Deepak Rajput

Transcription

1 Q Prov: (a than an infinit point lattic is only capabl of showing,, 4, or 6-fold typ rotational symmtry; (b th Wiss zon law, i.. if [uvw] is a zon axis and (hkl is a fac in th zon, thn hu + kv + lw ; (c that in th cubic systm th dirction [hkl] is paralll to th fac-normal (hkl; (d that in th cubic systm th angl φ btwn th fac-normals (h k l and (h k l is givn by ( hh + kk + ll cosφ [ h + k + l [ h + k + l ( that, whn using Millr-Bravais indics (hkil, h + k + i Ans (a Lt s considr a rgular polygon with n sids. ach vrtx acts lik a rotation point. According to trigonomtry of a rgular polygon, intrior angl subtndd by two succssiv π arms of a rgular polygon is always π radians. If m facs mt at vrtx, thn quation n that satisfis this condition is: π π π n m > + n m > m and n can only b, 4 or 6. It implis that a plan can only b filld with convx or rgular polygons which ar ithr quilatral triangls, squars, or hxagons (i.. n, 4, or 6 ot: n cannot b. It mans thr s no plan and polygon is actually a straight lin. And a straight lin cannot fill any plan. It nds to hav at last points to form a closd structur and fill th plan. (b Unit vctor passing through th fac plan (hkl is nothing but h k l Pˆ iˆ + ˆj + kˆ x y z and th Wiss zon axis vctor for its dirction [uvw] is r W uxiˆ + vyj ˆ + wzkˆ But ths ar actually prpndicular, and must satisfy th quation h i.. ˆ k ˆ l ˆ i + j + k. uxiˆ + vyj ˆ + wzkˆ x y z > hu + kv + lw r Pˆ.W (i.. dot product (c In th cubic systm, quation of unit vctor of th plan bcoms h k l Pˆ iˆ + ˆj + kˆ, whr a is th lattic paramtr a a a Th dirction vctor for (hkl is Th Univrsity of Tnnss Spac Institut, Tullahoma, Tnnss 788

2 r D haiˆ + kaj ˆ + lakˆ Thir cross product is Pˆ XD r h ˆ k ˆ l ( i + j + kˆ X ( haiˆ + kaj ˆ + lakˆ, which is qual to. It a a a implis that thy ar paralll to ach othr. (d Lt us considr th unit vctor passing through h k l and h k l b V ˆ d( h ˆ ˆ i + k j + lkˆ / a and V ˆ d ( hiˆ + k ˆ j + lkˆ / a rspctivly, whr d and d ar th intrplanar distancs and a and a ar th lattic paramtrs. Thir cross product isv ˆ. Vˆ Vˆ Vˆ cosφ, but thir mod is qual to as thy ar unit vctors. Hnc, th quation rducs to Vˆ.Vˆ cosφ Also, intrplanar distanc is rlatd to lattic paramtr and position vctor as a d h + k + l Hnc, cosφ d d ( hh + kk + ll / aa Rplacing a and a, and th quation rducs to ( h h + kk + ll cosφ.. provd h + k + l h + k + l ( A a/h O -a/i C a/k B Y X -U In th afordrawn figur, Ara of triangl OAB is qual to sum of aras of triangls of OAC and OCB. rom trigonomtry w know that ara of triangl is ½ ab sinc (whr a and b ar th lngths of th sids of triangl and C is angl subtndd btwn thos sids. Hnc, a a a a a a sin AOC sin BOC sin AOB, sin AOC sin BOC sin AOB * h i k i h k > + hi ki hk > h + k + i.. provd * (ot AOC BOC 6 o and AOB o, also sin o sin 6 o Th Univrsity of Tnnss Spac Institut, Tullahoma, Tnnss 788

3 Q Driv an xprssion for th rmi nrgy of a fr lctron mtal at zro tmpratur. Using th data givn, and any othr constants, valuat th rmi nrgy of th alkali mtals. Li a K Rb Cs Dnsity, gcm Atomic wight How would you masur th rmi nrgy xprimntally for ths mtals? Ans rmi nrgy is dfind as th highst nrgy occupid by an lctron at absolut zro. According to rmi-dirac statistics, th probability that a particl (frmion will hav nrgy ( is givn by f (, and at absolut zro its valu is -- ( ( / KβT + Suppos th volum (V contains lctrons. Thn, ( f ( d ( d Vm π h > g, but f ( from quation. Th quation rducs to: Vm g, but g( m π h V m > π h > > V m π h h π m V md d / / Th Univrsity of Tnnss Spac Institut, Tullahoma, Tnnss 788

4 Q Driv an xprssion for th ratio κ of th thrmal and lctrical conductivitis of a σ fr-lctron mtal. Calculat th valu of th Lorntz numbr κ L σ T Whr T is th absolut tmpratur. xplain th discrpancy btwn th calculatd valu and th following masurd valus of L for sodium at low tmpraturs. T, o K 6 L, W ohm dg -.7 x -8.7 x -8. x -8. x -8 Ans Thrmal conductivity (κ π nkβ Tτ m n τ lctrical conductivity (σ m Using ths two quations w can driv an xprssion for th ratio κ, which is σ κ π κ β > T --- ( σ κ π kβ κ T, it implis that T σ σ κ π kβ or LT, whr L, which is calld as Lorntz numbr. Its valu can b asily σ calculatd by substitution th valus of k and. ( k is.87 x - and is.6 x -9 π kβ.87x L > L π.6x > L.4x 8 W-ohm/K 9 β π kβ Lorntz numbr quation is L, which dosn t dpnd on Tmpratur i.. Lorntz numbr is tmpratur-indpndnt and its valu dpnds on th valus of Boltzmann constant and lctronic charg. But in rality Lorntz numbr dpnds on th rlaxation procsss for lctrical and thrmal conductivity bing th sam, which is not tru for all th tmpraturs, and bcaus of various assumptions takn (according to Drud s thory, thr is a discrpancy btwn th calculatd valus and th masurd valus. β 4 Th Univrsity of Tnnss Spac Institut, Tullahoma, Tnnss 788

5 Q 4 Assuming that silvr is a monovalnt mtal with a sphrical rmi surfac, calculat th following quantitis: (i rmi nrgy and rmi tmpratur (ii Radius of th rmi surfac (iii rmi vlocity (iv Cross-sctional ara of th rmi surfac (v Cyclotron frquncy in a fild of orstd (vi Man fr path of lctrons at room tmpratur and nar absolut zro (vii Orbital radius in a fild of orstd (viii Lngth of th sid of th cubic unit cll (ix Lngths of th first two sts of rciprocal lattic vctors in k-spac (x Volum of th Brillouin zon Dnsity of silvr. g cm - Atomic wight 7.87 g Rsistivity.6 x -6 ohm cm at 9 o K and.8 x -6 ohm cm at o K r s Ans 4 or silvr. (rfrrd pag 6, Ashcroft/Mrmin a.v (i rmi nrgy ε and rmi tmpratur T r a ( > rmi nrgy ε is.49 V and rmi tmpratur (ii Radius of th rmi sphr.6 r s a > Radius of th rmi sphr κ is. Å - (iii rmi vlocity v s ( r a κ Å x cm / sc > rmi vlocity is.9 x 8 cm/sc v s T 8. ( r a is 6.8 x 4 K s x 4 K (iv Cross-sctional ara of th rmi surfac A 4πκ Å - > Cross-sctional ara of th rmi surfac A is 8. Å - (v Cyclotron frquncy in a fild of orstd > ν c.8h x 6 Hz 4 x 9 Hz or 4 TraHz (vi Man fr path of lctrons at room tmpratur and nar absolut tmpratur [ ] > Man fr path is l Α Χ ( r 9 s a ρ μ > l RT.7 Å and l AZ.8 μ (RT is Room tmpratur and AZ is ar Absolut Zro o Th Univrsity of Tnnss Spac Institut, Tullahoma, Tnnss 788

6 (vii Orbital radius in a fild of orstd ω H c mc, and v rωc > ω c.9 x 6 s - and r.44 m (viii Lngth of th sid of th cubic cll Givn th dnsity of Silvr (. gcm - and Atomic wight (7.87 amu Atomic wight 7.87 x.66 x -4 g.79 x - g Dnsity of silvr. mass of on atom/volum of on atom.79 x - /(4πr / > r 4 x -4, and r.88 Ǻ As Silvr is a monovalnt lmnt, lattic paramtr is tims of radius > lattic paramtr (a 4.46 Ǻ (ix Lngths of th first two sts of rciprocal lattic vctors in k-spac It s nothing but π a > lngths.49 Ǻ - ach (x Volum of th Brillouin zon Volum of Brillouin zon is nothing but th volum of rciprocal lattic in k-spac i.. V B (π /V, whr V is th volum of th unit cll i.. a > V B 8π /V, and V is 88.7 Ǻ - > V B.796 Ǻ 6 Th Univrsity of Tnnss Spac Institut, Tullahoma, Tnnss 788

7 Q Show that th man nrgy pr particl at o K for lctrons obying rmi-dirac statistics is (, Whr ( is th rmi nrgy at T. Assuming th valu of this quantity at a finit tmpratur is π kt ( + ( ind th valu of th ratio (C v D /(C v Cl for an lctron gas with rmi nrgy 7 V, whr (C v D is th spcific hat of a gas of particls obying rmi-dirac statistics and (C v Cl is th spcific hat of a gas obying classical statistics. Ans Total nrgy of lctrons at o K can b xprssd as: total g( d quation ( Vm W know that g( m π h Thus quation ( rducs to Vm total md π h m V total d π h m V π h total m V / total d π h (quation But w know that at absolut zro rmi nrgy xprssion is: / h π (provd arlir, rfr solution to problm numbr m V / h π ---- quation ( / (m V quation can r-writtn as / m V / total π h m V / h π total m V / π h ( 7 Th Univrsity of Tnnss Spac Institut, Tullahoma, Tnnss 788

8 This quation finally rducs to total, whr is total numbr of lctrons man nrgy pr particl is nothing but Total nrgy dividd by total numbr of lctrons total i.. man, whr is at absolut zro i.. ( Hnc man ( As givn in th problm th xprssion for man nrgy pr particl at a finit tmpratur T is π kt man ( + ( W know that (C v D is nothing partial drivativ of man at constant volum, th quation for man (C v D is π kt ( C v D ( + T T ( ( C v D π ( k β T ( k β T ( Cv D ( π ( W know that spcific hat for gas obying classical statistics is givn by ( C v Cl k β T ( Cv D π kβt Th ratio, givn ( 7 V and k.867 x -4 β V/K ( Cv Cl ( ( Cv D 4.x T, whr T is th tmpratur in absolut scal ( Cv Cl Hnc th ratio of (C v D /(C v Cl for an lctron gas with rmi nrgy 7 V is 4. x - T 8 Th Univrsity of Tnnss Spac Institut, Tullahoma, Tnnss 788