Crystals: an introduction

some water vapor as on these chilled copper wires it grows in we can start by observing the growth of a crystal under the microscope we will use common salt sodium chloride dissolved in water first we place a drop of the solution on a glass slide water will evaporate and the salt will here it comes crystallizing in regular shapes all with rectangular outlines the fact that they're nearly Square in cross-section and stay that way as they grow means that the crystal is accumulating sodium and chlorine atoms about as rapidly on one face as on another if the crystals could keep on growing uninhibited by obstructions or lack of solution they would grow large like this one as early as 8 1997 crystallographers suspected that this regularity resulted atoms in the crystal then in 1912 this was proved through the use of x-rays which fall on the crystal are cooperatively scattered from the various atoms to give a defract beam a defract beam will occur only when the brag equation is satisfied here Lambda is the wavelength of the incident x-rays D the spacing between planes of atoms and Theta the angle made by the M with the atomic planes the defract beam flashes out when the planes make the angle Theta with the x-rays by measuring this angle we can determine the spacing between the planes of atoms x-ray defraction is the most powerful tool the crystallographer uses for studying crystal structure the patterns made by the diffracted x-rays are recorded by various types of defraction cameras this camera has inside it the sample of sodium chloride which has been placed in the center and and a strip of photographic film placed around the inside of the camera wall the X-ray beam is directed onto the sample and the defract x-rays from the various planes of atoms are recorded on the film here we're using a powdered sample any grain of the powder that happens to be in the right position to satisfy the brag equation contributes a defract ray the developed film shows the the defraction pattern this pattern depends on the arrangement of the atoms the crystallographer measures the positions and intensi on the film and with the aid of chemistry and some Ingenuity determines the arrangement of the atoms in the crystal this Arrangement is called the crystal structure that may be complicated and take years to determine or simple like that of sodium chloride to illustrate the structure of sodium chloride we will use a purple one to represent the sodium atom and a larger blue one for the chlorine atom these spheres are about 50 million times the actual size of the atoms as determined from x-ray defraction measurements although we're using spheres to represent the atoms we don't really know exactly what shape they are it's spherical the sodium and chlorine atoms in the crystal are are arranged in a regular array in this manner during actual growth these atoms are being packed together in a continuous three-dimensional structure whose front is progressing forward here at about 5,000 of a millimeter per second this means that orderly layers of atoms are being added to the crystal at the rate of about 10,000 layers per second we cannot hope to illustrate this rate with with the model although the model shows every atom lying quietly we know that in the real crystal each atom is vibrating a little around its average position because of thermal motion a crystal then is composed of a regular arrangement of atoms repeated in all directions for millions of atoms and surprisingly this is true of most solids there are very few substances which are non orderly or amorphus for example such materials as brass polyethylene and wood if we examine these by means of X-ray defraction we find patterns which indicate that their atoms are arranged in an orderly manner some of them more orderly than others is indicated by the varying sharpness of the patterns we can see the crystals in this brass wing nut by looking at it with a metallographic microscope the surface we're going to examine has been prepared by polishing and etching it with acid the wing nut is composed of many crystals a poly crystallin body the contrasting color is due to the acid acting differently on the differently oriented crystals each crystal is grown until it interfered with its neighbors to form an irregular boundary on an atomic scale a grain boundary might look like this the regular array of atoms in each Crystal fits poorly against that of its neighbor in some of the crystals we see straight lines which are not Crystal boundaries but twin boundaries within this single Crystal grain a twin or a twined crystal is formed when two adjoining individuals share a p of atoms that fits right into the structure of both forming a straight twin boundary this special relationship is so intimate that crystallographers usually think of it as a single Crystal with a twinning plane rather than two separate crystals a crystal may be thought of as made up of identical blocks each related to its neighbor by Straight translation such a block is the repeat unit for the pattern and is called the unit cell we can choose it anywhere as long as we can build up the crystal with it by translation for sodium chloride this could be the unit cell with chlorine atoms at the corners and face centers of the cube or this with sodium atoms at the corners and face centers of the cube either would serve as the unit cell of this face centered cubic Crystal now let's examine the symmetry of arrangement of atoms around a vertical axis through the center of the cube if we rotate the crystal around this axis through 360° we will find ourselves looking at exactly the same arrangement of atoms four times in the complete Revolution we will mark one face to keep track of where we are 1 2 3 4 and we're back where we started this is an axis of fourfold symmetry of the Crystal and therefore an axis of fourfold symetry of all its properties because they depend on the arrangement of the atoms consider for example the property of cleavage cleavage is the tendency of a crystal to come apart between adjacent planes of atoms when struck if we take this large sodium chloride Crystal place a blade on it at any position on the face so long as we choose the right direction and strike the back of the blade sharply we can cleave the crystal if this is where the crystal came apart then because of the four-fold Symmetry axis there should be another cleavage here at 90° to the 1 the blade is now at 90° to the first cleavage plane and there is the other cleavage suppose we try to cleave it in any other direction not at right angles to the first cleavage plane the crystal simply shatters this cubic structure has two other four-fold axes of symmetry here and here though some cubic crystals do not have four-fold axis of symmetry all cubic crystals have axes of three-fold symmetry such an axis goes through opposite corners of the cell like this if we turn the model so that that axis is vertical we can check the three-fold Symmetry call The Marked face one two three and we're back where we started we can choose an axis through these other opposite corners and again we have an axis of threefold symmetry and another through here and another here four of them all together some crystals also have planes of symmetry we can illustrate the planes of symmetry in this cubic structure by cutting the cell in half first along the diagonal now if we bring a mirror up to its cut face the cell appears to be complete again this is because the two halves of the cell are mirror images of each other the plane between them is called a mirror plane of symmetry or simply a plane of symmetry similarly this half and its Mirror Image complete the structure and there's therefore another plane of symmetry through here this cubic structure has nine such SYM symmetry planes every property of this Crystal must obey all of these symmetry planes and also all of the axes of symmetry because they describe the symmetry of arrangement of the atoms in the crystal for example the property of elasticity consider the elastic strain resulting when you apply a stress along the direction of nearest neighbor atoms and compare it with the elastic strain when the stress is applied along this direction you would suspect that they are different it is in fact so the elastic constants in these two directions are different and of course the elastic constant measured in this direction is the same as that measured in this direction because the arrangement of atoms is the same if we consider the growth rate of the sodium chloride Crystal we realize that it too must be different in different directions if it were not the crystal would be spherical but the exterior appearance of a crystal does not necessarily reveal its true symmetry this quartz crystal appears to have an axis of sixfold symmetry around which the same type of face is repeated six times but is it really sixfold since quartz is a pazo el electric Crystal mechanical deformation will cause it to be electrically charged at the edges between the faces we can check the Symmetry by the use of this property if we press the edge of the crystal on the contact of an electrometer we can determine whether the charge is positive or negative the electrometer is discharged the first Edge is negative rotating around the axis in question to the next Edge positive another 60° around negative 60° around again positive now negative and finally positive alternately negative and positive now we record this on a diagram of a cross-section of the six-sided Crystal our axis of rotation is perpendicular to the diagram the positively charged Corners occur three times around the axis and so do the negative Corners therefore therefore we have an axis of threefold symmetry according to this test well which is right sixfold or three-fold since the pzo electric charge in these three directions is positive and in these three directions is negative there must be a corresponding difference in the structure in those directions that is the arrangement of the atoms must have three-fold symmetry the pzo electric effect is just more sensitive to the difference than the growth of the crystal is we can check the Crystal's true Symmetry and still another way by etching the surface this section has been cut from a large quartz crystal perpendicular to its major axis it has been polished and etched for about an hour with hydrochloric acid the surface is pitted and the edges of each pit form a triangle revealing a again the three-fold Symmetry in many cases the way in which acid attacks a crystal is a sensitive test of its symmetry we can see this symmetry in a model which shows part of the structure of quartz with the atoms spread apart each black sphere which represents a silicon atom is surrounded by four red spheres which represent oxygen atoms and these groups are connected by their Corners in spiral array looking down the axis of such a spiral we can see the three-fold symmetry of the atomic arrangement to which the Ed pits and the Pio electric properties are sensitive in order to study the pazo electric effect in relation to structure we had to work with a single Crystal what happens if we a poly crystallin body like this Sandstone which is made up of quartz grains if we test it on the electrometer we get no indication of a charge this is because the crystals in it are randomly oriented and though each grain May acquire charge the total effect averages out to nearly zero a parallel experiment which we can perform with iron and a strong magnet demonstrates that the magnetic properties of a single Crystal also depend on the crystal symmetry we place this disc of iron cut from a single Crystal between the poles of a magnet now this is difficult to do because the field pulled so strongly on the dis in Iron the atoms are arranged like this the unit cell is a cube usually chosen with atoms at the corners and one in the center a body centered cubic structure here we have spread the atoms apart to make them easier to see each iron atom behaves like a compass needle a magnetic dipole and there's a preferred direction of alignment of these dipoles it is along a four-fold symmetry axis of the cube in either direction along any one of the three they're all equivalent crystallographically now the disc has been cut perpendicular to one of the axes of fourfold symmetry with the other two axes therefore lying in the plane of the disc there are thus four directions of preferred orientation of the dipoles in the disc between the poles of the magnet the disc tends to line up with all those oriented magnetic dipoles parallel to the field if we try to turn the disc it resists until we turn so far that all the magnetic dipoles switch to the next preferred Direction and the Crystal aligns itself in the new position so that the dipoles are again parallel to the field the earlier arrows show the various directions which the dipoles had before they switched so there are four preferred directions in the disc next we place a disc cut from a piece of poly Crystal and iron between the poles of the magnet Mark a point for reference as we turn this desk we find that it can be placed in any direction and it will stay there this is because the individual crystals in it are randomly oriented therefore it is impossible for any single preferred direction to be maintained throughout the disc clearly to understand the relationship between magnetism and crystal structure we also need to work with single crystals although the models we've been using show a perfect array with every atom in its proper position we know that real crystals are not quite as perfect as this small defects occur here and there in the arrangement of the atoms in the regular accumulation of atoms one may be left out creating a vacancy If part of a plane is left out the atoms adjust as well as they can the part where the misfit is localized is called a dislocation this is the kind of defect we now believe to be largely responsible for plastic flow in metals it makes it easier for the atoms to change Partners sometimes a crystal may include an atom where it doesn't belong if the atom is small enough it may go in interstitially here the interstitial atom is carbon in an iron Crystal the atoms of the crystal it may substitutional atom is phosphorus in a silicon crystal the substitution of controlled impurity atoms during the growth of the semiconducting crystals silicon and geranium determines whether their conductivity is in type negative andn about the way in which the arrangement of the atoms affects the properties of solids today in many Laboratories single crystals for research are being grown from fluxes from Water Solutions and from the Melt from studies of these crystals will come a better understanding between physical properties and