2 . general properties of ferrites . 2.1 . ferrite structure . apart from ferromagnetic metals , a number of chemical compounds ( e.g ferrites , garnets , plumbites and perovskites ) exhibit ferromagnetic properties . of these compounds ferrites have to date proved to be the most important from the standpoint of microwave applications . as the majority of ferrites crystallise with a cubic structure , similar to the mineral spinel , ( magnesium aluminate &amp;formula; ) , the term ferromagnetic spinel is sometimes used to describe those ferrites which exhibit magnetic properties . the general chemical formula of a ferrite is &amp;formula; where m represents a metallic cation . it is found that a spinel crystal structure is only formed if the ionic radius of the cation m is less than about 1 A&amp;ring; . if it is greater than 1 A&amp;ring; then the electrostatic Coulomb forces are insufficient to ensure the stability of the crystal . for example Ca++ ( ionic radius 1.06 A&amp;ring; ) does not form spinel crystals , while Mn++ ( ionic radius 0.91 A&amp;ring; ) does . the cation m is generally divalent , but other valencies are possible if the number of anions is doubled , e.g lithium ferrite &amp;formula; . the ions forming ferrites of practical importance are Ni++ , Mn++ , Fe++ , Co++ , Cu++ , Zn++ , Cd++ , Li+ , Mg++ . the spinel unit cell ( see fig 2.1 ) consists of a close packed cubic array of 32 oxygen anions , between which there are 96 spaces or interstices , 24 of which are filled with a cation , the remaining 72 being empty . the sites occupied by the cations are of two kinds known as tetrahedral or a sites and octahedral or b sites . the a sites of which eight are occupied , are surrounded by four oxygen anions and the b sites of which sixteen are occupied , are surrounded by six oxygen anions . when the chemical formula is written , the ions in the b sites are often enclosed in brackets to indicate their position , e.g Fe ( NiFe ) O4 for nickel ferrite . it might seem at first sight that the most likely arrangement of the cations would be with M++ ions on the a sites and Fe2+++ ions on the b sites but in practice three types of spinel can be distinguished . ( 1 ) . normal spinels in which M++ ions occupy the a sites and Fe2+++ the b sites . ( 2 ) . inverse spinels in which M++ ions occupy the b sites together with half the Fe+++ ions , the other half being on the a sites . ( 3 ) . random spinels in which both M++ ions and Fe+++ ions occur on the a and b sites . the preference of certain ions for a or b sites is of importance , as it is found that in general normal ferrite spinels are paramagnetic while inverse spinels are ferromagnetic . many ions show no strong preference for a particular site , this being especially true for those ions with a noble gas configuration such as Li+ , Mg++ , Al+++ and also those with a half-filled 3d electron shell e.g Fe+++Mn++ . where there is no strong site preference the most stable cation distribution can be calculated from a static model of charged spheres . of the remaining ions in ferrites which are of microwave interest Zn++ , has a preference for a sites while only Ni+++ and Cr+++ have a strong preference for b sites . when two or more cations are present , the distribution of ions with weak site preference may be affected by the presence of an ion with a strong site preference . most ferrite spinels can form solid solutions with each other in any proportion . this arises since there is a greater probability of a solid solution when two ferrite spinels are reacted together , than there is of the formation of separate crystals of the two spinels . a well-known example of a solid solution is nickel zinc ferrite , &amp;formula; , where a can take any value between 0 and 1 . unless great care is taken in the manufacture , the final ferrite formed is not exactly that corresponding to the proportions of raw materials used . this is because most ferrites can take up oxides into solution without forming a second phase and thus give rise to non-stoichiometric ferrite . in particular the ability of most ferrites to take up Fe2O3 in solution is important . in the preparation of ferrites the component oxides are reacted at high temperatures . during this sintering process there is a tendency for most ferrites to give off oxygen , as the equilibrium pressure in this reaction is often greater than one atmosphere and increases rapidly with temperature . this gives rise to an oxygen deficiency in the final product and to the formation of ferrous ions . the presence of ferrous ions in microwave ferrites is undesirable however , since it causes increased dielectric and magnetic loss as is discussed in this chapter and chapter 4 . for this reason , compounds are often made iron deficient , great care being taken to avoid loss of oxygen during sintering . 2.2 . preparation of ferrites . ferrites are prepared by a ceramic technique which involves sintering the component oxides at temperatures between 1000 &amp;deg; and 1450 &amp;deg; C . the stages in the preparation of ferrites are listed below : - raw materials &amp;symbol; decomposition to oxide &amp;symbol; milling &amp;symbol; presintering ( partial reaction ) &amp;symbol; remilling &amp;symbol; pressing and extruding to shape &amp;symbol; final sintering &amp;symbol; grinding to shape . a number of raw materials can be used in the manufacture of ferrites ; these include oxides , carbonates , oxylates and nitrates . the last three compounds decompose to oxides on heat treatment , and are thus prepared in situ at a temperature near to that at which solid state reactions commence . this process should favour the formation of good quality homogeneous materials . for example in the case of MgMn ferrites it has been reported that the use of nitrates gives rise to better microwave properties . an explanation is that the high decomposition temperature of the nitrates and the presence of nitrogen oxides help to prevent the formation of ferrous ions during the sintering process . the raw materials are first milled , usually in a steel ball mill , to give a homogeneous mixture of very fine particles . the process is generally carried out with the raw materials in a slurry of methylated spirit or any other liquid which is easily removed after milling . the evaporation of the methylated spirit is carried out rapidly to avoid any heavier particles separating out . the mixture of raw materials is then pre-fired at a temperature some 200 &amp;deg; C below its final firing temperature . this process causes partial reaction of the constituents and helps to reduce shrinkage during final sintering . the presintered powder is then remilled . two methods of moulding the powder into shape prior to the final sintering are commonly employed ; die pressing and extrusion . for die pressing a small quantity of binder is added to the powder so that when the sample has been pressed to shape , it can be handled relatively easily . to avoid the possibility of contamination of the sintered ferrite , distilled water has been used as a binder , although for certain shapes ( e.g rods ) organic wax emulsions have been found more satisfactory . gentle heating to remove the binder is necessary as violent volatilisation could cause the sample to crack . a moulding pressure of between 2 and 10 tons &amp;sol; sq in ensures a uniform end product without the risk of forming laminates in the pressed sample . for satisfactory extrusion a higher percentage of binder is required than for moulding . a solution of wax in petroleum has been used as a binder for extrusion and by careful choice of extrusion orifice very dense samples may be produced . as high a density as 99 % has been achieved under special conditions . extruded samples , in general , however are not as dense or uniform as those produced by die-pressing . the principal use of extrusion techniques has been for the manufacture of long thin rods , a shape often required in microwave applications . rods as long as 12 in x 0.04 in diameter have been produced by this method . the properties of the final product depend critically on the sintering process and the closest control of sintering time , temperature and atmosphere is required . generally , the sintering process is carried out at a temperature between 1000 &amp;deg; and 1450 &amp;deg; C for between 4 hours and 24 hours , depending on the ferrite . ferrites containing lithium and cadmium are usually sintered at lower temperatures due to the volatility of LiO and CdO2 while those containing nickel , cobalt and magnesium are sintered at the highest temperatures . by sintering for a long time at high temperatures , a uniform final product with a minimum of air pores can be obtained . the near absence of pores is a requirement for certain microwave ferrites . this is discussed in greater detail in chapter 3 . as already mentioned , however , the oxygen equilibrium pressure increases rapidly with increasing temperatures and this sets a limit on the maximum sintering temperature that can be used without reduction of ferric iron to ferrous iron . the porosity of a particular polycrystalline ferrite sample is usually quoted with reference to its X-ray or single-crystal density . the X-ray density is determined from measurements of the spinel lattice constant and table 2.1 gives values for a number of commonly used ferrites , for which the lattice constants are known . the density of typical polycrystalline pressed samples is between 80 % and 95 % of X-ray density , though figures as high as 99 % have been achieved . during sintering , shrinkage of the ferrite sample occurs . this may be controlled by careful preparation and by ensuring a uniform temperature over the sample , although the final shape may not have the tolerances required in practice . sintered ferrites , being ceramic in nature , require special methods of shaping . cutting can be carried out by use of a thin diamond slitting wheel or by use of an ultrasonic machine with a knife edge cutting head . an accurate finish can then be obtained by surface grinding with a carborundum wheel . the growth of single crystals of ferrite was originally of interest mainly to the physicist , as the crystals produced were too small for use in microwave applications . however , the development of non-linear devices employing small single-crystal samples has modified this situation , although they are still extensively used for the study of the fundamental properties of ferrites . two principal methods have been used for the formation of single crystals ; the borax melt and the flame fusion process . in the borax melt process , the constituent oxides of the ferrite are dissolved in a flux of molten borax by heating the mixture to between 1300 &amp;deg; and 1400 &amp;deg; C and maintaining this temperature for several hours . the melt is then cooled at a few degrees per hour until crystals start to form , or alternatively the flux is evaporated at a constant rate . a disadvantage of the method is that the borax vapour evolved is very corrosive and destroys most refractory materials , which necessitates the use of special furnace equipment . crystals of linear dimensions of about 1 cm can be obtained by this method . in the flame-fusion process constituent oxides are mixed in the correct proportions and sprinkled into an oxy-hydrogen flame . crystals of reasonable length , e.g 1-2 cm can then be grown on a refractory rod held in the flame . it is , however , very difficult to control the exact chemical composition of the crystal obtained by the flame-fusion process . 2.3 . magnetic properties of ferrites . the purpose of the following section is to provide an elementary account of the magnetic properties of ferrites , together with enough background material to enable the reader to place the section in perspective . it is stressed that since the object is to equip the microwave user of ferrites with a knowledge of their magnetic properties , the finer details of the subject must be sought in the bibliography provided . consideration will first be given to the origin of magnetism in electrons , atoms and ions , choosing as examples of the latter , elements which occur in ferrites . the mechanisms of para- , ferro- and ferri-magnetism will then be explained and reference made to the temperature behaviour of the saturation magnetisation of certain ferrites . in ferrites , one is principally concerned with the phenomenon of ferrimagnetism which will be treated in greater detail . 