatic magnetic refrigeration is not any exception. The classical refrigeration(Carnot refrigeration won’t damage either) and magnetic refrigeration are analogous except in literal sense there is difference in material used and the sheer source of energy supplied into the system. Below the each cycle is described one at a time and a schematic of the cycle is also displayed in figure2.1. Figure 2.1: Schematic showing basic working principal of magnetic refrigeration [26]. 1. Adiabatic magnetization: A magnetocaloric substance is placed in an insolated environment. The increasing external magnetic field (+H) causes the magnetic moments of the atoms to align, thereby decreasing the material’s magnetic entropy and heat capacity. Since overall energy is not lost(yet) and therefore total entropy is not reduced (according to thermodynamic laws), the net result is that the substance is 4 2.3. THERMODYNAMICS Magnetic Refrigeration [a\ heated by ∆T . 2. Isomagnetic enthalpic transfer: This raised temperature can then be dragged down by a fluid or gas or liquid helium. This casues system to loose few heat (−Q). Meanwhile the magnetic field is held constant to prevent the dipoles from reabsorbing the heat. Once sufficiently cooled, the magnetocaloric substance and the coolant are separated. 3. Adiabatic demagnetization(H = 0): The substance is returned to another adiabatic (insulated) condition so the total entropy remains constant. However, this time the magnetic field is decreased, the magnetic moments overcomes the field by absorbing heat in the process thus the sample cools, i.e., an adiabatic temperature change. Energy (and entropy) transfers from thermal entropy to magnetic entropy, measures of the disorder of the magnetic dipoles. 4. Isomagnetic entropic transfer: The magnetic field is held constant to prevent the material from reheating. The material is placed in thermal contact with the environment to be refrigerated. Because the working material is cooler than the refrigerated environment (by design), heat energy migrates into the working material (+Q). Once the refrigerant and refrigerated environment are in thermal equilibrium, the cycle can restart. To grasp both the similarity and the dissimilarity between a conventional gas compression method and a magnetic refrigeration process right below a schematic is provided. Figure 2.2: Schematic of two refrigeration technique for their correspondence. On the left is conventional gas compression and on the right is magnetic refrigeration techniques [12] 5 Chapter 3 Gadolinium’s Compound 3.1 The compounds Gadolinium demonstrates a magnetocaloric effect whereby its temperature increases when it enters a magnetic field, and decreases when it leaves the magnetic field. Following an experiment [23] carried out for magnetocaloric effect of GdLiF4 the relative entropy for it was between 20 and 60 percentage higher than of it’s counterparts. In addition to it one of the most notable examples of the magnetocaloric effect pertaining to gadolinium and some of its alloys is shown and published for Gd2 T i2 O7 [31]. Besides these any chemical property is ignored in this report. The unit cell of LiGdF4 is illustrated below. There are four lattice points belonging to Gd(III) ion within the unit cell. The four possibilities of spin orientation will be considered as the cases of ferromagnetic ordering, anti-ferromagnetic ordering and layered anti-ferromagnetic ordering in separate occasions. Figure 3.1: Positions of Gd+++ ions in Body-centered-tetragonal unit cell of LiGdF4 [3,21] There are four possible magnetic ordering or spin configurations for the R-element ions in LiRF4 . A quotation from reference [21] is succinct below. “....one ferromagnetic configuration, one antiferromagnetic configuration (spins 1,2 parallel, spins 3,4 anti-parallel), and two layered antiferromagnetic configuration [ (a) spins 1,2 parallel, spins 3,4 antiparallel; (b) spins 1,4 parallel, spins 2,3 antiparallel].” The eigenvectors of various configurations are φ(k, α) = q(k)φk (α)(k = 1, 2, 3, 4; α = 6 3.2. ELECTRONIC CONFIGURATION OF GD(III) IONS Gadolinium’s Compound [a\ x, y, z), φ(k)(α) are the eigenvalues and q(k) is 1 1 1 −1 q(k) = 1 1 1 −1 the eigenvector of Hamiltonian matrix. 1 1 1 −1 . −1 −1 −1 1 The first and second column are eigenvectors q(k) for ferromagnetic and antiferromagnetic and the last two are eigenvectors q(k) for layered antiferromagnetic. 3.2 Electronic Configuration of Gd(III) ions The distribution of electron in shell and subshell included is erratic among high atomic number elements and Gd is not any exception. The paramagnetic behavior of Gd(III) ion is due to properties pertaining to last three outermost shell, precisely n = 4, 5 and 6. It is clear paramagnetism is due to the presence of unpaired electrons, so all atoms with incompletely filled atomic orbitals are paramagnetic. Due to their spin, unpaired electrons have a magnetic dipole moment and act like tiny magnets. There are many rare Earth elements which demonstrate excellent magnetic properties but fails significantly in some aspect. To name one, isotropy is one the characteristic and Gadolinium is near to perfect and transcends it’s counterparts. The g-factor(It appears in the form of rank two tensor) for Gadolinium is a diagonal matrix with value 2. The Gd(III) ion is an excellent paramagnet. Gadolinium has atomic number 64 with the shorthand notation for it’s electronic configuration as [Xe] 4f7 5d1 6s2, note [Xe] is the symbol for element Xenon here it connotes electronic configuration. Gadolinium is believed to be ferromagnetic at temperatures below 20 C (68 F) [18] and it is strongly paramagnetic above this temperature [11]. In gadolinium, the 4f orbital is half full(seven electrons), and all the f-electrons are unpaired. Adding an electron to the orbitals would mean pairing up, which costs energy (electrons repel each other when paired due to their identical charge). It costs less energy to have the extra electron in an empty 5d orbital than to pair it with any of 4f electron. The remaining two electron is filled in 6s orbital. Moreover it is refreshing to know the 5s and 5p orbital of Xe atom is fully occupied. Now electronic configuration for oxidized Gd is imminent, the process of oxidization of Gd, knocks out three electrons. Without any further details, the configuration of Gd(III) is [Xe] 4f7. It should be vivid that seven electrons in shell 4 and orbital f is unpaired. Before evaluateing magnetic moment of Gd(III) ion it is safe to know by the virtue of Hund’s third rule the summed up angular momentum quantum number for shell 4 and orbital f is zero [5]. The spin contribution of each electron is 21 hence we obtain the total spin contribution for Gd(III) is 27 . The motive behind this detailing is to show the spin of electron is large enough to enforce accounting of dipole-dipole interaction in the specimen. Shortly, the energy expression due to mentioned interaction will appear. 7 3.3. G-FACTOR Gadolinium’s Compound [a\ Figure 3.2: An illustration of electronic configuration of the Gd atom [1]. 3.3 g-factor The report is solely concerned by spin properties of a valence electron in Gd(III) ion. This is since it’s orbital quantum number is zero, one should refer to hund’s rule for such inference. The following discussion will pertain to both the spin momenta and the angular momenta and the physics that follows it. These two momenta when added gives another momenta termed as total angular momenta. Moving on, the discussion is going to be short along with sufficient mathematics. This section talks from spin to Landé g-factor, where the notion of total angular momenta is essential. 3.3.1 Rudiments Spin is an intrinsic form of angular momentum carried by elementary particles, atomic ~ this vector quantity in turn produces magnetic nuclei, and so on. [15]. It is denoted by S, moment of electron. This phenomenon has a counterpart known as orbital angular momentum, the revolution of an electron around the nucleus, gives rise to the orbital angular ~ and in turn it produces orbital magnetic dipole moment, The moment, denoted by L, ratio between each aforementioned magnetic moment and angular momentum is termed as gyromagnetic ratio. It is a value and denoted by γ. Moreover, the addition of these two angular momenta produces a new angular momentum called total angular momentum denoted by J~ Mathematically, µs ~ , µ~s = γs S S γs = (3.1) gs q (3.2) 2m In equation 3.1 γs is the gyromagnetic ratio and in equation 3.2 gs is the g-factor, q and m are the electric charge and the mass of the particle respectively. Note that gs is a dimensionless quantity. Similarly for the orbital angular momentum, Mathematically, γs = γl = µl ~ , µ~l = γl L L 8 (3.3) 3.3. G-FACTOR Gadolinium’s Compound [a\ gl q (3.4) 2m In equation 3.3 γl is the gyromagnetic ratio and in equation 3.4 gl is the g-factor, q and m are the electric charge and the mass of the particle respectively. Note that gl is a dimensionless quantity. Note that in equation 3.2 and equation 3.4 the gyromagnetic ratios can be expressed B in terms of Bohr’s magnetron(µB ). It is simply expressed like γ = 2πgµ Furthermore h the above expressions and can be written for total angular momentum once J~ has been obtained after addition of spin and orbital angular momenta. This is written below. γl = γj = µj , µ~j = γj J~ J (3.5) gj q (3.6) 2m In equation 3.5 γj is the gyromagnetic ratio and in equation 3.6 gj is the g-factor, q and m are the electric charge and the mass of the particle respectively. Note that gj is a dimensionless quantity. γj = 3.3.2 Landé g-factor The Landé g-factor [9] is a particular example of a g-factor, particularly for an electron with both spin and orbital angular momenta. The expression for Landé g-factor [9] is given below gJ = 3 S(S + 1) − L(L − 1) + 2 2J(J + 1) (3.7) In equation 3.7 gJ is the Landé g-factor and S, L and J are standing for spin, orbital and total angular momenta of an electron. The Gd(III) valence orbital (shell 4 and orbital f) is exactly half filled and holds L = 0 and S = 21 , this in turn produces J = 12 . Evaluation in equation 3.7 yields gJ = 2. Another way around is, simply put L = 0 and J = S in equation 3.7 to obtain the identical result! 9 Chapter 4 Demagnetization 4.1 Rudiments The demagnetizing field is the magnetic field (H-field) generated by the magnetization (Density of magnetic dipole moment in a specimen) in a magnet [9]. The total magnetic field in a region containing magnets is the sum of the demagnetizing fields of the magnets and the magnetic field due to any free currents or displacement currents. The term demagnetizing field reflects its tendency to act on the magnetization so as to reduce the total magnetic moment. Since the magnetization of a sample at a given location depends on the total magnetic field at that point, the demagnetization factor must be used in order to accurately determine how a magnetic material responds to a magnetic field. The demagnetizing field of an arbitrarily shaped object is very difficult to calculate even for the simple case of uniform magnetization. For the special case of ellipsoids (which includes spheres) the demagnetization field is linearly related to the magnetization by a geometry dependent constant called the demagnetizing factor [9]. 4.2 Introductory explanation ~ ·B ~ = 0. This means there is neither According to famous Maxwell’s equation we have ∇ ~ ·A ~ > 0) nor any sink(∇ ~ ·A ~ < 0), subsequently the existence of magnetic any source(∇ monopoles is impossible. If there is a source then there must be a sink, they pair up so that magnetic dipoles are inevitable. For further discussion pull in the case of Gauss’s law ~ ·E ~ = ρ , how electric field E ~ has either a source or a sink which in simple words means ∇ o electric monopoles. ~ ) process inside a ferromagnetic specimen reaches to top When the magnetization (M surface(for the moment) it has to stop suddenly exposing a magnetic pole either North or South, this is same for the bottom surface only for magnetic pole to incur opposite polarity. Following bullets deal with this phenomena in successive manner. 1. Either end of surface acts as contrasting magnetic poles. This makes up one surface end acting as a source and the other end as a sink. So then at least by the virtue of ~ ·B ~ 6= 0. specimen’s one surface at a time ∇ 2. This non-zero divergence compels the existence of an anti-parallel demagnetization ~d inside and outside the specimen, for the field outside of the specimen it is field H called stray field. ~d ) is proportional to the magnetization (M ~ ). 3. The demagnetization field(H 10 4.3. MATHEMATICAL BACKGROUND [?] Demagnetization [a\ 4. The expression of field [22] for sphere is ~ γM 4π , γ is called demagnetizing factor and 4π 3 ~ equals eventually the field for spherical specimen is M 3 . In the case of ellipsiodal specimen the demagnetizing factor is distinct for each axes and hence there are three ~k M , k ∈ {x, y, z}. of them following it the expression in this case is γk4π As it has been shortly revealed that geometrical constituent to contribute to this field is surface of the specimen, further special cases can be devised from the case of ellipsiod. The other important cases are an infinite plate (an ellipsoid with two of its axes going to infinity) which has γ = 4π in a direction normal to the plate and zero otherwise and an infinite cylinder (an ellipsoid with one of its axes tending toward infinity with the other two being the same) which has γ = 0 along its axis and 2π perpendicular to its axis [24]. Verbal explanation apart following figures will help boost the existing concept in demagnetization field. Figure 4.1: Illustration of demagnetization field inside an elliptical specimen. [27] Figure 4.2: Illustration of demagnetization field inside a thin film specimen [9]. 4.3 Mathematical background [33] Mathematics is the foundation of Physics, the phenomena in Physics up to now are all described and inferred by mathematical grounds. This section is here to forward a simple mathematical insight on the subject of demagnetization field. The following discussed mathematical techniques are subject of acknowledgment. So that said there is one important note to hold which is highlighted here there exists no magnetic monopoles and ~ satisfies Maxwell’s equation. the magnetic field B 4.3.1 Non-zero field The famous Maxwell’s equation [14] for magnetic field is ~ ·B ~ = 0, In addition we get B ~ = µo (H ~ +M ~) ∇ (4.1) ~ refers to the magentization of specimen and H ~ is called the magnetic field, this magnetic M field can be specifically field due to free electric current inside specimen but for the context 11 4.3. MATHEMATICAL BACKGROUND [?] Demagnetization [a\ ~ and M ~ in hand it turns out to be demagnetization field. From equation (4.1) the field H can be written as ~ ·H ~ = −∇ ~ ·M ~ ∇ (4.2) This is implying the idea of magnetic monopole which is acting as a non-zero sink (due to ~ Since then the equation is as tidy as the negative sign) for the field H. ~ ·H ~ = ρ, such that ρ = −∇ ~ ·M ~ ∇ (4.3) But never buy this illusion! It is not an physical implication of monopole density. Just as a reminder this has been already discussed in above section and the culprit are the surfaces of specimen where there’s an exposure of poles. 4.3.2 ~ Solution for H For the case of no free electric currents and applied electric field the fourth Maxwell ~ ×H ~ = 0. Moving forward, it should yield a equation or Ampere’s circuital law gives ∇ magnetic potential namely Φ such that ~ = −∇Φ. ~ H (4.4) Eventually the equation (4.2) and equation (4.4) leads to the Poisson equation as below and the negative gradient of it’s solution is exact expression of demagnetization ~ field H. ~2 Φ = −ρ. ∇ (4.5) From here the solution is very easy to obtain especially when the method is using Green’s function G(r). This is three dimensional case and we can consider ρ as an unit point charge. The Green’s function is shown below G(r) = 1 . 4π|r| (4.6) The solution of equation (4.5) is convolution of equation (4.6) with the R.H.S of equation (4.5), the term is named as force function or source. Z Φ(r) = 0 0 0 dr G(r − r )ρ(r ) = Z 0 ρ(r ) dr = 4π|r − r0 | 0 Z ~0 · M ~0 −∇ dr . 4π|r − r0 | 0 (4.7) It’s noticeable that equation (4.7) could be solved for Φ either directly by substituting ρ ~ . The suitable case can be followed. In this case it is or evaluateing the divergence of M ~. suitable to let the gradient act on Green’s function instead of magnetization field M Z 1 0 0 ~ 0. ~ Φ(r) = dr ∇ ·M (4.8) 4π|r − r0 | There should had been a surface term but the surface is taken outside of specimen, its contribution is null. The gradient should be performed on Green’s function and below there is the expression for it. ! Z 0 r−r 0 ~ 0. Φ(r) = dr ·M (4.9) 4π|r − r0 |3 The term inside small bracket in equation (4.9) is analogous to a new Green’s function ~ − r0 ), seems to be interlocked as convolution with magnetization and is termed as K(r ~ only following the vector dot product. Moreover a simple relation between the field M 12 4.4. THE DEMAGNETISING FIELD INSIDE A CYLINDER Demagnetization [a\ ~ radial component of K(r) and the Green’s function of equation (4.5) given by equation (4.6). In short mathematical way it turns out as following. ~ = K(r) Kr (r) = − ~r . 4π|r|3 (4.10) 1 dG(r) = . dr 4πr2 (4.11) ~ is easy to formulate, attention should be on the geometrical shape of This last form of K specimen and the chosen co-ordinate system which brings variation on it, even afterwards it gives a naive way to evaluate Φ(r) as given by equation 4.8. Already an enigmatic magnetic monopole density concept ρ offers cumbersome method to solution. The final ~ has been formulated is to take it’s dot product with magnetization step to do once K ~ field M , finally perform the gradient of the outcome. Hereby one gets demagnetizing field, ~d . general notation is H An example of calculation of demagnetization is demonstrated below from reference [33]. It is noteworthy that the main concern would be to calculate Φ as expressed in equation 4.9. The first term in the dot product is quite tricky and is usbject to change with choice of co-ordinate system. 4.4 The demagnetising field inside a cylinder First, consider a magnet of length L along the z axis, which is the longitudinal axis of a cylinder. The upper end lies at z = +δ, the lower end at z = δ, so that the length is L = 2δ, and z = 0 is at the middle of the cylinder. The cross-section could be a circle of radius R, for most simplicity, but it doesnt absolutely have to be. For the circular cross section, there is no need yet to make any special assumption about the radius R compared to the cylinder length L. Longitudinal magnetization Mz Initially, suppose the cylinder is magnetized in the z direction, that is, along its axis of symmetry. Then M = Mz ẑ. This places surface charge densities of σ = ±Mz at z = ±δ, respectively. So the top end has positive charge, the bottom end has negative charge. To find the potential at an observer point r = (x, y, z), inside the magnet, consider the pos0 0 0 0 0 0 itive source charges at r = (x , y , δ), and the negative source charges at r = (x , y , −δ). From the Green function integral over charge density, one has now only surface integrals on the ends, Z Mz 1 1 0 0 q q Φ(x, y, z) = dx dy − (4.12) 4π re2 + (z − δ)2 re2 + (z − δ)2 Note that re2 = (x − x0 )2 + (y − y 0 )2 If we want to just find the field in the center of the cylinder, it is not so difficult, putting ~ z can be found as a function of z. In this case there is dependence here x = y = p 0. Then H 0 02 02 only on r = x + y and only a radial integration is needed (dx0 dy 0 → dθ0 r0 dr0 ), Φ(x, y, z) = Mz 4π Z dθ0 Z r0 dr0 1 1 q −q r2 + (z − δ)2 r2 + (z − δ)2 13 (4.13) 4.4. THE DEMAGNETISING FIELD INSIDE A CYLINDER Demagnetization [a\ The integration is quite simple. For the cylindrical base a circular cross-section can be taken with radius R and for the angular part θ ranges from 0 to 2π, integral yields Mz Φ(x, y, z) = 2 q q 2 2 2 2 R + (z − δ) − R + (z − δ) − |z − δ| + |z + δ| (4.14) The resulting field has to be an even function of z. Thus it can be calculated for z > 0; the result for z < 0 will be symmetrical. Indeed, for any z ∈ (−δ, δ), this is q q Mz 2 2 2 2 Φ(x, y, z) = (4.15) R + (z − δ) − R + (z − δ) + 2z 2 Then the field on the axis of the cylinder is imminent, dΦ z − δ 1 z + δ 1 Hz = − − q = Mz 1 + q dz 2 2 R2 + (z − δ)2 R2 + (z + δ)2 (4.16) Note the somewhat surprizing result. At z = 0, the last terms equal each other and combine, to give ( R2 if δ R, −Mz 2δ δ 2 Hz (0) = −Mz 1 − √ ≈ (4.17) 2 2 if δ R, −Mz (1 − Rδ ) R +δ ~ , which is why it is demagnetization. Further, its The field points opposite to M strength depends on the aspect ratio of the cylinder. Note the limiting behaviors. When the cylinder is long and thin, the longitudinal field at its center gets very small. On the other hand, if the cylinder is short and wide, the longitudinal field at its center is maximized, nearly equal to the strength of its magnetization. This latter case corresponds to the strongest demagnetization that can take place. 14 Chapter 5 Dipolar interaction energy 5.1 The energy The expression for the energy is X j Jijαβ = X X S 2 µ2 g αα g ββ B j αβ 3 rij (δαβ − β α 3rij rij 2 rij ) (5.1) The spin term can be well observed in quadratic form in energy expression. This is the reason, mentioned recently, to take into the account of spin due to electrons in orbital 4f7 for Gd(III). The additional reason it is no more negligible is the specimen is at millikelvin temperature order. Indeed a quotation from a book Stephen Blundell, Magnetism in Condensed Matter; Oxford University Press(2001) p. 74 Section Magnetic dipolar interaction will clarify the negligibilty of dipole interaction energy. “We can easily estimate the order of magnitude of this effect for two moments each of µ2 −23 J which is equivalent to µ ≈ 1µB separated by r ≈ 1Å to be approximately 4πr 3 ≈ 10 about 1K in temperature. Since many materials order at much higher temperatures(some around 1000K), the magnetic dipolar interaction must be too weak to account for the ordering of most magnetic materials. Nevertheless, it can be important in the properties of those materials which order at millikelvin temperatures.” The term inside the small bracket is termed as lattice sum and is easily evaluated by computer via fortran or C or equivalent platform. An ion (indexed by i) is chosen as an origin and fixed afterwards summation (on index j) can be done for appropriate neighbors to ion i. Moreover there should be a sum running for α and β as well, they run over x, y and z axes. Indeed this parameter termed g α,β gets dramatically varied as the R-element varies in the unit-cell whereas the term S 2 µ2B is constant and the differences in values of inter-ionic distance r for different R-element can be rendered negligible. In the interest of this report is to take Gd as R-element. The properties of Gadolinium element have been extensively presented in preceding chapter. Further reading for the elements other than Gadolinium can be done here [21]. Next in the queue is calculation of the energy term (equation (5.1)). There are few hidden concept related to the energy eigenvalues value of this interaction! Below they are in bullets 1. The energy eigenvalues are distinct in distinct magnetic orderings or spin configurations of Gd(III) ions, the one with lowest energy is ground state configuration. 2. The fact that energy eigenvalues depends upon distance between two interacting dipole eventuate in it’s change according with the specimen’s geometrical shape. For instance assume a sphere of specimen holding unit cells (say 100), the maximum 15 5.2. HAMILTONIAN MATRIX Dipolar interaction energy [a\ distance between two unit cell can’t surpass the diameter of the sphere. Now place all those unit cells in a mono-layered unit cell structure, the maximum intercell distance can be easily way off from the beforehand diameter of sphere. In addition there exists ample amount of distinct unit cells separated by distance greater than the diameter. 3. Following above point the cases for different geometric shape should be considered and the outcome should be analyzed. Obviously one of them will stand the most suitable shape. Nonetheless there are the cases for identical convergence between cylindrical object and ellipsiodal object with different dimensions! 4. By observing dipole interaction energy term (equation (5.1)), in all geometrical object thus far considered, the only term that survives summation series, for α = β, P P 3rα 2 −r2 α ∈ x, y, z). All the cases for otherwise produces series term are j α ijr5 ij , (rij ij α rβ rij ij 5 αβ rij P P j is nullified due the reflection symmetry of the chosen geometrical ob- ject. 5. The equation is not only slowly converging but also conditionally convergent in 3D. In other words, the value of the sum is not well defined unless one specifies the ways and cases to sum up the terms(The geometric shape of specimen should be mentioned like spherical, cubic, cylindrical, etc.). An alternative approach to this direct summation is Ewald summation which is both fast and accurate in convergence. During the internship dipole interaction energy was evaluated for chosen ions they are tabulated in coming pages. Refer to fig 3.1, the lattice summation was on sublattice 1-1, 1-2, 1-3 and 1-4, in addition each of these ionic combination has three components namely along axes x, y and z. The geometrical shape considered were sphere, ellipsiod and cylinder. In sufficiently huge limit the case of ellipsiod converges to the case of cylinder. In addition for the case of both ellipsiod and cylinder the value of summation converges with respect increment in z-axis intercept or the z-axis cut off. Besides there was a necessity for magnetic configuration of ions and the unit-cell parameters for various LiRF4 . These details were extracted from the reference here [21], this reference is useful for insights in regards of preceding concepts as well. 5.2 Hamiltonian matrix The Hamiltonian expression is already provided and it was evaluated by computer program. The importance should go to indices of the term in expression. The detailed studies lie in LT method [20]. Here it will be as simple as it can, the indices tell us the Hamiltonian is tensor of rank four. However such cumbersome case can be easily overcomed. The idea is to limit it to square matrix which is a definite rank two tensor. Here as a short reminder in equation 5.1, for ions an index (index i in this case) is fixed, situated in a chosen unit cell where about the origin lies as well. The next index(index j in this case) is varied along 1, 2, 3 and 4. Keep in mind j is varied for each and every unit cell inside the cut off dimensions(Boundaries) of geometrical object. Next is indices for component of position vector to the ion (indicated by j) namely α and β, they vary along x, y and z. It is rather easy to note that the term S 2 µ2B g αα g ββ is out of summation pertaining to the ions(indicated by indices i and j) and once it is calculated it was denoted by p (Unit is in S.I. divided by Boltzmann constant) it can be recorded for all goodness. Only the remnant can be considered for summation. 16 5.3. GEOMETRICAL SHAPES Dipolar interaction energy [a\ 1. Only for α = β the lattice sum survives otherwise the sum is zero. This is due to reflection symmetry, check preceding section for bullet number 4. 2. Basically the matrix comes out to be a square twelve by twelve and more importantly symmetric. Henceforth the matrix will yield real eigenvalues. 3. It is constructed for all the components of four sublattice sums and features three block diagonal submatrices within it. It will be expressed in coming pages. 4. Once eigenvalues and it’s corresponding eigenvectors are known the lowest eigenvalues will be the ground state energy and the eigenvector will depict orientation of four ions along the x, y, and z components. The cases of ion Terbium and Holmium in compound LiT bF4 and LiHoF4 respectively show nullified component of g along x-axis and y-axis hence turns out to be easiest of all. Only the component along z-axis of g is non-zero. Following the multiplication of prefactor(denoted by F) the Hamiltonian gets a more simplified matrix and boils down to four by four square matrix instead, and this matrix is sufficient for inferences. Once there’s a Hamiltonian matrix a simple knowledge of algebra leads to the inferences. The matrix can be used to extract it’s eigenvalues and corresponding eigenvectors. The eigenvalues tell the magnitude of energy for it’s corresponding eigenvector. Eigenvectors clarify the spin configuration of four ions embedded in an unit cell of the compound. 5.3 Geometrical shapes As it should be acknowledged by now that geometrical shape determines the value of direct summation (over all the ions) present in energy term alongside the demagnetization field. The aforementioned demagnetization factor pertaining to geometrical shapes, is already extensively dealt in preceding chapter. It will not be discussed in this section any further. Here three shapes were conceived namely spherical, ellipsiodal and cylindrical. Among these it is observed cylindrical shape is the best fit for business. The least appropriate comes out to be spherical. This inference appears following a skim over the least energy values necessary for system to exist. Succinctly saying, the shapes with ground state energy for the cystal LiRF4 in descending order is spherical, ellipsiodal and cylindrical. Moreover the cylindrical shape has both fastest and sharpest convergence record, hence so relatively smaller values for the cylindrical dimensions can lead to accurate value of summation series. The calculations and it’s outcome will be presented, tabulated and discussed shortly. 5.4 The ions In the compound LiRF4 , R connotes distinct elements. It can stand for Terbium, Holmium, Dysprosium, Erbium, Gadolinium, so on. Since the accurate physical and chemical parameters for mentioned elements are known from various sources it is a very good opportunity to test the coded Fortran program for it’s outcome. Parameters like lattice constants do not significantly change for these elements and as we know from equation (5.1) the prefactor holds spin value, a constant(Bohr’s magneton) and the g-factor, this leaves lattice summation term which we ignore for a moment here since it has been extensively discussed in preceding sections. Spin value differs for different ionic element for instance Gd(III) has spin value 72 . The Bohr’s magneton is harmless and the turn is for g-factor. g-factor is a very important term so to be technical it is a rank two tensor. Anyway! it can be expressed as a diagonal matrix. This g-factor also characterizes 17 5.4. THE IONS Dipolar interaction energy [a\ element to be isotropic or anisotropic. The ion Gadolinium (Gd(III)) is isotropic [9] and has the 3 × 3 square diagonal matrix with 2 as the element along it’s diagonal. Simply put for Gd(III) g αβ = 2 for all α = β else if α 6= β then g αβ = 0. In addition for some elements like Terbium and Holmium g-factor can have zero value for the components along both x-axis and y-axis but not along z-axis, simply it is expressed like g zz 6= 0 and g αα = 0 for α ∈ {y, z}. This g-factor can really put great differences between elements on their matter of ground state energy and corresponding configuration. 18 Chapter 6 Preamble for energy evaluation 6.1 Prelude The calculations were carried out via Fortran 95, a simple codes were composed and modified as per necessity and library for linear algebra was used as well. For convergence test compound LiT bF4 was taken furthermore the summation case was also isolated to 1-1 sublattice summation (refer to reference here [23]) and for position indices α = β = z. ~ =0 As for the other indices pertaining to ions, i = 1 and j ∈ 1, 2, 3, 4 but in the case of R summation is not done to be sure it does not interact with itself! The tasks that were done are checking the convergence of the direct summation with respect to increment in the chosen axis(in this report z-axis was chosen) for all the geometrical shape. This will give idea about the most favorable values for all three intercepts in order to keep the lattice summation series as close as possible to point of convergence. More to it is that spherical shape starts a long distant value than other two. It hardly change over it’s radius. The case for spherical shape was dropped not far from when the radius reached the value5000Å. Discussion about two other geometrical shapes are in following paragraphs. Following it energies(energy eigenvalues) pertaining to all the configurations(eigenvectors) were calculated. Basically a sphere with radius 500Å was taken and it’s radius was varied up to 5, 000Å, it is actually difficult to note on the graph since it appears as a short line on the top left corner in Fig 6.1. Thereafter the z-axis intercept was stretched up to 200, 000Å and 400, 000Å once at a time which becomes the case for ellipsiod(note that the x and y principal axes are both set to 500Å, 1, 000Å and 2, 000Å). For last case is for cylinder and only equation describing it in the code was modified to bring ellipsiodal case to cylindrical case. In addition to it the value of base radius was varied along 500Å, 1, 000Å and 1, 500Å. For more details refer to appendix of this report. 6.2 The convergence The convergence of summation series is acknowledged so here in this section the speed and the value the series converges is presented and analyzed. The appendix displays the Fortran95 coding and it is quite helpful to glance over it, especially for the insights of details. As a tautology to preceding section above this convergence case with particular conditions on particular ion chosen is the representative for any other ion and the conditions, i.e. the convergence phenomena is identical regardless of chosen ion and condition. The index on the graph is encrypted which connote geometrical shape and the value(for x-axis and y-axis intercept or principal axes). Such that cy, el and spn connote cylinder, ellipsiod and sphere respectively. Whilst srxy is connoting the x and y principal axes with the number that following corresponds to it’s value. 19 6.2. THE CONVERGENCE Preamble for energy evaluation [a\ The values of lattice summation wherever they appear are hundred times more than the exact evaluated values. This ensures tidy appearance Fig 6.1 depicts the the convergence of direct lattice summation with respect to the increment in z-axis intercept. Fig 6.1 is zoomed in and displayed in Fig 6.2 3.6 cylinder, R=500Å cylinder, R=1000Å cylinder, R=1500Å ellipsoid, XY principle axes=500Å ellipsoid, XY principle axes=1000Å ellipsoid, XY principle axes=2000Å sphere, Radius=2000Å 3.4 lattice sum 3.2 3 2.8 2.6 2.4 2.2 0 50000 100000 150000 z intercept [Å] 200000 Figure 6.1: Convergence of lattice sum(Direct method) with respect to the z-axis intercept goes upto 200, 000Å . 20 6.2. THE CONVERGENCE Preamble for energy evaluation [a\ lattice sum 2.1915 2.191 2.1905 cylinder, R=500Å cylinder, R=1000Å cylinder, R=1500Å ellipsoid, XY principle axes=500Å ellipsoid, XY principle axes=1000Å ellipsoid, XY principle axes=2000Å sphere, Radius=2000Å 2.19 2.1895 100000 120000 140000 160000 180000 200000 z intercept [Å] Figure 6.2: Zoomed picture of above graph, it shows z-axis intercept from around 100, 000Å upto 200, 000Å . This is the zoomed depiction of above graph. The portion zoomed has z-axis intercept beginning from 100, 000Å. Notice the lattice sum for ellipsiod with principal axes on x-axis and y-axis equal to 2000Å is just protruding on top right corner It is observable that direct summation series sharply converges for cylindrical shape then closely following it is ellipsiodal shape and finally sphere is actually out of discussion. This is so because other two shapes have unfolded much lower convergence point for relatively similar dimensions. It is favorable to choose, for the satisfactory convergence condition, the values of three principal axes such that any change on them, either increment or decrement, has no impact on the summation series value up to 4th decimal place. The graphs above depicted variation up to 200, 000Å in z-axis intercept, this was actually not sufficient for the case of ellipsiod to meet the convergence condition contention. Thereafter for this case variation in z-axis intercept was extended up to 400, 000Å. Moreover even in the case of cylinder when it’s x-axis and y-axis intercepts were equal to 1500 the lattice sum was off by 1 in it’s 4th decimal place, compared to when intercepts equals to 500Å and 1000Å. So for this particular case of cylinder as well z-axis intercept was extended up to 400, 000Å. To be more succinct and be comprehensive a graph and a table is presented in Fig 6.3. 21 6.3. JUXTAPOSITION Preamble for energy evaluation [a\ 2.1903 2.1902 lattice sum 2.1901 2.19 2.1899 cylinder, R=500Å cylinder, R=1000Å cylinder, R=1500Å ellipsoid, XY principle axes=500Å ellipsoid, XY principle axes=1000Å ellipsoid, XY principle axes=2000Å sphere, Radius=2000Å 2.1898 2.1897 2.1896 2.1895 200000 250000 300000 350000 z intercept [Å] 400000 Figure 6.3: A different perspective to above graph, it shows z-axis intercept from around 200, 000Å upto 400, 000Å . (The tables below are incomplete for the complete data please look for appendix ) Remember that srxy is connoting both x-axis and y-axis intercepts srz is same only for z-axis srz Lat. Sum(srxy = 500) Lat.Sum(srxy = 1000) Lat. Sum(srxy = 1500) 180000 190000 200000 380000 390000 400000 2.189618 2.189617 2.189615 - 2.189668 2.189661 2.189656 - 2.189751 2.189736 2.189723 2.189635 2.189634 2.189632 Table 6.1: The summation series for the case of cylinder . 6.3 Juxtaposition The graph and table were very useful in grasping the slow convergence of series. Necessity lies on singling out values of principal axes to obtain best converged series. Following bullets are just for this purpose. 1. The narrower or thinner the cross section of object is, the faster the series convergence is. The contrary is true for broader cross section of the object. 22 6.3. JUXTAPOSITION Preamble for energy evaluation [a\ srz Lat. Sum(srxy = 500) Lat.Sum(srxy = 1000) Lat. Sum(srxy = 2000) 180000 190000 200000 380000 390000 400000 2.189721 2.189709 2.189699 2.189629 2.189627 2.189626 2.190303 2.190241 2.190187 2.189786 2.189778 2.189770 2.191844 2.191643 2.191468 2.190200 2.190173 2.190147 Table 6.2: The summation series for the case of ellipsiod . 2. The more elongated is the height of object, the more accurate is the series convergence. 3. Unsurprisingly the best convergence among available data is for cylinder with base radius 500Å and height 200000Å 4. Henceforth for upcoming calculations, to pick cylindrical specimen with small base area and elongated height is compelling. Technicality apart actually ellipsiodal specimen isn’t that far off from it’s cylindrical counterpart. The difference starting from 5th decimal place is neglected henceforth a quick skim over above data is enough to say ellipsiodal specimen with x and y principal axes both equal to 500Å and z principal axis equal to 400000Å concurrently produces identical result alongside the cylindrical specimen with base radius 1500Å and height 400000Å. So the inference, hereby, shall be cylindrical specimen with base radius 1500Å and height 400000Å is chosen. Well discussion has not died here, the detail in data for convergence further tells about equivalence between distinct shapes and with their dimension. The result produced by cylindrical specimen with base radii 500Å and 1000Å and height 200000Å is almost identical to that of cylindrical specimen with base radius 1500 and height 200000Å. The last note for this discussion is, the ellipsiodal object with x and y principal axes 500Å and z principal axes 400000Å has identical convergence along with cylindrical object with base radii 500Å, 1000Å and 1500Å for first two, height is 200000Å and for the last, height is 400000Å. 23 Chapter 7 Ground state configuration and energy 7.1 Approach The data in preceding chapter are talking a lot! The nature of convergence was observed in all cases, by the virtue of which the geometrical object and it’s cut-off dimensions was chosen. Three compounds are considered. First two will be the case for replication of reference [21]. The first two R-element in LiRF4 are Terbium (Tb) and Erbium (Er). Moving onto it, the specimen conceived was sphere with radius 500Å. The dipolar energy values and the configurations were sought thereafter ground state configuration and energy can be isolated from results. An important note here is The energy was calculated by neglecting the demagnetization field Hd extensively discussed in chapter 3. This imitation is to testify the output/results of codings provided in appendix section is infallible/impeccable/unerring or simply put error-free. For such case the output must be concurrent to that displayed in referenceskm. For this report the illustrations of energy calculation via Fortran95 is in Appendix B and for complete set of data refer to Appendix A. Once the results for first two compounds is found to be concurrent to that of mentioned reference then Gadolinium can be placed as R-element. For that matter, a cylinderical specimen of LiGdF4 was conceived with dimensions like base radius equal to 1, 500Å and the height equal to 400, 000Å. So that being said following section will display the outcomes of codings alongside short remark. During calculations lattice parameters and g-tensor values are required and are tabulated right below. In this opportunity let it be cleared that the factor denoted by p, S 2 µ2B g αα g ββ clinging in front of dipolar interaction energy term in equation (5.1) was calculated. The table 7.1 contains useful values of important parameters of energy calculations. Relevant to it are bullets below as well. 1. Whenever equation (5.1) expressed in for S.I. units should bear magnetic permeability(for vacuum) factor µo in denominator. 2. Next note is the lattice summation was done in Å units and is necessary to give a heed. 3. In addition the energy values after calculation is expressed in terms of ◦ K, this forces division by Boltzmann constant. 4. The term ’S’ stands for the electron spin quantum number and it varies significantly among elements. The data is tabulated in Table 7.1. 24 7.2. CASE OF TERBIUM Ground state configuration and energy [a\ Compound c(Å) a(Å) g|| g⊥ Shell S L Terbium Erbium Gadolinium 10.873 10.70 10.97 5.181 5.162 5.219 17.7 3.137 2.0 0.0 8.105 2.0 4f 8 4f 11 4f 7 3 3 6 0 3 2 7 2 Table 7.1: The data can be acquired from references [6, 17, 21, 28]. 7.2 Case of Terbium To be consistent with reference [21] the specimen LiT bF4 compound is spherical in shape with radius 500Å. The following data are extracted from the coding done for this report and should be in agreement with the mentioned reference to be correct. Note that the spin value was taken 21 instead of 3 as shown in table 7.1 since the reference as well, takes the former value. 7.2.1 Data H12,12 0 .. . .. . = 0 0 0 0 ··· .. . .. . ··· ··· ··· ··· ··· .. . .. . 10 −5 −8 −5 ··· .. . .. . ··· 0 · · · · · · · · · −5 −8 −5 10 −5 −8 −5 10 −5 −8 −5 10 ··· The Hamiltonian matrix after multiplying with respective g-factor and be careful not by the prefactor p! Config. 1 2 3 4 5 6 7 8 9 10 11 12 Eigenvalues -1.43492 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.13804 3.03110 3.03110 Table 7.2: Twelve configurations and it’s corresponding energy value in ◦ K . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Eigen-Vector = 0 0 0 0 0 0 0 0 −50 49 −50 −49 −49 −50 −49 50 −49 50 49 50 −50 −49 50 −49 The four columns of above matrix are four eigenvectors corresponding to four non-zero eigenvalues in table 7.2. 25 7.3. CASE OF ERBIUM Ground state configuration and energy [a\ 7.2.2 Inference The data for Terbium, calculated for this report produced ground state with • Energy Eigenvalue without account of demagnetization field -1.43492◦ K. On the other hand the reference [21] provides -1.6824 ◦ K the missing account of demagnetizing field in this report should be the difference. • Energy Eigenvector along negative Z-axis. The ground state configuration is ferromagnetic ordering This conclusion is in agreement with the reference [21]. In addition there are two degenerate states with layered anti-ferromagnetic configuration along Z-axis and one anti-ferromagnetic state. 7.3 Case of Erbium To be consistent with reference [21] the specimen LiT bF4 compound is spherical in shape with radius 500Å. The following data are extracted from the coding done for this report and should be in agreement with the mentioned reference to be correct. Note the spin value for this case too was 21 instead of 32 . 7.3.1 Data The Hamiltonian matrix after multiplying with respective g-factor and be careful not by the prefactor p! As it is stated before the Hamiltonian matrix is 12 × 12 and features block diagonal matrices. There are three blocks so their dimension is 4 × 4. Three blocks represent for the x, y and z axis. They are presented below. −18 −13 13 30 −13 −18 30 13 H4,4 = 13 30 −18 −13 30 13 −13 −18 −18 30 13 −13 30 −18 −13 13 H4,4 = 13 −13 −18 30 −13 13 30 −18 5 −2 −4 −2 −2 5 −2 −4 H4,4 = −4 −2 5 −2 −2 −4 −2 5 Config. 1 2 3 4 5 6 Eigenvalues -0.75396 -0.75396 -0.22127 -0.22127 -3.56830E-02 6.6296E-02 Table 7.3: First six configurations and their corresponding energy value in ◦ K . 26 7.4. CASE OF GADOLINIUM Ground state configuration and energy [a\ Config. 7 8 9 10 11 12 Eigenvalues 9.59106E-02 9.59106E-02 0.11362 0.11362 0.1190 0.1190 Table 7.4: last six configurations and their corresponding energy value in ◦ K . 0 500 0 −499 0 499 0 500 0 −499 0 −500 0 −500 0 499 −499 0 −500 0 500 0 499 0 Eigen-Vector = 0 −499 0 500 −499 0 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The four columns of above matrix are the first four eigenvectors corresponding to the respective first four lowest eigenvalues in table 7.3. 7.3.2 Inference The data for Erbium, calculated for this report produced ground state with • Energy Eigenvalue without account of demagnetization field -0.75396 ◦ K. On the other hand the reference [21] provides -0.3654 ◦ K the difference between them is obscure. • Energy Eigenvector along both Y and X axes. The two possible configurations are layered anti-ferromagnetic ordering. The ground state with energy value -0.75396 ◦ K has degeneracy of 2 henceforth there are two distinct corresponding eigenvector, in physical terms two distinct ground state configurations. Both of them are ’Layered antiferromagnetic’ along X axis and Y axis in separate occasions. 7.4 Case of Gadolinium So far two distinct R-element ions have been scrutinized and the inference was concurrent to previous research works. Hereby Gadolinium was chosen as R-element ion and this time the spin value for Gd(III) ion was taken as 27 (This should change the factor p). A cylindrical specimen with base radius 1, 500Å on XY plane and the height 400, 000Å along Z-axis. Followings section reveals the outcomes. 7.4.1 Data The Hamiltonian matrix is 12 × 12 and features block diagonal matrices. There are three blocks so their dimension is 4 × 4. Three blocks represent for the x, y and z axis. They 27 7.4. CASE OF GADOLINIUM Ground state configuration and energy [a\ are presented below. −327 −135 H4,4 = 600 1063 −327 1063 H4,4 = 600 −135 655 −928 H4,4 = −1200 −928 −135 600 1063 −327 1063 600 1063 −327 −135 600 −135 −327 1063 600 −135 −327 −135 600 −135 −327 1063 600 1063 −327 −928 −1200 −928 655 −928 −1200 −928 655 −928 −1200 −928 655 Config. 1 2 3 4 5 6 Eigenvalues -2.40104 -2.12704 -2.12704 -0.65606 -0.65606 0.27078 Table 7.5: First six configurations and their corresponding energy value in ◦ K . Config. 7 8 9 10 11 12 Eigenvalues 0.27078 1.20052 1.20052 1.31213 1.85626 1.85626 Table 7.6: last six configurations and their corresponding energy value in ◦ K . 0 500 0 500 0 500 0 −499 0 −500 0 499 0 −499 0 −500 0 0 −499 0 0 499 0 0 Eigen-Vector = 0 499 0 0 0 0 −499 0 −500 0 0 0 −500 0 0 0 −500 0 0 0 −500 0 0 0 The four columns of above matrix are first four eigenvectors corresponding to respective first four lowest eigenvalues in table 7.5. 7.4.2 Inference The data for Gadolinium, calculated for this report produced ground state with • Energy Eigenvalue without account of demagnetization field -2.40104 ◦ K. 28 7.4. CASE OF GADOLINIUM Ground state configuration and energy [a\ • Energy Eigenvector along negative Z-axis. The ground state configuration is ferromagnetic ordering. In addition there are two degenerate states with layered anti-ferromagnetic configuration along both X-axis and Y-axis. Finally one anti-ferromagnetic state along X-axis. 29 Summary The magneto-caloric effect of Gadolinium compounds have been well observed and tested for, this report has incorporated it’s concept. Since the total spin, which in turn produces magnetic moment of the ion, of Gd(III) ion is significant the inter-ionic dipole-dipole interaction was into account. There must be distinct amounts of energy for distinct configurations of the magnetic dipoles. The objective of internship was to unfold the ground state configuration and it’s associated energy value of Gadolinium compound, in the interest of this report the compound is GdLiF4 . This was to be acheived by studying the relevant subject matters. Preliminaries have been already presented in Chapter 1 of this report. To continue, the study progressed through magneto-calotic effect, Gadolinium’s compound and dipolar interactions. The codings was done for calculations of summation series (It appears in the expression of energy due to inter-ionic dipolar ineractions, it’s clear in chapter 5). This series converges upon conditions imposed, details are in chapter 6, and hence the graphs help to optimize dimensions and geometrical shape of the specimen for minimum ground sate energy. In chapter 7, two trials were carried out to scrutinize the codings. it’s outcomes were compared to a well published reference [21]. Afterwards the codings were ran for Gadolinium compound (GdLiF4 ). At the end, the ground state was observed to be ferro-magnetic ordering along negative z-axis and the energy value was -2.40104 ◦ K. 30 Appendix A Program Coding The programming language used was Fortran95. The codings were done for convergence of lattice sum along the chosen Z-axis dimension of the geometrical objects. Moreover for construction of energy matrix subsequently the calculation of energy eigen values and energy eigenvectors. There are respective terminologies and the constraint to comply in order to grasp the notion of summation series. Firstly the terminologies are mentioned within the coding, just search for exclamation sign ’ !’ or take a glance over coming bullets below. Secondly the ions subjected for summation series must be inside the geometrical object hence a the geometrical constraint should be formulated and complied on. Obviously only the raw equation of object in 3D should be used for the shake of constraint for example the equation of sphere x2 + y 2 + z 2 ≤ r2 should be used in the case of sphere. Next, for all the studied geometrical object the Z-axis is chosen to be subject of increment in order to incur convergence in summation series. In the case of sphere only magnitude of the radius is increased and then convergence can be observed, it’s as simple as it can be. The terminologies and the constraints to satisfy pertaining to the respective geometric object 1. For sphere the radius is expressed by ’sr’ and the only geometrical constraint to satisfy is x2 + y 2 + z 2 ≤ r2 . 2. For ellipsiod the x and y principal axes are expressed by ’sr’ and the z principal axis sr by ’srz’. In addition to this ’zc’ is stands for the ratio srz . The only geometrical 2 z 2 2 2 constraint to satisfy is x + y + zc2 ≤ sr . 3. For cylinder the base on XY plane has radius expressed by ’sr’ and the height along z-axis by ’zr’ and the only geometrical constraint to satisfy is x2 + y 2 ≤ r2 and z ≤ zr. A.1 A.1.1 Lattice summation series Sphere program lattice implicit none real*8, parameter :: sr =500.0 ! the radius of Sphere real*8, parameter :: a = 5.181 ! the lattice constant along x and y directions real*8, parameter :: c = 10.873 ! the lattice constant along z direction real*8, dimension(3) :: t ! This is the translation vector running through the la 31 A.1. LATTICE SUMMATION SERIES Program Coding [a\ ttice points integer i,j,k,nx,ny,nz,l real*8 tot,zc,srz,s,bm,an,bol,p,mp open(1,file =’etbz.txt’, status =’unknown’) tot =0.0 ; t = 0.0; nx = int(sr/a); ny = int(sr/a); nz = int(srz/c) do l = 1, 5 sr = l*1000. do i = -nx, nx t(1) = i*a do j = -ny, ny t(2) = j*a do k = -nz, nz t(3) = k*c if((t(1)**2 +t(2)**2+ t(3)**2) .le. sr**2) then if( i .eq. 0 .and. j .eq. 0 .and. k .eq. 0) then cycle else end if tot =tot + ((-3)*t(3)*t(3) +(dot product(t,t)))/((sqrt(dot product(t,t)))**5) !tot = tot + (t(1)**2-t(2)**2)/(sqrt(dot product(t,t))**5) !tot = tot +((t(1)*t(2))/(sqrt(dot product(t,t))**5)) should be 0.0 !tot = tot +((t(2)*t(3))/(sqrt(dot product(t,t))**5)) should be 0.0 !tot = tot +((t(1)*t(3))/(sqrt(dot product(t,t))**5)) should be 0.0 end if end do end do end do write(1,*) sr, tot tot =0.0 end do end program lattice A.1.2 Ellipsiod program lattice implicit none real*8, parameter :: sr = 500.0 ! sr stands for the x and y principal axes of ellipsiod real*8, parameter :: a = 5.181 ! the lattice constant along x and y directions real*8, parameter :: c = 10.873 ! the lattice constant along z direction real*8, dimension(3) :: t ! This is the translation vector running through the lattice points integer i,j,k,nx,ny,nz,l 32 A.1. LATTICE SUMMATION SERIES Program Coding [a\ real*8 tot,zc,srz ! srz stands for z principal axis of ellipsiod open(1,file =’etbz.txt’, status =’unknown’) tot =0.0 ; t = 0.0; nx = int(sr/a); ny = int(sr/a) do l = 1, 20 srz = l*10000.0 ;zc = srz/sr; nz = int(srz/c) do i = -nx, nx t(1) = i*a do j = -ny, ny t(2) = j*a do k = -nz, nz t(3) = k*c if((t(1)**2 +t(2)**2+ t(3)**2/(zc**2)) .le. sr**2) then if( i .eq. 0 .and. j .eq. 0 .and. k .eq. 0) then cycle else end if tot =tot + ((-3)*t(3)*t(3) +(dot product(t,t)))/((sqrt(dot product(t,t)))**5) !tot = tot + (t(1)**2-t(2)**2)/(sqrt(dot product(t,t))**5) !tot = tot +((t(1)*t(2))/(sqrt(dot product(t,t))**5)) should be 0.0 !tot = tot +((t(2)*t(3))/(sqrt(dot product(t,t))**5)) should be 0.0 !tot = tot +((t(1)*t(3))/(sqrt(dot product(t,t))**5)) should be 0.0 end if end do end do end do write(1,*) srz, tot tot =0.0 end do end program lattice A.1.3 Cylinder program lattice implicit none real*8, parameter :: sr =1500.0 ! The base radius of cylinder real*8, parameter :: a = 5.181 ! the lattice constant along x and y directions real*8, parameter :: c = 10.873 ! the lattice constant along z direction real*8, dimension(3) :: t ! This is the translation vector running through the lattice points integer i,j,k,nx,ny,nz,l,n real*8 tot,s,srz,zr ! zr stands for the height of cylinder along the chosen dimension Z 33 A.2. HAMILTONIAN MATRIX Program Coding [a\ open(1,file =’cysrxy1500.txt’, status =’unknown’) tot =0.0 ; t = 0.0; zr =0.0; nx = int(sr/a); ny = int(sr/a) do l = 1, 20 zr = l*10000.0; nz = int(zr/c) do i = -nx, nx t(1) = i*a do j = -ny, ny t(2) = j*a do k = -nz, nz t(3) = k*c if((t(1)**2 +t(2)**2) .le. sr**2 .and. t(3) .le. zr) then if( i .eq. 0 .and. j .eq. 0 .and. k .eq. 0) then cycle else end if tot =tot + ((-3)*t(3)*t(3) +(dot product(t,t)))/((sqrt(dot product(t,t)))**5) !tot = tot + (t(1)**2-t(2)**2)/(sqrt(dot product(t,t))**5) !tot = tot +((t(1)*t(2))/(sqrt(dot product(t,t))**5)) should be 0.0 !tot = tot +((t(2)*t(3))/(sqrt(dot product(t,t))**5)) should be 0.0 !tot = tot +((t(1)*t(3))/(sqrt(dot product(t,t))**5)) should be 0.0 end if end do end do end do write(1,*) zr, tot tot =0.0 end do end program lattice A.2 Hamiltonian Matrix The lattice summation series should have a nearly converging value for the specified indices in equation (5.1). The program code to yield the summation series values is provided in following section. Here the program code for construction of matrix along with calculation of energy eigenvalues and eigenvectors is presented. program ter real*8, allocatable,dimension(:,:) :: m real*8, allocatable,dimension(:) :: n real*8, allocatable,dimension(:) :: W real*8, allocatable,dimension(:) :: WORK integer i, j, LWORK, INFO CHARACTER, PARAMETER :: jobz=”V” CHARACTER, PARAMETER :: uplo=”U” real*8 s,bm,an,bol,p,mp 34 A.2. HAMILTONIAN MATRIX Program Coding [a\ ! The GLOBAL PREFACTOR FOR THE CALCULATION OF ENERGY VALUE bm = 9.274009*(1E-24); an = 1E-10; bol = 1.380650*(1E-23); mp = 1E-7; s = 0.5 ! FOR GADOLINIUM SPIN MUST BE S = 7/2.0 p = mp*(bm*s)**2/((an**3)*bol) write(*,*) ’the prefactor is’, p LWORK = 50 allocate(m(1:12,1:12)) allocate(n(1:12)) allocate(W(1:12)) allocate(WORK(1:LWORK)) n = 0.0; m = 0.0 open(10, file = ’v.txt’, status = ’new’) ! It is eigen vector of H matrix open(20, file = ’m.txt’, status = ’new’) ! It is H matrix open(30, file = ’e.txt’, status = ’new’) ! It contains eigen values of H matrix in ascending order ! FOR DIAGONAL ELEMENTS do i = 9, 12 do j = 9, 12 if(i == j) then m(i,j) = 10.860413220309056 end if end do end do !———————————————– m(9,10) = -5.7357941217199366 m(9,11) = -8.6030275279961259 m(9,12) = -5.7357941217197874 m(10,11) = -5.7349347936914663 do j = 10, 11 if(j == 10) then !m(j,j+1) = m(9,12) m(j,j+2) = m(9,11) else if(j ==11) then m(j,j+1) = m(9,10) end if end do !———————————————– !SYMETRIZATION OF MATRIX !n = TRANSPOSE(m) !m = m + n do i = 1, 12 do j = 1, 12 !if(i .eq. j) then m(j,i) =m(i,j) !m(i,j) = m(i,j)/2.0 35 A.2. HAMILTONIAN MATRIX Program Coding [a\ !end if end do end do m = m*p !———————————————– do i = 1, 12 write(20,*) (int(m(i,j)), j = 1, 12) end do !LIBRARY USE call DSYEV(jobz,uplo,12,m,12,W,WORK,LWORK,INFO) !———————————————– ! FOR FURTHER ANALYSIS OF MATRIX WE MAKE OUPUT AND SEE THE RESULT OF [email protected]@@@@@@@@@@@@@@@@@@@@@ do i = 1, 12 write(30,*) i, W(i) write(10,*) (int(100*m(i,j)), j = 1, 12) end do !———————————————– do j = 1 , 12 n(j) = m(j,1) end do write(*,*) ’dot product is’, dot product(n,n) write(*,*) ’The energy of lattice interaction in degree K is,’, W(1) write(*,*) INFO end program ter 36 Bibliography [1] Electronic configuration of Gd. https://pilgaardelements.com/Gadolinium/ AtomProperties.htm. Retrieved: 22/03/2017. [2] Magnetic Refrigerator Successfully Tested ames laboratory news release, ames laboratory. https://www.eurekalert.org/features/doe/2001-11/dl-mrs062802.php, 2001. Accessed: 2017-06-23. [3] Gadolinium atom properties michael pilgaard. https://pilgaardelements.com/ Gadolinium/AtomProperties.htm, 2016. Accessed: 2017-06-28. 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