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# IntroductiontoCondensedMatterPhysics PHYS624

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Chem 5939250 Character Tables for Point Groups Each point group has a complete set of possible symmetry operations that are conveniently listed as a matrix known as a Character Table As an example we will look at the character table for the C2V point group In 02V the order is 4 1 E1C121csvand1cs V Representations are subsets of the complete point group they indicate the effect of the symmetry operations on different kinds of mathematical functions Representations are orthogonal to one another The Characteris an integer that indicates the effect of an operation in a given representation E Chem 59 250 Character Tables for Point Groups The effect of symmetry elements on mathematical functions is useful to us because orbitals are mathematical functions Analysis of the symmetry of a molecule will provide us with insight into the orbitals used in bonding T twill it Notes about symmetry labels and characters A means symmetric with regard to rotation about the principle axis B means antisymmetric with regard to rotation about the principle axis Subscript numbers are used to differentiate symmetry labels if necessary 1 indicates that the operation leaves the function unchanged it is called symmetric 1 indicates that the operation reverses the function it is called antisymmetric Chem 59250 Symmetry of orbitals and functions A pZ orbital has the same symmetry as an arrow pointing along the zaxis y 39 39y E x C2 x 6V xz oquot yz No change symmetric 1 s in table Chem 59250 Symmetry of orbitals and functions A pX orbital has the same symmetry as an arrow pointing along the xaxis z z y I y E No change 0 E X 6 X2 X symmetric V 1 s in table Z Z y I 02 Opposite X X antisymmetric 6 Z V y 1 s in table x2y2z2 Xy xz Chem 59250 Symmetry of orbitals and functions A py orbital has the same symmetry as an arrow pointing along the yaxis z z CDC y E No change E X X s mmetric 6 Z V V y 1 s in table Z Z y 02 Opposite X G xz X antisymmetric V 1 s in table X2y2z2 Xy xz Chem 59250 Symmetry of orbitals and functions Rotation about the n axis Rn can be treated in a similar way The z axis is pointing out of the screen X X If the rotation is still In E No change the same direction eg C symmetric counter clockWise 2 1 s in table then the result is y y considered symmetric If the rotation is in the opposite direction ie X gt X clockWise then the 6V X2 Opposite result is considered antisymmetric antisymmetric G V yz 1 s in table x2y2z2 Xy xz Chem 59250 Symmetry of orbitals and functions cl orbital functions can also be treated in a similar way The z axis is pointing out of the screen x gt x E No change 02 symmetric 1 s in table y y X X 5V xz Opposite 6 Z antisymmetric V y 1 s in table x2y2z2 Chem 59250 Symmetry of orbitals and functions cl orbital functions can also be treated in a similar way The z axis is pointing out of the screen So these are representations of the view of the dZ2 orbital and dx2y2 orbital down the zaxis Lry symmetric y y 1 s in table E C2 X X No change G v yz Chem 59250 Symmetry of orbitals and functions Note that the representation of orbital functions changes depending on the point group thus it is important to be able to identify the point group correctly E z2 lt X2 22 X2 yz xy 4 Chem 59250 Symmetry of orbitals and functions D3h E 2 C3 3 02 oh 2 S3 3 6V A 1 1 1 1 1 1 1 x2 y2 z2 A 2 1 1 1 1 1 1 RZ More notes about symmetry labels and characters E indicates that the representation is doublydegenerate this means that the functions grouped in parentheses must be treated as a pair and can not be considered individually The prime and double prime in the symmetry representation label indicates symmetric or antisymmetric with respect to the Oh 0051200 05 y y y sin120 087 087 1 C2 x Cs 1 x x gt x 1 1 0 087 087 05 05 1 y Chem 59250 Symmetry of orbitals and functions oh E 803 602 604 302 L 684 886 36h 65d 042 A19 1 1 1 1 1 1 1 1 1 1 x2 y2 22 A29 1 1 1 1 1 1 1 1 1 1 E9 2 1 0 0 2 2 0 1 2 0 222 x2 yz x2 yz T1g 3 0 1 1 1 3 1 0 1 1 RX Ry R2 T29 3 0 1 1 1 3 1 0 1 1 xz yz xy A1U 1 1 1 1 1 1 1 1 1 1 A2U 1 1 1 1 1 1 1 1 1 1 Eu 2 1 0 0 2 2 0 1 2 0 T1U 3 0 1 1 1 3 1 0 1 1 x y 2 T2U 3 0 1 1 1 3 1 0 1 1 More notes about symmetry labels and characters T indicates that the representation is triplydegenerate this means that the functions grouped in parentheses must be treated as a threesome and can not be considered individually The subscripts g gerade and u ungerade in the symmetry representation label indicates symmetric or antisymmetric with respect to the inversion center i 3 Chem 59250 Character Tables and Bonding We can use character tables to determine the orbitals involved in bonding in a molecule This process is done a few easy steps 1 Determine the point group of the molecule 2 Determine the Reducible Representation F for the type of bonding you wish to describe eg o 7 75L 7 The Reducible Representation indicates how the bonds are affected by the symmetry elements present in the point group 3 Identify the lrreducible Representation that provides the Reducible Representation there is a simple equation to do this The lrreducible Representation eg 2A1 B1 Bz is the combination of symmetry representations in the point group that sum to give the Reducible Representation 4 Identify which orbitals are involved from the lrreducible Representation and the character table Chem 59250 Character Tables and Bonding Example the G bonding in dichloromethane CHZCIZ The point group is 02V so we must use the appropriate character table for the reducible representation of the sigma bonding 1 6 To determine 1 6 all we have to do is see how each symmetry operation affects the 4 6 bonds in the molecule if the bond moves it is given a value of 0 if it stays in the same place the bond is given a value of 1 Put the sum of the 1 s and 0 s into the box corresponding to the symmetry operation The E operation leaves everything where it is so all four bonds stay in the same place and the character is 4 1111 The 02 operation moves all four bonds so the character is 0 Each 6V operation leaves two bonds where they were and moves two bonds so the character is 2 11 Overall the reducible representation is thus CZV E 02 6V xz G v yz F G 4 0 9 9 0 Chem 59250 Character Tables and Bonding We now have to gure out what combination of symmetry representations will add up to give us this reducible representation In this case it can be done by inspection but there is a simple equation that is useful for more complicated situations sz E 02 6v X2 G v yz F 4 0 2 2 G Because the character under E is 4 there must be a total of 4 symmetry representations sometimes called basis functions that combine to make 1 6 Since the character under 02 is 0 there must be two ofA symmetry and two of B symmetry The irreducible representation is 2A1 B1 BZ which corresponds to s pz pX and py orbitals the same as in VBT You can often use your understanding of VBT to help you in nding the correct basis functions for the irreducible representation Chem 59250 Character Tables and Bonding sz E 02 6v X2 G v yz F 4 0 2 2 G The formula to gure out the number of symmetry representations of a given type is mi zraerzatiw ii H tiiifitis fihj39ii mi it 3959 14fquotswit rzti iii Thus in our example lt1gtlt4gtlt1gtlt1gtltogtlt1gtlt1gtlt2gtlt1gtlt1gtlt2gtlt1gti gilt1gtlt4gtlt1gtlt1gtltogtlt 1gtlt1gtlt2gtlt1gtlt1gtlt2gtlt 1gt1 1 Z14110112 112 1 gilt1gtlt4gtlt1gtlt1gtltogtlt 1gtlt1gtlt2gtlt 1gtlt1gtlt2gtlt1gti Which gives 2 As 0 A2 s 1 B1 and 1 BZ Chem 59250 Character Tables and Bonding Example the G and 7 bonding in 803 The point group is D3h so we must use the appropriate character table to nd the reducible representation of the sigma bonding 1 6 rst then we can 39 go the representation of the n bonding 1quot To determine 1 6 all we have to do is see how each symmetry operation affects the 3 6 bonds in the molecule The E and the Oh operations leave everything where it is so all three bonds stay in the same place and the character is 3 1 11 The C3 and 83 operations move all three bonds so their characters are 0 The 02 operation moves two of the bonds and leaves one where it was so the character is 1 Each 6V operation leaves one bond where it was and moves two bonds so the character is 1 Overall the reducible representation for the sigma bonding is D3h E 203 ac2 6h 283 36 V F G 3 0 1 3 0 1 p39 39o8 b Chem 59 250 x2 z2 x2 y2 xy XLVZ nAi 131201311131201311l nA1 1 A 131201311131201311l A5 0 nE 132201310132201310l nE 1 We can stop here because the combination A 1 E produces the 1 6 that we determined None of the other representations can contribute to the o bonding ie nAu1 nAu1 and nEu are all 0 The irreducible representation A 1 E shows us that the orbitals involved in bonding are the s and the pX and py pair this corresponds to the sp2 combination we find in VBT Chem 59250 Character Tables and Bonding Now we have to determine F for the 7 bonding in 803 S5 To determine 1 7E we have to see how each symmetry operation affects the 7 systems in the molecule The treatment is similar to what we did 039 for sigma bonding but there are a few signi cant differences 1 Pi bonds change sign across the internuclear axis We must consider the effect of the symmetry operation on the signs of the lobes in a 7 bond 2 There is the possibility of two different 7rtype bonds for any given obond oriented 90 from each other We must examine each of these This means that we have to nd reducible representations for both the 7 system perpendicular to the molecular plane Til vectors shown in red and the pi system in the molecular plane TEN vectors shown in blue T Note These are just vectors that are associated with each sigma bond not with any particular atom they could also be placed in the middle of each 80 bond The vectors should be placed T to conform with the symmetry of the point group eg the blue vectors conform to the C3 axis Chem 59250 Example the G and 7 bonding in 803 First determine the reducible representation for the pi bonding perpendicular to the molecular plane PM Ox T 9 8 0 The E operation leaves everything where it is so all three vectors stay in the same place and the character is 3 O The C3 and 83 operations move all three vectors so their characters 1 5h 02 are 0 The 02 operation moves two of the vectors and reverses the sign of the 0 other one so the character is 1 8 0 The Oh operation reverses the sign of all three vectors so the character 0 is 3 Each 6V operation leaves one vector where it was and moves the two others so the character is 1 Overall the reducible representation for the perpendicular n bonding is D3h E 203 302 ch 283 36V PM 3 o 1 3 o 1 Chem 59 250 w x2 y2 xy XLVZ 1312013 111 31201311l o lt1gtlt3gtlt1gtlt2gtltogtlt1gtax 1x4lt1gtlt 3gtlt 1gtlt2gtltogtlt 1gtlt3gtlt1gtlt1gt g 1 13220 1BX 001 3 2201310l g1 Going through all the possibly symmetry representations we nd that the combination N2 Equot produces the Fmthat we determined The irreducible representation shows us that the possible orbitals involved in perpendicular TE bonding are the pZ and the dxZ and dyZ pair This is in agreement with the TC bonding we would predict using VBT Chem 59250 Example the G and 7 bonding in 803 First determine the reducible representation for the n bonding in the molecular plane PM 0 8 0 The E operation leaves everything where it is so all three vectors stay in the same place and the character is 3 O The C3 and 83 operations move all three vectors so their characters av 02 are 0 The 02 operation moves two of the vectors and reverses the sign of the other one so the character is 1 Mus The Oh operation leaves all three vectors unchanged so the character is 3 Each 6V operation reverses the sign one vector where it was and moves the two others so the character is 1 Overall the reducible representation for the parallel n bonding is D3h E 203 302 ch 283 36V PM 3 o 1 3 o 1 Chem 59 250 x2 y2 xy XLVZ m 131201311131201311 m 0 my l131201311131201311 m2 1 nE 132201310132201310 nE 1 Going through all the possibly symmetry representations we find that the combination A 2 E produces the I Mthat we determined The possible orbitals involved in parallech bonding are only the dX2y2 and dXy pair The A 2 representation has no orbital equivalent Note Such analyses do NOT mean that there is 1 bonding using these orbitals it only means that it is possible based on the symmetry of the molecule Chem 59250 Character Tables and Bonding Example the G and 7 bonding in CIO439 6c6 The point group is Td so we must use the appropriate character table to nd quot I the reducible representation of the sigma bonding 1 6 first then we can go 1919 the representation of the n bonding 1quot The E operation leaves everything where it is so all four bonds stay in the same place and the character is 4 Each C3 operation moves three bonds leaves one where it was so the character is 1 The 02 and 84 operations move all four bonds so their characters are 0 Each 6d operation leaves two bonds where they were and moves two bonds so the character is 2 Td E 803 ac2 6S3 66d r 4 1 o o 2 G 3 Chem 5925O Td E 8 C3 3 02 6 s3 6 Cd z2 E 222 X2 y2 X2 y2 T1 391 Rx Ry R2 T2 3 0 1 1 x y 2 xy xz yz The irreducible representation for the 6 bonding is A1 T2 which corresponds to the s orbital and the pX py pz set that we would use in VBT to construct a the sp3 hybrid orbitals suitable for a tetrahedral arrangement of atoms To get the representation for the n bonding we must do the same procedure that we did for 803 except that in the point group Td one can not separate the representations into parallel and perpendicular components This is because the threefold symmetry of the bond axis requires the orthogonal vectors to be treated as an inseparable pair Chem 59250 Example the G and 7 bonding in CIO439 The analysis of how the 8 vectors are affected by the symmetry operations gives Td E 803 302 683 66d r 8 1 o o o X2y222 E 222 X2 y2 X2 y2 T1 Rx Ry R2 T2 X y 2 ny XZ 32 The irreducible representation for the n bonding is E T1 T2 which corresponds to the dX2y2 and dxy pair for E and either the px py pz set or the dxy dxz dyz set for T2 since T1 does not correspond to any of the orbitals that might be involved in bonding Because the px py pz set has already been used in the 6 bonding only the dxy dxz dyz set may be used for n bonding Review of Statistical Mechanics 31 Microcanonical Canonical Grand Canonical Ensembles ln statistical mechanics we deal with a situation in which even the quantum state of the system is unknown The expectation value of an observable must be averaged over ltOgt ltZ39 lOl Z39gt 31 1 where the states form an orthonormal basis ofH and w is the probability of being in state The ws must satisfy Eu 1 The expectation value can be written in a basis independent form ltOgt Tr po 32 where p is the density matrix In the above example p Eiwim The condition Eu 1 ie that the probabilities add to 1 is Tr p 1 33 We usually deal with one of three ensembles the microcanonical emsemble the canonical ensemble or the grand canonical ensemble In the microcanonical ensemble 18 Chapter 3 Review of Statistical Mechanics we assume that our system is isolated so the energy is xed to be E but all states with energy E are taken with equal probability p06H7E 34 C is a normalization constant which is determined by 33 The entropy is given by S7lnC In other words SE lnlt of states with energy E Inverse temperature 6 1kBT 35 5 t P i as kBT av E Pressure P where V is the volume First law of thermodynamics s as dEWdV dS dE kBTdS i PdV Free energy F E 7 kBTS Di erential relation dF ikBSdTi PdV 1 6F le 35 36 37 33 39 310 311 312 313 Chapter 3 Review of Statistical Mechanics 20 P i 2 5gtT 314 while 7T2 315 In the canonical ensemble we assume that our system is in contact with a heat reservoir so that the temperature is constant Then p O e H 316 It is useful to drop the normalization constant C and work with an unnormalized density matrix so that we can de ne the partition function Z Tr p 317 or Z 254 318 The average energy is E 1 2E e Ea Z a a 6 i 1 Z 35 n 6 7 7 2 7 k3 T 6T an 319 Hence F ikBTan 320 The chemical potential M is de ned by 6F 1 321 W Chapter 3 Review of Statistical Mechanics 21 where N is the particle number In the grand canonical ensemble7 the system is in contact with a reservoir of heat and particles Thus7 the temperature and chemical potential are held xed and p 0 WWW 322 We can again work with an unnormalized density matrix and construct the grand canonical partition function Z Ze WEa rN 323 Na The average number is 6 N ikBTm an 324 while the average energy is E alZkTalZ 325 7 n n as M B at 32 BoseEinstein and Planck Distributions 321 BoseEinstein Statistics For a system of free bosons7 the partition function Z Z e WEa rN 326 EMN can be rewritten in terms of the single particle eigenstates and the single particle energies 6139 Ean0 0n161 Z Z 63E i r2im m H Zei wertmiv i Chapter 3 Review of Statistical Mechanics 22 1 33928 1 ltmgt 6 5im 7 1 329 The chemical potential is chosen so that N ltnigt l 1 EW1 lt330gt The energy is given by L Z emw 1 331 139 N is increased by increasing M M S 0 always Bose Einstein condensation occurs when N gt Z ltnigt 332 0 In such a case7 ltnogt must become large This occurs when M 0 322 The Planck Distribution Suppose N is not xed7 but is arbitrary7 eg the numbers of photons and neutrinos are not xed Then there is no Lagrange multiplier M and l 6quot 7 1 ltmgt 333 Consider photons two polarizations in a cavity of side L with 6k hick lick and k 2 quotlam77712 334 E 2 Z memmz ltnmxmymzgt mac gtmy mz Chapter 3 Review of Statistical Mechanics We can take the thermodynamic limit L 7 00 and convert the sum into an integral Since the allowed 137s are 2quot mhmy m1 the g space volume per allowed I is 271393L3 Hence we can take the in nite volume limit by making the replacement Hence For hckm gtgt 1 and For hckm ltlt 1 and EN k E E 0V 1 3 fkA 2W3d3Efl 2 kmdgk Wk 0 2703 E h dk 7 1 2 kmaxd3 k hck 0 2703 E mk 7 1 Vk T4 kamax x3 dx 7T2hc3 0 ex 7 1 Vk T400 xgdr 7T2hc3 0 61 7 1 4Vk 3 xsdr 3TA 7T2hc em 7 1 ng E max k T 3712 B V163ka CV 3712 33 FermiDirac Distribution For a system of free fermions the partition function Z Z 5719EVMN EMN 336 337 338 339 340 341 342 Chapter 3 Review of Statistical Mechanics 24 can again be rewritten in terms of the single particle eigenstates and the single particle energies 6139 Ean0 0n161 but now 711 01 344 so that Z Zei 2ini5i Eini m 1 H ei werwz39 i m0 Hlt1 WkD 345 i 1 3 46 ltnlgt 7 6 6im 39 The Chemical potential is Chosen so that N i 1 3 47 Egg mw 1 39 The energy is given by 6i lt348gt 34 Thermodynamics of the Free Fermion Gas Free electron gas in a box of side L hzkz 349 6k 2m with k f mmmwmz 350 Chapter 3 Review of Statistical Mechanics Then7 taking into account the 2 spin states7 AtT0 25 E 2 Z memymz ltnmxmymzgt mxmymz km dSk 2 Vo 2703 ltwwgt 6 2m 1 351 km d3k 1 N 2 V 3 2 2 0 2w 6 ltwgt H 352 1 2 1sz 19 1 7 353 Nag mt 2m 6 m 1 All states with energies less than M are lled all states with higher energies are empty We write For kBT ltlt 6F N V 3 1 771325 3 1 771325 3 W271 ths oodee 0 n d66 0 V 277WT0 kF h 7 6F MT0 354 N W dsk kg 2 355 V 0 2703 3712 E kF d3k 71sz V 0 2703 2m 1 52k 356 5 V 6F d3k m2 1 2 ma 7T2 d662 357 1 mew 1 3 1 1 777527 M 1 1 0d662 emew 1 71gt 3 1 777222 3 thg thg 00 l 1 A de 62 mew 1 Chapter 3 Review of Statistical Mechanics 26 Qm mg Md 1 m2 0d 1 i e e e e 37rzh M 757511 0 e W W 1 thg 3 em 1 2 2 00 k Td m2 3 m2 B 95 MMBTz MikBTyc 0lt 0 i 2 37r2h3 thg em1 lt gt 3 m3 z 2771 gt 1 P 5 f k T d 37r2h3w7r2h3n 1ltBgt 2 2n7111 lt 2ngt0 ew1 g 3 kBT 2 4 i I O T 358 thu 2 M 1 with 00 k 359 1 d k 0 z611 We will only need 7T2 I 360 1 12 lt gt Hence7 g 3 kBT 2 4 6F2 W 1 5 11OT 361 To lowest order in T7 this gives u 6F 17 g 0T4gt 6F 1 7 71L 0T4gt 362 E mg Wd 1 66 V 0 1 1 3 1 777327 M g 777325 M g 1 777327 00 g 1 7T2h3 0 deez thg 0 deg emew 1 7 1 7T2h3M dew 6m 1 Qm m2Md 1 m2 0d 1 7 66 66 57rzh 757111 0 e m t 1 thg a em 1 2m 771523 kBTd2 2 3 5w2h3M2 q2h30 gw1 ltWkBTgt2 ikBTmO Olt5 m 5 2m g 270 0 2 72 1 Plt gt 00 2M 7 2 k T 2 d 5n 1 23213 M 2n71lflt72ngt0 ew1 Chapter 3 Review of Statistical Mechanics 27 2 2m g 15 kBT mat 1 7 7 11 064 3N 5w kBT 2 1 0 T4 363 5VEFlt12ltEFgt Hence the speci c heat of a gas of free fermions is 2 k T CV 7T NkB B 364 2 EF Note that this can be written in the more general form CV const k3 96F kBT 365 The number of electrons which are thermally excited above the ground state is N g 6 kBT each such electron contributes energy kBT and hence gives a speci c heat contribution of k3 Electrons give such a contribution to the speci c heat of a metal 35 Ising Model Mean Field Theory Phases Consider a model of spins on a lattice in a magnetic eld H igprZSf 2 21125 366 with Sf i12 The partition function for such a system is h N Z lt2 cosh kB Tgt 367 The average magnetization is 1 h S t h 368 2 an M lt gt The susceptibility x is de ned by X 369 h0 Chapter 3 Review of Statistical Mechanics 28 For free spins on a lattice 1 1 ENkB T A susceptibility which is inversely proportional to temperature is called a Curie suc X 370 septibility In problem set 3 you will show that the susceptibility is much smaller for a system of electrons Now consider a model of spins on a lattice such that each spin interacts with its neighbors according to 1 Z Z H 75215 s 371 W This Hamiltonian has a symmetry 51 a is 372 1 For kBT gt J the interaction between the spins will not be important and the susceptibility will be of the Curie form For kBT lt J however the behavior will be much different We can understand this qualitatively using mean eld theory Let us approximate the interaction ofeach spin with its neighbors by an interaction with a mean eld h H 72th 373 i with h given by h J2ltSfgt 374 i where z is the coordination number In this eld the partition function is just 2 cosh and 81 tanh kBLT 375 Using the self consistency condition this is J2ltSzgt S th ltgt an kBT 376 Chapter 3 Review of Statistical Mechanics 29 For kBT lt J2 this has non zero solutions 51 31 0 which break the symmetry Sf a 75 In this phase there is a spontaneous magnetization For kBT gt J2 there is only the solution 51 0 In this phase the symmetry is unbroken and the is no spontaneous magnetization At kBT J2 there is a critical point at which a phase transition occurs Crystalline Solids Symmetry and Bonding Branislav K Nikolic Depar men f o f Physcs and As franc71y Universn o f Desaware U 5 A PHYS 624 Introduction to Solid State Physics httpwwwphysicsudeledubnikolicfeachingphy5624phy5624hfml my problem 1021 aTomscm3 essenTially a Thermodynamic limiT possible EIThe TranslaTionally invarianT naTure of The periodic solid and The facT ThaT The core elecTrons are very Tigthy bound aT each siTe so we ignore Their dynamics makes approximaTe soluTions To manybod EIThe simplesT model of a solid is a periodic array of valance orbiTals emb The Qhe Figure is ofTen equivalenT To solving The who le sysTem edded in a maTrix of aTomic cores Solving The problem in one of irreducible elemenTs of The periodic solid eg one of The sphe PHYS 624 CrysTulline Solids SymmeTry and Bonding EITranslaTional symmeTr39y of The laTTice Ther39e exisT a seT of basis vecTor39s 5i b 5 such ThaT Theanomic sTrucTune remains invar39ianT under39 Tr39anslaTions Through any vecTor39Rn 11167 112 n15 wher39e ml12196 Z O O O O 63 gt HO O OOO EIOne can go from any locaTion in The laTTice To an idenTical locaTiongby following paTh composed of inTeger39 mulTiples of The vecTor39s c and b CINOTE Keep in mind ThaT basic building blocks of periodic sTr39ucTur39es can be more complicaTed Than a single aTom eg in NaCl The basic building block is compOSed of one Na and one Cl ion which is r39epeaTed in a cubic paTTer39n To make The NaCl cr39ysTalline sTr39ucTur39e PHYS 624 Cr ysTalline Solids SymmeTr y and Bonding EIThe 1D sys rems can have only discre re Transla rional symme rry EIIn 2D and 3D cases a crystal can also have other symmefr ies roTa rions around axes reflec rions on planes and combina rion of These opera rions among Themselves and wi rh translnfions Thu are not Iu ice vecfors DSPACE GROUP The comple re seT of rigid body mo rions Tha r Take cr s ral in ro i rself A Y S t R u 6 PHVS 624 Crystallme Sollds Symmetry and Bondmg Square Rectangular EIZD Ia Hices are and inferfaces o Wigner 52in l Iexngonal Oblique o u u o a i u g u o I a m c2 Centered Rectangular 1 39 EIInversion Symme rry is obeyed by all Ia ices Invariance under r gt r noT ma rhema rical fiction lmy na rurally appear as surfaces 1 real crys rals PHYS 624 Crysf allimz Solids Symmetry and Bonding CIA collecTion of poinTs in which The neighborhood of each poinT is The same as The neighborhood of every oTher poinT under some TranslaTion is called a Bravais aTTice EIThe primiTive uniT cell is The parallel piped in 3D formed by The primiTive aTTice vecTors which are defined as Those aTTice vecTors ThaT produce The primiTive cell wiTh The smallesT volume QP z xU 5 EIThere are many differenT primiTive uniT cells common feaTures Each cell has The same volume and conTains only one siTe of Bravais aTTice WignerSei l39z cell gt single siTe siTs in The cenTer of The cell which is invarianT under all symmeTry operaTion ThaT leave The crysTal invarianT EINonprimiTive uniT cell LaTTice region which can conTain several siTes and u3ually has The same PoinT Group symmeTry as The aTTice iTself which produces The full crysTal upon repeTiTion PHYS 624 CrysTaIIine Solids SymmeTry and Bonding Ii cxugmm a1 2 a a O PHYS 624 Cr ys ralline Solids Symme rr y and Bonding lAl Bl EITiling of The plane Cells are free ma To have r39aTher39 peculiar shapes as long as They fiT TogeTher properly CIPr39imiTive uniT cells puT end To end fill The cr39ysTaI DSince They conTain only a single par TicIe The volume QPUC of The primiTive cell is exacTIy The inverse of The densiTy of The cr39ysTaI PHYS 624 CrysTalline Solids SymmeTry and Bonding DMos l of la H ices occurring in na l39ur e ar e no l39 Br avais la H ices bu l39 ar e la H ices wi l39h a basis cons l39r uc l39ed by beginning wi l39h a Br avais la H ice and puHing a each la H ice si l39e an identical assembly of par Ticles r a l her l39han a single r o l39a l39ionally invar ian l par l39icle Sfar39f wifh hexagonal 2D Iaffice A and I replace single poinf in The cenfer wifh a pair of poinfs B EIThe do H ed line is aglide line l39he la H39ice is invar ian l when l39r ansla l39ed ver l ically by 12 and r eflec l ed abou l This line bu l39 i1 is no l39 invar ian l under ei l39her oper a l ion separa l39elyl PHYS 624 Crysmiim Salids Symmmy and Banding EINonPrimiTive EIemenTary UniT Cell Minimal volume of The crysTal conTaining several parTicles which has The same poinT group symmeTry as The crysTal iTSelf and which produces The full crysTal upon repeTiTion k O O Primitive Unit Cell Non rimitive I nit Cell C O O O O O O C O C PHYS 624 CrysTalline Solids SymmeTry and Bonding Pay aTTenTion To 45 roTaTion around axis passing Through The yellow atom o 0ZRE Ewe311 mam373 39 pge01iLzk EITo accounT for more complex sTr39ucTur39es like molecular solids salTs eTc one also allows each laTTice poinT To have sTr ucTur e in The form of a basis a good example of This in 2D is The Cqu planes which char39acTer39ize The cupraTe high Temperafure superconducfors lt5gt77 77 7 7 l l l l Basis l l39he basis is composed Primmve of Two oxygens and 0 cs one copper39 aTom laid down on a simple l l 1 l 397 square laTTice wiTh The Cu aTom cenTer39ed on The laTTice poinTs 4 E l l l 4 4 lt3 r g PHYS 624 Crystalline Solids SymmeTr39y and Bonding immoral limt21t1gulai EIThe cenTered laTTice is special since if may also be c xv 39 c as ice composed of a TwocomponenT basis and u ecTangular uni r ce PHYS 624 Cr ys ralline Solids Symme rr y and Bonding EIOnce we decoraTe la r rice wiTh a basis i rs symmeTries will change EIAdding a basis does noT au roma rically desTroy The roTaTional and reflecTion symmeTries of The original la r rice 111411 444444 1444414 iii i i 2544442 444444 EIDecoraTion of a Triangular la r rice wiTh chiral molecules will preserve Qo ra rional symmeTries buT will desTroy reflecTion symmeTries my original IaTTice PHYS 624 Crys ralline Solids Symme rry and Bonding EITwo aTTices are The same if one can be Transformed conTinuously inTo The oTher39 wiThouT changing any symmeTr39y oper39aTions along The way Mirror plane blIkequot md l39EstDIEd In deforming The I I I I I I I I I I I I I I I I I I I I quot rectangular III Iu I H39 quotquotquot quot aTTiceinToThe I I I I I I I I I I I I I I I I I I I I I d I I I I I I I I I I I I I I I I I I I I I Cardere I I I I I I I I I I I I I I I I I I I I I r39ecTangular39 I I I I I I I I I 39I I I I I I I I 39 I quot 39 39 IOTTlCC r39CfICCTlOH I I I I I I I I I I I I I I I I I I I I I Symmerriy aboul I I I I I I I y I I I I I I I I I I I I I I I The y axis is 39Rggmngumr L x animal Rectangular deSTr oyed EITwo aTTices ar39e equivalenT if Their39 space groups 5 and 539 are The same up To linear change of coordinaTe sysTems ie Ther39e exisT a single maTr39ix 0 r39epr39esenTing The change of coor39dinaTe sysTem 1 1 I a ORO Ot R 1 PHYS 624 Cr ysTaIIine Solids SymmeTr y and Bonding 17 DisTincT 2D LaTTices EIThe various planar paTTerns can by classified by The TransformaTion groups ThaT leave Them invarianT Their symmeTry groups A maThemaTical analysis of These groups shows ThaT There are exachy 17 differenT plane symmeTry groups v t V h 9 39 l PHYS 624 CrysTalline Solids SymmeTry and Bonding Q I 9PM 7C QC Cubic Tetragoual Onhurhombic WTOIID39EIiHiE lbr Hba r r bg c ag ba c a 90 a3 90 al8ry90 u 939 39 90 b Slmple 6 Base Centemd Body C entemd Q PUC 2 Face Centered QPUC Tl iclinic Hexagonal Rhombahedml zy bg c ub5 c abc a 5 5 g 9039j 213510 1 31 90 r 120 5 ELL 139 in C a g 2 PHYS 624 Cr ys ralline Solids Symme rr y and Bonding EISymmeTry TransformaTions form GROUPS EIGr39oup So is defined as a seT E A B C which is closed under39 a binary opero rion oSxS gtS Ao B e S and so risfies The following axioms The binar39y oper a rion is associa rive A o B o C A o B o C There exis rs an iden ri ry E o A A o For39 each A e S Ther39e exisT an inver39s elemenT A391 e S such ThaT A0 A391 A39lo A E EISpace Grou elements TranslaTions E11136 inversions reflecTions roTaTions im oper39 roTaTions inversion and refle TionUi 0 P and es operaTion o is composiTion of S6 elem nTsU fFUFe S Ullao zl FUIZUFEIAU2Z UlelUimaeS QUlf 1U 1 U 1res E0eS PHYS 624 Cr ys ralline Solids Symme rr y and Bonding WU ltIgtgtllt PIltDgt H ilberT spaces or more properly space of rays are represenTed by uniTary or anTiuniTary Timereversal ta tja afgL a Z k a i EIWigner 1926 SymmeTry operaTions operaTors EIIn The HilberT space of coordinaTe wave funcTions 2mm PU KW ElGroup Representation Group of UniTary OperaTors U UV isomorphic To The sace group A UnmUwzlfz EIThe represenTaTion is reducible if There exisT a nonTrivial subspace of The HilberT space which is invarianT for all UW and ireducible if uch subspace does noT exisT PHYS 624 CrysTaIIine Solids SymmeTry and Bonding z nlths mum m mmpmiunn small mm fth symdfy 39 m 2 n 11 Armagnan emu m mum mm 3 mm hmtald m 4 41mm Dhnscmmmmg m mum ans EIThe point group is no enough 1390 defermine H39Ie Ia H39ice opera ons CIFor example if we choose any a reflected abou f H39Ie x or y axis and each is invariant under a 180 7 r o fcrfionsl wJ Asx wuwns 24 ErysmHme Sahds Symmwy and Bandmg O l FDOEO Schb nflies Notation gtCm quotcyclicquot when There is a single axis of raTaTian and number m indicaTes The mfald symmeTry around The axis ab d hedralquot when There are Twofold axes aT The righT angles To anaTher axis QT TeTrahedral when There are four seTs of raTaTian axes of Three fold symmeTry as in TeTrahedran gtO acTahedral when There is a fourfold raTaTian axis combined wiTh perpendicular Twa fald raTaTian axis as in acTahedran InTernaTional Tables of CrxsTallographx Example 6m is mirror plane con Taining a sixfold axis 6m is a mirror plane perpendicular To a sixfold axis PHYS 624 CrysTallimz Solids SymmeTry and Bonding q Cubic Tric i In I Mount linic Unlin Trlgonnl Tcxmgunal Homganal PHYS 624 CrysTalline Solids SymmeTry and Bonding CISTereograrn Twodimensional projecTions of a poinT ThaT does naT belong To any special axis or plane of symmeTry on The surface of a sphere and iTs images generaTed by acTing on The sphere wiTh various symmeTry operaTions of The poinT group Pick a poinT on a sphere CIThe space group always has only Translations as its subgroup T ltS EIIn general it cannot be formed from these subgroups because 0 the g ide planes translation reflection and screw translation rotation lag 0 39l e i H Ri eSf oElfeTS PgtltT a EIIn general Point Group leaves the Bravais lattice invariant but not the crystal itself recall that the crystal is defined as the Bravais lattice and the atomic basis in each unit cell EH57 NonSymmorphic Groups include glyde planes and screw axes D73 Symmorfic Groups P ltS S P AT 01651 Magnetic Shubnikov or Color Groups for the lattices whose oints are decorated with quantummechanical spin PHYS 624 Crystalline Solids Symmetry and Bonding EISharp peaks in Xray diffracTion paTTern are exclusively resulT of laTTice symmeTries The inTensiTy of The peaks depends on deTails EIOrder parameTar has To have The same symmeTry as The laTTice iTself gt FerroelecTrics have dipole momenT rule ouT any symmeTry group ThaT has perpendicular mirror plane or roToinversion axis because dipole changes sign under These operaTions Cn Cnv C1 EIPiezoelecTric such as quarTz do noT have dipole momenT buT acquire one upon mechanical deformaTion in some direcTion 1 a u a u 3 e a gt P 2 B e 3 3 3 a 2 Br Bra y w a 7 0 Qh crysTals cannoT be cenTrosymmeTric excludes poinT groups Dnh and y which rules ouT possibiliTy of large effecT in a huge number of compounds PHYS 624 CrysTalline Solids SymmeTry and Bonding DH 0 HamilTonian H is invarianT under39 cerTain symmeTry operaTions Then we may choose To classifyjrhe ener39gy eigens ra res as sTaTes of The symmeTry operaTion and H will noT connecT sTaTes of differen r symmeTry U ma H ujgtltui PHYS 624 Cr ys ralline Solids Symme rr y and Bonding uigt ltuj EIEffecTive singleporTicle HomilTonion of a solid commuTes wiTh The representations o space group operaTiolis H U 0 gt UUlfH Pg 17 UUlfgnk 1 g17 Ulf A A U gt A n gt n gt EIOne can furTher re uce compleany from solvnng Schrodinger equoTIon in a uniT cell secured by TronsloTionol symmeTry To finding band dispersion and Bloch eigensToTes only in a porTion of The BZ enltUI gtenltI gt Ty 17 2 Egg 2 Pgarlf U4 EISelecTion rules Wiqner EckerT Theorem 11 0w2gt 0gtr era 211163 PHYS 624 CrysTalline Solids SymmeTry and Bonding Uesgt Tab Cy 1 Ly m Ly Jr yl m by my xi 3 J J x y E C C2 3 rJ Q C2 C3 x v C3 C3 E n C E Ci gt x y In Us 0y 4r AX y a 01 ax 4 x m ax 03 ltgt y x a a 01 Ur av 0 m I F 0 0 which has the full symmclry of lhc pninl group E aquot a m 6 2 M 1 na which also has lhu l39ull symmetry 0th poinlgmup CA UV 0 U m 3 X l DNTrn which has he symmetries E C3 lb 03 C3 01 7 a a i A k07ra 0 lt k lt l which has 119 symmeln39cs E 0 a E C1 C3 C4 5 3 kklIIa0 lt k lt l which has he symmclrics E m 3 C2 C4 C3 6 Z l umu lt k lt l which has he symmclrics E a 0 c4 cg E cl ax c3 C4 C2 E PHYS 624 Crys ralline Solids Symme rr y and Bonding El Solids are composed of elemenTs wiTh lelTiple orbiTals ThaT produce lelTiple bonds Now imagine whaT happens if we have several orbiTals on each siTe spd eTc as we reduce The separaTion beTween The orbiTals and increase Their overlap These bonds increase in widTh and may evenTLlally overlap forming bands EIValance orbiTals which generally have a greaTer spaTial exTenT will overlap more so Their bands will broaden more El EvenTLlally we will sTop gaining energy from bringing The aToms closer TogeTher dLle To overlap of The cores 5 Once we have reached The opTimal poinT we fill The sTaTes 2 parTicles per LlnTil we run oLlT of elecTrons EIElecTronic correlafions Vl ly complicaTe This simple picTLlre of band formaTion since They sTrive To keep The orbiTals from being lelTiply occupied PHYS 624 Crystalline Solids Symmetry and Bonding quantum 1111111ba393e1395 n i39i llll r39l witlit 4r 3 13 IJIe LLBE B N Na lg A 1 A 1 KC 1 transition metals EEC 211 G Kr Rib l Lil armiti n 1119315 Y v 3d 111 Csan Rare Earths ILauthanidesj CCALLI Tl39al39lsiliilj39l39 I39I39IE39LRIS La v Hg TI Rn PHYS 624 Cr ys ralline Solids Symme rr y and Bonding For39 large n The or39biTals do noT fill up simply as a funcTion of n as we would expecT from a simple Hydrogenic model wiTh 4 Ine 2 l quot 2712142 EILevel crossings due To aTomic screening The poTenTial felT by sTaTes wiTh large I are screened since They cannoT access The nucleus Thus or39biTals of differ39enT pr39inciple quanTum numbers can be close in energy For39 example in elemenTaI Ce 4f15d1652 boTh The 5d and 4f or39biTals may be considered To be in The valence shell and form meTalIic bands However39 The 5d or39biTals are much larger and of higher39 symmeTr39y Than The 41 ones Thus elecTr39ons Tend To hybr39idize move on or39 off wiTh The 5d or39biTals mor39e Qec rively The Coulomb r39epulsion beTween elecTr39ons on The same 41 or39biTal will be sTr39ong so These elecTr39ons on These or39biTals Tend To form magneTic momenTs PHYS 624 Cr39ysTalline Solids SymmeTr39y and Bonding EIWan der Waals bonding formed by aToms ThaT do noT have valence elecTr ons available for sharing The noble elemenTs and crysTaIs or organic molecules flucTuaTing dipolequot zeropoinT moTion of van der Waals inTer acTion is due To slighT polarizaTion of The elecTr onic wave funcTion in one aTom due To The surrounding aToms EILower39ing of zero poinT energy by dipole dipole inTer acTion J 2g 2 30 4 mag PHYS 624 Cryslalllne Sullds Symmeh39y and Banding AE ND bonding Bonding CICovalen r Bonding formed when elec rrons in well defined direc rional orbi rals which can be rhough r as linear combina rions of The original a romic orbi rals have s rrong overlap wi rh similar orbi rals in neighboring a roms solids produced in This way are semiconduc rors or i nsula rors PHYS 624 Crys ralline Solids Symme rry and Bonding 6D 75H e Ionic Bonding Formed when Two differenT V Type of aToms are combined one ThaT prefers To lose some of iTs valence elecTrons and 39 31 7 become posiTive ion and one ThaT prefers To grab elecTrons from oTher aToms and become a negaTive ion Bonding Then occurs by Coulomb aTTracTion beTween The ions CombinaTions of such elemenTs are I VII II VI and III V In The firsT case bonding is purely ionic in The oTher Two There is a degree of covalenT bonding presenT EIThe energy per molecule of a crysTal of I N sodium chloride is 79 5136 eV64eV lower Than The energy of The separaTed neuTral aToms The cohesive energy wiTh respecT To separaTed ions is 79eV per coulomb molecular uniT All values on The figure Repulsion bemezn are exper39memal39 Modelling constant A4148 for MCI elecfrlon clouds PHYS 624 Crystalline Solids Symmetry and Bonding pans EIIonic and covalenT bondings ar39e Two limiTing cases of which only The laTer39 can exisT in solids compOSed of a single Type of aTom In majoriTy of cases The bonding is of an inTer39mediaTe naTur39e r39epr39eSenTing a mixTur39e of The Two exTr39emes N 510 1 Z 053 Cl 50 w W 5 2 l 0 5 U3 a u Nu PHYS 624 Cr39ySTalline Solids SymmeTr39y and Bonding MeTallic Bonding formed when elecTrons are shared by all aToms in The solid if producing a unifrom quotseaquot 3d P i m K is auuupcp V 7 Q 39I I J y of negaTive charge CIIn meTallic Ni FCC 3d84sz The 45 and 3dorbiTals are almosT degeneraTe and Thus boTh parTicipaTe in The bonding HoweverThe 4s orbiTals are so large compared To The 3dorbiTals ThaT They encompass many oTher laTTice siTes forming nondirecTional bonds In addiTion They hybridize weakly wiTh The d orbiTals The differenT symmeTries of The orbiTals causes Their overlap To almosT cancel which in Turn hybridize weakly wiTh each oTher Thus whereas worbiTals form a broad meTallic band The dorbiTals form a narrow one PHYS 624 CrysTulline Solids SymmeTry and Bonding CIIn con l39r39asf 1390 covalen l39 bonding elec l39r39onic wave functions in metals are very ex l39ended compared 1390 The separa on between atoms 4 Ni 4323d5 Amplitude 05 110 1 5 I r A Fig 19 The amplitude of he 3d wavefuncion and the 4 5394vavefuucli0n of Ni l4 The Imlfdismnccs m the LLI SI second and third nearev n ighburs rr1 and 7393 are shown for comparison PHYS 624 Cr ys ralline Solids Symme rr y and Bonding quotT Hydrogen bond A Covalent bond 711 EIHydrogen Bonding formed when H is present and bounded to two other atoms organic molecules DNA and water ice this is a special bond due to its lack of core electrons bare proton left after sharing ls electron its th mass must use quantum mechanics for eg zeroj point proton motion and high ionization energy PHYS 624 Crystalline Solids Symmetry and Bonding ElThe Type of bond ThaT forms beTween Two orbiTaIs is dicTaTed largely by The amounT ThaT These orbiTals overlap relaTive To Their separaTion a Bond Overlan Lattice constituents Ionic very small lt a 39thGSt unfrurtrzited li b39llllll l39 packing Covalent small w 1 determined by the similar structure of the orbitals b letallic Very large gt L 39lObGSt packed Iiirlilleil Valence PHYS 624 Crystalline Solids Symmetry and Bonding orbitals EIBor39nO penheimer39 Approximation P171 m d we 2 mam3901 gvm 139zzm 13 gt lt leclrongtzEltI71gtml9 39 39 gtnucl gtnucl gtnucl gtnucl VeeVenVl amrN Vmrl awrN 2 Q Pelectron l T V E electron 2m N CIVIrIal Theorem Velma gt 2 E Z N gtnucl n E Z rk VkE k1 Els139abili1yzE0 lt Em V IZEOYIWCC I Daniel 10 0 Electron gt0 lt121eclrongtoo Eoltgt gt 0 c Elecmon mmo ltVgtm 2E0 E lt0 Veg Kn are decreasing PHYS 624 Cr ys ralline Solids Symme rr y and Bonding gt Vnn are increasing CIMany elemen rs adopf muH ipIe crysfal sfr ucfur es between 0 K and Their melfing Temperafur e CIPquonium has particularly elaborafe phase diagram Transformmiou Phase Structure atoms per unit Density gcc Temp C cell 112 monoclinic 16 198 185 fc 1110110511111 34 178 ft onhorhombic 8 171 fee 4 159 ft telragonal 2 160 late 2 16 5 PHVS 524 CrysmHme Suhds Symmevry and Bundmg Crystalline Phases of Carbon Diamond PHYS 624 CrysmHme Suhds Symmwy and Bundmg AB AB AB AB S 3 m 2 AR AB 13 AB 3 ipx ipy pz AB AB AR AZ S ipx py ipz AB 4 A I I NIH NIH NIH NIH AB AB AB A ig in on in 1 anion in 1 K d gE iA iB9lPilb d gE iA iB PHYS 624 Crystalline Solids Symmetry and Bonding Conduction band Valence ban PHYS 624 Crystalline Solids Symmetry and Bonding EIIn 139he cryS l39al The bonding and an l39ibonding sfa l es acquire dispersion Snag which leads To The forma rion of The valence and conduction energy bands wi l39h a gap Egg be l39ween The Two manifolds of sfa res m lt11 4 gt43 s Cr ysfalline Phases of Carbon Graphi re between Hayers PHYS 624 Cr ys ralline Solids Symmetry and Bonding Selecting a Uni139 Cell in Hexaonal La H ice of Grahene 77 Mom 7 O 0 AW 23 Us Unii Cells Atom 1 z 93 13 luom 2 Iq 23 mtme c mak As afrach sf all dJmensinn 19 me 41mm we 4mm y PHVS 524 rys aHme Sahds Symmwy and Bandmg Haney Ul l9 ll Plbonding IA 18 Illamtibonding IA 18 J PHYS 624 Crystalline Solids Symmetry and Bonding 8 4 7 O 4 4 o o 0 333 39039 39039 333 4 4 2 7 HimE 1 K mm Chiral angle is be rween 4ai 7al a H39I It l wan1 x a ix y 7 2 2 1KK EIZigzag nano rubes correspond To n 0 or 0 m and have a chiral angle of 0 armchair nano rubes have n n and a chiral angle of 30 while chiral nano rubes have general n m values and a chiral angle of be rween 0 and 30 According The Theory nano rubes can ei rher be me rallic green circles or semiconduc ring quoti i 39 armchairquot PHYS 624 Crysfalline Solids Symmetry and Bonding Crystalline Phases of Carbon Buckminsterfullerene EIFullerenes are the only form of molecular carbon known The stability of fullerenes come from the giant delocalized electron system which in the case 5 of the 660 contains 12 pentagonal and 20 hexagonal rings but no pentgonal faces will share a side an effect known as isolated pentagon rule K AQ K Q K4 iquot BUCKMINSTERFU LLERENE L41 L 39 HWN PHYS 624 Crystalline Solids Symmetry and Bonding CIA Copper Cu atom is surrounded by six Oxygen O atoms which form an octahedron the CuO atoms are bonded by strong covalent bonds quot 397 V M The crystallographic structure of hightemperature 39 v u v superconductors HTSC comprises two basic func O tional elements layered quasitwo dimensional Cqulattice planes and ii interplanar regions quot 0 mostly containing metallic spacer atoms in a certain II LuSrJ A concentration ratio eg La1Srz The essential I lt gt quot quot role of these spacer regions is to provide the 01102 V 6 planes with a suitable amount of mobile charge carri ers At a favourable charge carrier concentration and 0 0 for suf ciently low temperatures the planes enter a superconducting state with highly unusual properties SKA 39 EIThe empty space between CuO octahedra can accommodate atoms which are easily ionized to produce mixed covalent ionic structure PHYS 624 Crystalline Solids Symmetry and Bonding CICaTiO3 PbTiO3 BaTiO3 PbZr39O3 can behave as PiezoelecTr39ics or39 FeroelecTrics d212 1 Pt 1 TemperoTure PHYS 624 Cr39ysTalline Solids SymmeTr39y and Bonding CIThe px py pZ or39biTaIs of The Three 0 aToms and The dx d3222 or39biTaIs of The B aTom par TicipaTe in The for39maTion of covalenT bonds in The ocTahedr39on CIComplexiTy STr39ong covalenT bond beTween 30 as well as ionic characTer39 of The bond beTween BO uniTs and The A aToms A aToms provide The necessary number39 of elecTr39ons To saTisfy all covalenT bonds ElecTriC Field Pyroelectric Piezoelectric STress Thermal Expansion J 39 ElasTooptic ThermoopTic OpTic Field Notes on Density thctional Theory 1 Basic Theorenls The energy E of a system with a given Hamiltonian H is a functional of the normalized manyeparticle wave function 1 We write this functional as El l lt I lHl I gt 1 and you recall that the meaning of Eq 1 is quottake 1 act on it with H multiply by the complex conjugate of 11 and integrate over all spacequot Consi er now a system of N electrons moving in a xed potential V7 de ned for example by some atomic nuclei and interacting with each other via the Coulomb interaction The Hamiltonian is then 13 1 52 H V v 2 V 2m22m a n 21 N 1 Hezemnigas Z Wu 3 21 N where Helmmwkgas is the part which is independent of V Observe that the ground state energy Egg is a wellede ned functional of V7 Egg cI VHVT The slightly convoluted de nition is take V solve the Schroedinger equation for the ground state 1 and put this 11 in Eq 1 Thus you give me V and this uniquely up to obvious degeneracies xes Egg The particle density is de ned by nrd37 1d3TN 117 1TN lt Z 6T 7 1 Tl 7 N 4 21 N and using this de nition we have E lt 139l 2 p 12 62 WINL13 V lt 5 7 21 N2m Qi nirj r rnr Hohenberg and Kohn showed that Egg is a unique functional The argument is deceptively simple assume the contrary then there must be two energies corresponding to the same nr in other words there must exist two li s which are different by more than an overall phase hence have different energies but lead to the same particle density Call these 111 and 1 2 Because they are different they must solve the Schroedinger equation with different V s say V1 and V2 thus Helectronigas V1 1 1 E1 1 1 Helectronigas V2 1 2 E2 1 2 19 Now consider the energy you would get if you used 112 as a variational wave function for Heleatmngas V1 Becase 112 is not the ground state of this Hamiltonian you must get an energy greater than E1 thus lt 1 2lHezemon7gas Vll l 2 gt E12 gt E1 8 But by writing V1 V2 V1 7 V2 and using Eqs 57 and the assumption that the two wave functions lead to the same density one nds E 7 E2 13 var 7 V20 me so that E2 7 E1gt MW 7 WWW 9 Now repeat the argument interchanging the roles of 1 and 2 One obtains E21 E1 d3r Mr 7 V1Tnr gt E2 in other words d3r var 7 mm m gt E2 7 E1 10 Eqs 9 10 cannot both be true unless V2 VQED We have thus established that the ground state energy is a functional of the density nr Egg Again the recipe for nding CI is rather convoluted for given nr nd the V7 which leads to a ground state with this density solve the Schroedinger equation and insert the resulting wave function into Eq 1 This 7de nition7 makes it obvious that in fact CI is minimized when nr is the ground state density a density other than the ground state density would lead to a wave function other than the ground state wave function and therefore to an energy higher than the ground state energyifor a given external potential Summary the ground state energy of a system of N electrons in an exter7 nal potential V7 is determined by minimizing a functional This functional may be written q electronigaslin ll l dSTVTnT 11 with elecwon7gas given by 132 1 52 bezemwas nani lt m Z i 5 Emmi gt 12 21 N i j Z 7 with 11 the exact ground state wave function for the given V The virtue of Eq 11 is that ltIgtezeamngas makes no explicit reference to V7 it is a universal functional determined only by the properties of the electron gas If one can obtain a reasonable approximation to it and carry out the minimization then one can use it with my V r 2 Application The actual applications of the formalism follow from a second brilliant insight of Kohn this time with Lu Sham To understand it it is helpful to step away from the problem brie y and consider a problem of N noninteracting electrons moving in some potential VKS7 VT Veff Because it is noninteracting electrons we know exactly what to do solve the Schroedinger equation h2 7 V21MT WT VeffT iT ENMT 13 and nd the N lowest eigenvalues and eigenfunctions The wave function liKS is then the properly antisymmetrized product of the N lowest eigenfunctions the charge density is the sum of the charge density in each occupied eigenfunction WU Z wwwm 14 i1 N the energy is E 2 E 15 21 N The key insights of Kohn and Sham are 1 there exists some Veffr which is a property only of the electron gas and which of course is a functional of nr and w ich when used in 13 reproduces the exact ground state density and energy for each external potential V 2 reasonable approximations to Veff may be constructed The simplest such approximation is the 7local density approximation in which Veff is approximated by the value which would give the right answer for a uniform electron gas at density The answer for the uniform electron gas is known from detailed numerical studies The simplest approximation which describes the electron gas at moderate densities is 2 I VLDA1T d3r frrl 7 vxcmm 16 The rst term in Eq VLDA1 is simply the electric potential which would be produced by the charge density The second term is a correction coming from more complicated interactingeelectron physics Muc effort over the years has gone into developing improved functionals both to more accurately represent the energy of the uniform electron gas and to produce functionals account for nonfuniform densities 3 The algorithm The algorithm adapted to the units we have used in Class for solving the hy drogen atom is For a system of N electrons moving in the external potential QZ V 7 1 u u lt 7 1 Guess a density 2 Compute the potential veffu 2 131 62W chnu 18 luiu l Because we will deal only with spherically symmetric distributions of charge7 we know that the electrostatic potential at radius u is given by taking the charge at u lt u and concentrating it at the origin7 so Veffu u Zdu nu 7VXOnu 19 You will use two choices of chnu One based an old preedensity functional theory approximation of Slater is very simple to write down and is i 3 E7441 20 The other is based on a more detailed parametrization of the properties of the uniform electron gas and requires a bit of notation De ne rs 713 21 recall we measured distances in units of the Bohr radius en 05 7 1g 1 grs VPgngw 1 17 s 27 s 1 17 s 27 s A 2 2D 7 C Vpemew 05 Aln7 s B 7 Crs ln7 s T73 7 s lt l 3 23 rs gt 1 22 with numbers 7 701423 24 51 1029 25 52 03334 26 A 00311 27 B 70048 23 C 00020 29 D 700116 30 3 Obtain the radial eigenfunctions gi corresponding to the N lowest eigene values of the radial schroedinger equation 12 du2 Wu VEffu9iu Ei9iu 31 4 Get the density recall 9 V 9 109M 32 5 go back to 2 and repeat until the density and ground state energy cease to change signi cantly 6 Compute the ground state energy from E 2 E 33 i1 N Br39anislav K Nikoli Depaquotfmenf of Physics and As39fronomy Um39vequots39l39fy of Deaware 151 PHYS 624 Introduction To Solid Sfa re Physics httpwwwphysicsudeledubnikolicTeachingphy5624phys624h39rml Weakly correlaTed elecTron liquid Coulomb inTeracTion effecTs When local perTurbaTion poTenTial is swiTched on some elecTrons as 3 m a 39 a G will leave This region in order To ensure Q o consTanT 8F 3 U chemical poTenTial is a Q Q a Q 0 my Thermodynamic poTenTial Therefore in 5 3 65128 equilibrium iT musT be homogeneous a ThroughouT The crysTal lt9 lt9 Z Szlficcthafgc i 332132ch assume e5Ur ltlt 8F F f8 T a 0 68F 8 65U PHYS 624 Quick and DirTy InTroducTion To MoTT InsulaTors EIExcepT in The immediaTe viciniTy of The per39Tur39baTion char39ge assume ThaT5Ur is caused 8510 by The induced space charge gt Poisson equaTion V2 Ur 8 rrTF 0 V2 2 r12 Bar r2 Bar r aer 16 2 1V 47h 80 r a 80 F 2 a 0 me2 rTF 2D 23 3 Cu 6 8F nCu 28510 cm rTF 055A in vacuum D8F O 6Ur q a 47229 0 3 1 2 23 h2 D8F mlt37r2n 8F37I2n 28F PHYS 624 Quick and Dir Ty InTr oducTion To MoTT InsulaTor s 23 2r 3 MoTT MeTCllIHSUlG39l39OI TransiTion l bound states w free states DBelow The criTical elecTron concenTraTion The poTenTial well of The screened field exTends far enough for a bound sTaTe To be formed screening lengTh increases so ThaT free eecTrons become localized MoTT InsulaTors EIExamples transition metal oxides glasses amorphous semiconductors PHYS 624 Quick and DirTy IrrTroducTion To MoTT Insulm ors Ohm law I jam 60 2 06mm cow30160 3 EITheoretical Definition of a Metal Re039a TOwaODc T 13 7z1 272 7me2 1 6M Re039a T 00 gt 02quot gt 0 2 DC W 6a Drude DC a m ClTheoretical Definition of an Insulator 1T1 1310 IllilgloRe0a qa 0 PHYS 624 Quick and Dirty Introduction to Mott Insulators th vs Insulator Experiment Fundamental requirements for electron transport in Fermi systems 1 Quantummechanical states for electronhole excitations must be available at energies immediately above the ground state no gap since the external field provides vanishingly small energy 2 TheSe excitations must deSCribe delocalized charges no wave function localization that can r PHYS 624 Quick and Dirty Introduction to Mott Insulators contribute to transport over the macrOSCop ic sample sizes J Mott Insulators due to electronelectron interaction manybody physics leads to the Insm f fors ue lo 3 cr quot39i f39 gap in the charge excitation spectrum interaction Singleparticle physms EIMottHeisenberg antiferromagnetic order of O39Band Iquot U9T P3 eledmn 39mel39ads the preformed local magnetic moments below ith a periodic potential of the Ions gt N el Temperamm ap in the single particle spectrum EIMottHubbard no longrange order of local CIPeierls InSUlators electron interacts magne c momems ith static lattice deformations gt gap CIMottAnderson disorder correlations EIWigner Crystal Coulomb interaction dominates CIAndersonInSUlators electron a1 IOW dens of Charge PS 2DEHEF interacts With the disorderSUch as nS12ns33 or r 3D67 Thereby localizing impurities and lattice imperfections eledrons info a Wigner laln ce I PHYS 624 Quick and Dirty Introduction to Mott Insulators Elec rr on in a periodic po ren rial crys ral gt energy band gk 2tcoska 1D Tigh rbinding band N1 N2 N4 N8 i a Nco 4 PHYS 624 Quick and Dirty Introduction to M0 Insulators Band BlochWilson Insula ror39 Wilson39s r39ule 1931 par rially filled energy band a me ral otherwise a insula ror39 metal insulator semimetal Counter example TransiTionmeTal oxides halides chalcogenides Fe me ral wi rh 3d64sp2 FeO insula ror39 wi rh 3d6 PHYS 624 Quick and Dir fy In rr39oduc rion ro Mo r r Insula ror s Anderson Insula ror39 Inability edge density of states extended states localized States critical region PHYS 624 Quick and Dir fy Introduction To Mo H Insula i or s MoTT InsulaTor A solid in which sTrong repulsion beTween The parTicIes impedes Their flow gt simplesT carToon is a sysTem wiTh a classical ground sTaTe in which There is one parTicle on each siTe of a crysTalline laTTice and such a large repulsion beTween Two parTicIes on The same siTe ThaT flucTuaTions involving The moTion of a parTicle from one siTe To The nexT are suppressed i From weakly correlaTed Fermi liquid To sTrongly correlaTed MoTT insulaTors INSULATOR n STRANGE METAL 211 c 1 L 1ETAL STRONG CORRELATION WEAK CORRELATION PHYS 62439 Quick and DirTy InTroducTion To MoTT InsulaTors Mo r r Gedanken Experimen r 1949 iim ieg all 2H r if39 70an 39 allergy O O O Q d atomic distance a e oo a romic limi r no kine ric energy gain insula ror39 de 0 possible me ral as seen in alkali me rals Competition between M22f and U a MefalInsuafor Transi on eg V203 NiSSe2 PHYS 624 Quick and Dir fy In rr39oduc rion ro Mo r r Insula ror s EIBand insulaTor including familiar semiconducTors is sTaTe produced by a sulee quanTum inTerference effecTs which arise from The facT ThaT elecTrons are fermions NeverTheless one generally accounTs band insulaTors To be quotsimplequot because The band Theory of solids successfully accounTs for Their properTies EIGenerally speaking sTaTes wiTh charge gaps including boTh MoTT and Bloch Wilson insulaTors occur in crysTalline sysTems aT isolaTed occupaTion numbers where V is The number of parTicles per uniT cell V V ElAlThough The physical origin of a MoTT insulaTor is undersTandable To any child oTher properTies especially The response To dopingVV are only parTially undersTood DMoTT sTaTe in addiTion To being insulaTing can be characTerized by presence or absence of sponTaneously broken symmeTry eg spin anTiferromagneTism gapped or gapless low energy neuTral parTicle exciTaTions and presence or nce of Topological order and charge fracTionalizaTion PHYS 624 Quick and DirTy InTroducTion To MoTT InsulaTors Periodic Table with the Au Outer Electron Conn gurations of Neutral es h s m Thequot Ground 51m Li 59 39l lw mmquot mm m dume he elvvlrouuv xnnhuuruunn n ulnlnx Cquot N 0 F i m an hwd H411 pun pmm Y Tm ktun s p 1 Hg 4 xx 2v IL 1 m n mp Mgu Ipurx hlunrv m primawn q r cm A A IRN up m n rxghl Ktnon I r n n gt IL 1 K Sc Ti cw Mn Co m4 Cu Zn As Sequot 23ml2 1 3d 7 31quotquot 3dquot 3d T 3dquot 11 quot39 3d 39 4 a 48 4 45quot 4 4539 4x 4 4324 4A4p 4524 4 4 4554 4mm nbw 5r yu Zr Nbquot Ma Tc Ru nnw Inquot 5n 39 Te F Xe 4d 4m 4m 4d7 4m 5 5s 53 55 as 5 35 511 as syn 555 5 H pun Pc39 4 5m x 539 r 63 6pln39up u up L P u up39 u V Fm Ha 02 BY Hob Er Tm Yb Lun Gd 1 4f 4 y 4 4111 4 4 7V 739 731 5d 551 56 61 cs 4 cm 6 Es Thy cw ES Fm Mum Nam L PHYS 624 Quick and Dir Ty InTr oducTion 1390 Mo 39 Insulafor s Theoretical modeling Hubbard Hamiltonian Hubbard Hamiltonian 19605 onsite Coulomb interaction is most dominant H H frCilyC a ilrciallljz nlln39l 17 l band structure correlation eg U 5 eV W 3 eV for most 3d transitionmetal oxide such as MnO FeO CoO NiO Mott insulato A Hubbard s solution by the Green s function decoupling method a 3 a insulator for all finite Uvalue Al3d39 pl interaction U o Lieb and Wu s exact solution for the ground state of the 1D Hubbard model PRL 68 k a insulator for all finite Uvalue pband PHYS 624 Quick and Dirty Introduction to Mott Insulators g I s E m a a E g PHVS 624 Qumk and DWyImruduchun m Munlnsumurs Dynamical MeanField Theory in PicTures gt meanfield soluTion becomes exacT CIHubbard model gt singleimpuriTy Anderson model in a meanfield baTh CISolve exachy in The Time domain gt quotdynamicalquot meanfield Theory EIIn ooD spaTial flucTuaTion can be neglecTed Dynamical meanfield Theory DMFT of correlaTedelecTron solids replaces The full laTTice of aToms and elecTrons wiTh a single impuriTy aTom imagined To exisT in a baTh of elecTrons The approximaTion capTures The dynamics of elecTrons on a cenTral aTom in orange as iT flucTuaTes among differenT aTomic configuraTions shown here as snapshoTs in Time In The simplesT case of an s orbiTal occupying an aTom flucTuaTions could vary among I0 IT ll or ITl which refer To an unoccupied sTaTe a sTaTe wiTh a single elecTron of spinup one wiTh spin down and a doubly occupied sTaTe wiTh opposiTe spins In This illusTraTion of one possible sequence involving Two TransiTions an aTom in an empTy sTaTe absorbs an elecTron from The surrounding reservoir in each TransiTion The hybridizaTion VV is The quanTum mechanical ampliTude ThaT specifies how likely a sTaTe flips beTween Two differenT configuraTions PHYS 624 Quick and DirTy InTroducTion To MoTT InsuIaTors DSTa ric Har rr39eeFock or39 Densi ry Func rional Theory mm pm I Vextrgtpltrd3r I VKS P 2 1quot mom11531 013 Eexchangewn 2L m r s r r I 6E W e 2 VmpltrgtVmltrgt Jlff39gl 3r a ppmpr2f l 1 nl EIDynamic Dynamical MeanField Theory GAa Z co 2mm 1k 1 1 2Aa E Aa 1GAay rprgtG EmprG1nxtrgtprd3r 3 prpr 3rd3rlEeXChan epr 2 r r g PHYS 624 Quick and Dir fy In rr39oduc rion ro Mo r r Insula ror s DOS 1 t mZw t 9 J1 0 0 o m fer mionic quosipor ticles Frequency t PHYS 624 Quick and Dirty Introduction to Mott Insulators NOTE DOS welldefined even though there are no Model Mobile spin electr39ons interact with frozen spin electrons J W KN gt W e39 Ekko N particle N 1particle EN EfN 1 P 139 gt gt If gt PHYS 624 Quick and Dir fy In rr39oduc rion ro Mo r r Insula ror s surfacelayer thickness 10 12 cleavage plane side View O 0 Vanadium Oxygen tOp View PHYS 624 Quick and Dir ry In rr39oduc rion To Mo r r Insula ror39s 300 K Jim 700 e39V Elva Ell EV NTENSETY larbitxary units Phys Rev Lett 90 186403 2003 Photoemission spectrum of metallic vanadium oxide V203 near the metal insulator transition The dynamical meanfield theory calculation solid curve mimics the qualitative features of the experimental spectra The theory resolves the sharp quasiparticle band adjacent to the Fermi level and the occupied Hubbard band which accounts for the effect of localized d electrons in the lattice Higherenergy photons used to create the blue spectrum are less surface sensitive and can better resolve the quasiparticle peak PHYS 624 Quick and Dirty Introduction to Mott Insulators I Er RpmMAJEEI11 II 4 004 0232 3304 39 I I I 1 I I ll 39 n 11 MM INSUL TfJR MEFAL 30E P ESSUlTE EEIP quot3920 i n arms cr 0 0035 n TEMPE RATURE KII h 1 A D NTlFEFIRCM IGPJETIC I N E ULATD39H LI I l I I 39 Mun INCREASINQ PHESQ MRE n51 hb r DWISIUH EEHO PRESSURE POINT MET13955 WITH TD SCALE PHYS 624 Quick and Dir fy In rr39oduc rion ro Mo r r Insula ror s Wigner CrysTal Since The mid1930s TheorisTs have predicTed The crysTallizaTion of elecTrons If a small number of elecTrons are resTricTed To a plane puT inTo a liquidlike sTaTe and Squeezed They arrange ThemSelves inTo The lowesT energy configuraTion possiblea Series of concenTric rings Each elecTron inhabiTs only a small region of a ring and This bull39seye paTTern is called a Wigner crysTal Only a handful of difficulT experimenTs have shown indirecT evidence of This phenomenon gt 0 v 7 Li 9w 39 l3quot Q i quot a 0 Q galquot 31 3 o Q ll a Q 6 Q G 9 0 0 PHYS 624 Quick and DirTy InTroducTion To MoTT InsulaTors Beyond Solid S ro re Physics Bosonic Mo r r Insulo ror39s in Op ricol Lo r rices DEVOLUTION sUpem luid sToTe with coherence m l 2 ll f and superfluid sToTe offer resToring The coherence PHYS 624 Quick and Dir ry In rr oduc rion To Mo r r Insulators

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