fiziks Institute for NET/JRF, GATE, IIT JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics CONDENSED MATTER PHYSICS

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1 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics CONDENSED ATTER PHYSICS NET/JRF (JUNE-) Q. A nrrow be of X-rs with wvelength.5 Ǻ is reflected fro n ionic crstl with n fcc lttice structure with densit of. gc -. The oleculr weight is 8 AU (AU =.66 - g) A. The lttice constnt is () 6. Ǻ (b).56 Ǻ (c). Ǻ (d).56 Ǻ () - Solution: Given n, = 8 kg,.g c = kg -, e f f N A 6. 6 tos kd ne f N f A B. The sine of the ngle corresponding to () reflection is 6. 6.A () (b) (b) Solution: According to Brgg s lw 8 d sin, sin where d d (c) (d) 8 for () plne h k l.5a sin. 6A 6 8 Q. A flu untu (fluoid) is pproitel eul to -7 guss-c. A tpe II superconductor is plced in sll gnetic field, which is then slowl incresed till the field strts penetrting the superconductor. The strength of the field t this point is 5 guss. A. The penetrting depth of this superconductor is () Ǻ (b) Ǻ (c) Ǻ (d) Ǻ () 7 Solution: Given Fluoid guss c H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co

2 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics H c guss First Criticl field 5 The reltion between first criticl field nd penetrtion depth is H c H c 7. 5 c 6 c A B. The pplied field is further incresed till superconductivit is copletel destroed. 8 5 The strength of the field is now guss. The correltion length of the superconductor is () Ǻ (b) Ǻ (c) 68 Ǻ (d) Ǻ None of the options is tched. 8 H c guss. The reltion between second criticl field nd correltion length is H c. 7 c 6 c 5A H c 8 5 Solution: Given second criticl field 5 Q. The two diensionl lttice of grphene is n rrngeent of Crbon tos foring honecob lttice of lttice spcing, s shown below. The Crbon tos occup the vertices. (A). The Wigner-Seit cell hs n re of d c c () (c) (b) 6 (d) d b b (d) Solution: Priitive lttice vectors re b nd b b i 6 j b i ˆj ˆ ˆ ˆ ˆ i j, A bb H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co

3 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics (B). The Brvis lttice for this rr is () rectngulr lttice with bsis vectors d nd d (b) rectngulr lttice with bsis vectors c nd c (c) hegonl lttice with bsis vectors nd (d) hegonl lttice with bsis vectors b nd b (c) Solution: The Brviss lttice for this rr is the Hegonl lttice with bsis vectors nd NET/JRF (DEC-) Q. The potentil of ditoic olecule s function of the distnce r between the tos is b given bv r. The vlue of the potentil t euilibriu seprtion between 6 r r the tos is: () / b (b) / b (c) / b (d) / b (d) b dv Solution: Given r V r. At euilibriu rdius, 6 r r dr rr dv r dr 6 7 r b r r b b 7 r 6 r 6 b The vlue of potentil t euilibriu isv r 6 r b r b b. b Q5. If the nuber densit of free electron gs in three diensions is incresed eight ties, its Feri teperture will () increse b fctor of (b) decrese b fctor of (c) increse b fctor of 8 (d) decrese b fctor of 8 () Solution: The reltion between Feri energ nd electron densit is E n / F. / 8n EF ' EF. H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co

4 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics Q6. The ecittions of three-diensionl solid re bosonic in nture with their freuenc ω nd wve-nuber k re relted b ω k in the lrge wvelength liit. If the cheicl potentil is ero, the behviour of the specific het of the sste t low teperture is proportionl to () T / (b) T (c) T / (d) T (c) s Solution: If the dispersion reltion is k in lrge wvelength. Then the specific het is C v T / s. Given / k C v T NET/JRF (JUNE-) Q7. Consider sste of non-intercting prticles in d diensionl obeing the dispersion s reltion Ak, where ε is the energ, k is the wvevector, s is n integer nd A is constnt. The densit of sttes, N(ε), is proportionl to s d d s s d () (b) (c) (d) (b) Q8. The eperientll esured trnsission spectr of etl, insultor nd seiconductor thin fils re shown in the figure. It cn be inferred tht I, II nd III correspond, respectivel, to d s () insultor, seiconductor nd etl (b) seiconductor, etl nd insultor (c) etl, seiconductor nd insultor (d) insultor, etl nd seiconductor () Trnsission (%) Wvelength (n) Q9. The energ reuired to crete lttice vcnc in crstl is eul to ev. The rtio of the nuber densities of vcncies n( K)/n( K) when the crstl is t euilibriu t K nd K, respectivel, is pproitel () ep (b) ep 5 (c) ep 5 (d) ep (d) H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co

5 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics Solution: The eution for nuber densit of vcncies to for vcncies, N: densit of lttice sites E / k E BT e kb T T e E / kbt e n Ne n, K k e n B e n K Q. The dispersion reltion of phonons in solid is given b E / k BT E E n k k k k k The velocit of the phonons t lrge wvelength is B where E: Energ reuired () / (b) (c) (d) / (d) Solution: For lrge, k, k, k re sll. k d k k k vg. dk e k k k k k k NET/JRF (DEC-) Q. A gnetic field sensor bsed on the Hll Effect is to be fbricted b iplnting As into Si fil of thickness μ. The specifictions reuire gnetic field sensitivit of 5 V/Tesl t n ecittion current of A. The iplnttion dose is to be djusted such tht the verge crrier densit, fter ctivtion, is () (b).5 - (c). - (d). - (b) IB Solution: n.5 where V H V / T. tev.6 5 B H Q. In bnd structure clcultion, the dispersion reltion for electrons is found to be k k k, k where β is constnt nd is the lttice constnt. The effective ss t the boundr of the first Brillouin one is () 5 (b) 5 (c) (d) H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co 5

6 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics (d) Solution: k k k, Effective ss k Brilliouin one boundr is t k d k d k, k, k. d Hence k d k,,. Q. The rdius of the Feri sphere of free electrons in onovlent etl with n fcc structure, in which the volue of the unit cell is, is () / (b) / (c) / (d) () Solution: Rdius of Feri sphere is k F N V / / k F, EF n / N For fcc solid k F V. NET/JRF (JUNE-) Q. Using the freuenc-dependent Drude forul, wht is the effective kinetic inductnce of etllic wire tht is to be used s trnsission line? [In the following, the electron ss is, densit of electrons is n, nd the length nd cross-sectionl re of the wire nd A respectivel.] () A / ne (b) ero (c) / ne A (d) A / ne (c) Q5. The phonon dispersion for the following one-diensionl ditoic lttice with sses nd (s shown in the figure) K H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co 6

7 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co 7 is given b sin K where is the lttice preter nd K is the spring constnt. The velocit of sound is () K (b) K (c) K (d) K (b) Solution: For sll vlue of (i.e. long wvelength pproition liit). We hve sin sin For Acousticl brnch: Velocit of sound is v g

8 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics Q6. The electron dispersion reltion for one-diensionl etl is given b k k sin sin k 6 where k is the oentu, is the lttice constnt, is constnt hving diensions of energ nd k. If the verge nuber of electrons per to in the conduction bnd is /, then the Feri energ is () / (b) (c) / (d) 5 / () Q7. If the energ dispersion of two-diensionl electron sste is E uk where u is the velocit nd k is the oentu, then the densit of sttes D depends on the energ s () / E (b) E (c) E (d) constnt (c) Solution: In two diensionl sste, the nuber of llowed k-sttes in rnge k nd E k dk is g k dk L kdk. Given dispersion reltion is E uk k E de dk u u g E L de E E de u u g de L E E. de u L u EdE NET/JRF (DEC-) Q8. The phsicl phenoenon tht cnnot be used for eor storge pplictions is () lrge vrition in gnetoresistnce s function of pplied gnetic field (b) vrition in gnetition of ferrognet s function of pplied gnetic field (c) vrition in polrition of ferroelectric s function of pplied electric field (d) vrition in resistnce of etl s function of pplied electric field (d) H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co 8

9 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics Q9. The energ of n electron in bnd s function of its wve vector k is given, where E, B nd re constnts. The b Ek E B k k k effective ss of the electron ner the botto of the bnd is () (b) (c) (d) B B B B (d) Solution: Ner the botto of the bnd the k k k, k k, k k E k E B k k k E B k k k E B k k k E B B k * Effective ss of the electron is d E / dk B Q. A DC voltge V is pplied cross Josephson junction between two superconductors with phse difference. If I nd k re constnts tht depend on the properties of the junction, the current flowing through it hs the for evt evt () I sin (b) kv sin (c) kv sin (d) I sin kv () Q. A unifor liner onotoic chin is odeled b spring-ss sste of sses seprted b nerest neighbour distnce nd spring constnt reltion for this sste is. The dispersion k k () k (b) k sin k k (c) k sin (d) k tn (c) Solution: The dispersion reltion for unifor liner ono-toic chin of tos is k k sin H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co 9

10 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co NET/JRF (JUNE-) Q. The pressure of nonreltivistic free Feri gs in three-diensions depends, t T, on the densit of ferions n s () 5 / n (b) / n (c) / n (d) / n () Solution: The Feri energ in three diension is defined s / / n V N E F Where, n is the electron concentrtion or densit of free Feri gs. The totl energ of free Feri gs in D is / 5 5 V N N NE E F The pressure of nonreltivistic free Feri gs is defined s 5 / / 5 V N N V E p N F 5/ / / n n n ne F Q. Consider n electron in b.c.c. lttice with lttice constnt. A single prticle wvefunction tht stisfies the Bloch theore will hve the for r ik r f. ep, with r f being () (b) (c) (d) (b) Solution: The priitive trnsltionl vector for BCC is

11 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics Bloch function defined s r u ik. r ik. r e f r e ˆ ˆ ˆ ˆ ˆ ˆ,, ˆ ˆ ˆ ' i jk b ' i jk c ' i jk k k r r Here f is toic wvefunction, which hs the periodicit of the lttice i.e. u r u r k k Given Bloch function f ( r) ' f ( r ) f ( r ' ) ' f ( r ) ' f ( r ) f ( r ) Siilrl, ' ' f ( r b ) f ( r ) nd f ( r c ) f ( r ) f ( r ) Other functions do not stisf the periodicit Q. The dispersion reltion for electrons in n f.c.c. crstl is given, in the tight binding pproition, b k k k k k k k where is the lttice constnt nd is constnt with the diension of energ. The - coponent of the velocit of the electron t,, is () / (b) / (c) / (d) / H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co

12 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics (d) Solution: Group velocit of electron in dispersive ediu is epressed s v d d d d iˆ ˆ j kˆ viˆ v ˆ j vkˆ dk dk dk dk k k k k sin sin ˆ k k k k i sin sin ˆj v k k k k sin sin k ˆ At,, sin sin ˆ v i sin sin ˆ j sin sin kˆ v iˆˆjkˆ iˆˆjkˆ v ˆ ˆ ˆ i v jv k v, v, v The - coponent of velocit is v NET/JRF (DEC-) Q5. When lser light of wvelength flls on etl scle with engrvings t gring ngle of incidence, it is diffrcted to for verticl chin of diffrction spots on screen kept perpendiculr to the scle. If the wvelength of the lser is incresed b n, the ngle of the first-order diffrction spot chnges fro () 6.6 (b) 5. (c) 5 to 5.8 (d) (c) Solution: The condition of i pek in grting is bsin ;,,,, where b is the width of slit or width of engrving, wheres is the order of diffrction nd is the ngle of diffrction For st order diffrction: bsin (i) H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co

13 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics When wvelength of incident light incresed to n, let s ssue the st order pek ppers t bsin (ii) Subtrcting eution (i) fro eution (ii), we get bsin bsin 9 sin sin sin sin b sin sin sin sin Q6. Consider the crstl structure of sodiu chloride which is odeled s set of touching spheres. Ech sodiu to hs rdius r nd ech chlorine to hs rdius r. The centres of the spheres fro siple cubic lttice. The pcking frction of this sste is r r () (b) r r r r r r r r r (c) r r r r r (d) r r (b) Solution: This uestion cn onl be solved b solving ech option b ssuing r r nd copring result with the pcking frction of siple cubic which is. 6 r r Option (): rr rr r r r Option (b): r r 8r 6 r r r Option (c): r r 8r r r r Option (d): r r 8r 8 Thus, correct option is (b) H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co

14 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics Q7. Consider two crstlline solids, one of which hs siple cubic structure, nd the other hs tetrgonl structure. The effective spring constnt between tos in the c -direction is hlf the effective spring constnt between tos in the nd b directions. At low tepertures, the behviour of the lttice contribution to the specific het will depend s function of teperture T s () T for the tetrgonl solid, but s T for the siple cubic solid (b) T for the tetrgonl solid, nd s (c) T for both solids (d) (d) T for both solids T for the siple cubic solid Solution: The specific het of solid in three diensions is proportionl to T nd it is independent of crstl structure. In D : CV T In D : CV T In D : CV T Q8. A superconducting ring crries sted current in the presence of gnetic field B norl to the plne of the ring. Identif the incorrect stteent. () The flu pssing through the superconductor is untied in units of hc / e (b) The current nd the gnetic field in the superconductor re tie independent. (c) The current densit j nd B re relted b the eution j B, where is constnt (d) The superconductor shows n energ gp which is proportionl to the trnsition teperture of the superconductor () Solution: The flu untition in superconducting ring is n o hc where o in CGS units nd e h o in KS units. e H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co

15 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics NET/JRF (JUNE-5) Q9. X -r of wvelength is reflected fro the plne of siple cubic lttice. If the lttice constnt is, the corresponding Brgg ngle (in rdin) is () (b) (c) (d) 6 8 Ans. (c) Solution: According to Brgg s Lw d sin where d for plne h k l sin d sin Q. The criticl gnetic fields of superconductor t tepertures K nd 8 K re A / nd 5.5 A/ respectivel. The trnsition teperture is pproitel () 8. K (b).6 K (c).9 K (d) 5. K Ans. (b) Solution: The reltion between criticl field nd criticl teperture is T HC T H TC Let t T T, H T, T T, H T H T C Thus we get C T T HC T H, HC T H TC TC T H C T T T HCT TC HC T T C HC T T HC T T HC T C where T k, T 8k, H T A nd C C / T C 8.6 H T 5.5 A/ C H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co 5

16 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics Q. The low-energ electronic ecittions in two-diensionl sheet of grphee is given b E k vk, where v is the velocit of the ecittions. The densit of sttes is proportionl to E () E (b) (c) Ans. () Solution: The nuber of k - sttes in rnge k nd k gkdk L kdk E k de dk The densit of stte is E (d) dk in two diension is E L E de L g E de EdE g E de L E E E E de Q. A He Ne lser opertes b using two energ levels of Ne seprted b.6 ev. Under sted stte conditions of opticl puping, the euivlent teperture of the sste t which the rtio of the nuber of tos in the upper stte to tht in the lower stte will be, is pproitel (the Boltnn constnt 5 k 8.6 ev / K ) () K (b) 8 K (c) Ans. (d) Solution: According to Boltnn reltion N E ep N kt N kt E ep T N E N k ln N N E ev k ev K N.6, 5 B 8.6 /, 6 K (d) B E E K E N N.6 T.877 K ln T K H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co 6

17 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics NET/JRF (DEC-5) Q. The first order diffrction pek of crstlline solid occurs t scttering ngle of when the diffrction pttern is recorded using n -r be of wvelength.5n. If the error in esureents of the wvelength nd the ngle re.n nd respectivel, then the error in clculting the inter-plnr spcing will pproitel be (). n (b). n (c).5 n (d). n Ans.: () d d Solution: Brgg s Lw for n, dsin d, sin sin sin Error in d cn be clculted s d d d sin sin d sin sin d sin sin sin d d d d tn tn where,.5,., o o.5 d.5 o sin sin Thus,..5 8 d.5 tn d n H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co 7

18 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics Q. The dispersion reltion of electrons in -diensionl lttice in the tight binding pproition is given b, k k k k where is the lttice constnt nd,, re constnts with diension of energ. The effective ss tensor t the corner of the first Brillouin one,, is () / / / (b) / / / (c) Ans.: (c) / / (d) / Solution: The effective ss s tensor untit cn be written s * * * * * * * ij where * * * * ij E kik since k k k k j / / / * kk, k * k k * k, other ters re ero k Now, t,, ;,, * * * * ij / / / H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co 8

19 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics Q5. A thin etl fil of diension contins electrons. The gnitude of the Feri wvevector of the sste, in the free electron pproition, is () c 7 (b) c 7 (c) 7 c (d) c 7 Ans.: (b) Solution: This is the cse of two diensionl etl bo. The Feri wve vector of electron in D is N kf n ; L c L k F 7 c c c Q6. For n electron oving through one-diensionl periodic lttice of periodicit, which of the following corresponds to n energ eigenfunction consistent with Bloch s theore?, () Aepi (b) Aepi (c) Aepi ih Ans.: (b) Solution: According to block theore, Aep i (d) Aep i Aep i Aepi i Aep i H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co 9

20 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics NET/JRF (JUNE-6) Q7. Consider electrons in grphene, which is plnr onotoic ler of crbon tos. If the dispersion reltion of the electrons is tken to be k ck (where c is constnt) over the entire k -spce, then the Feri energ F depends on the nuber densit of electrons s () F (b) F () Solution: In D, densit of stte is L g k dk (c) F kdk, where ck k nd c L. d g d L d c c c Now, nuber electrons t T K is (d) d dk c F L F L N N g d d F F c c c c L c F / / F Q8. Suppose the freuenc of phonons in one-diensionl chin of tos is proportionl to F the wve vector. If n is the nuber densit of tos nd c is the speed of the phonons, then the Debe freuenc is () cn (b) cn (c) cn (d) (d) Solution: Given k ck (c is velocit of phonon) L d L g d d d/ dk c Now L L D c c D Also D N g d d N N N D c cn, n L L cn fd. Best nswer is (d). cn H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co

21 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics Q9. The bnd energ of n electron in crstl for prticulr k -direction hs the for k A B k, where A nd B re positive constnts nd k. The electron hs hole-like behviour over the following rnge of k : () k (b) k (c) k (d) k () d, d Bsink, B k dk dk * Effective ss d / dk B k * * Effective ss of electron e nd effective ss of holes h re opposite in sign i.e., * * h e. * Now, in the rnge k, is positive * While in the rnge k, is negtive Thus, electron hs hole like behviour in the region k Solution: k A B k NET/JRF (DEC-6) Q. Consider hegonl lttice with bsis vectors s shown in the figure below. If the lttice spcing is, the reciprocl lttice vectors re (),,, (b),,, (c) Ans. : (),,, (d),,, H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co

22 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics Solution: Fro the figure, we cn write ˆ ˆ, ˆ, ˆ Now V. ˆ ˆ. ˆˆ ˆ ˆ. ˆ (let us ssue) Also, ˆ ˆ ˆ ˆˆ Reciprocl lttice vectors re * ˆ ˆ ˆ ˆ ˆ V ˆ ˆ * ˆ ˆ ˆ ˆ V * for : ˆ ˆ *, ˆ ˆ Q. Consider one-diensionl chin of tos with lttice constnt. The energ of n electron with wve-vector k is k k, where nd re constnts. If n electric field E is pplied in the positive -direction, the tie dependent velocit of n electron is (In the following B is the constnt) ee () Proportionl to B t (b) proportionl to E ee (c) independent of E (d) proportionl to sin B t Ans. : (d) Solution: In the presence of electric field E, we cn write dp dk F ee ee ee dt dt H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co

23 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics ee k t k t Integrtion gives, d k The group velocit v dk dk Since, k k Thus, v sin k, k k sin k Tie dependent velocit of electron is vt sin kt sin ee k t ee sin k t ee vt sin B t Q. A thin rectngulr conducting plte of length nd width b is plced in the -plne in two different orienttions s shown in the figures below. In both cses gnetic field B is pplied in the -direction nd current flows in the direction due to the pplied voltge V. b B B V V b Ans. : (d) V If the Hll voltge cross the -direction in the two cses stisf V V the rtio : b ust be () : (b) : (c) : (d) : Solution: Since, Hll voltge is given b VH V IB, where w is width of conducting plte. w H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co

24 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics l Since, in cse (I), V IR nd R A b b V I I b bv Then, V V w H IB bvb bvb w w l b And lso in cse (II), R A b V V V IR I R IB VB Then, VH V w b Since, V V : b : b NET/JRF (JUNE-7) Q. The energ gp nd lttice constnt of n indirect bnd gp seiconductor re.875 ev Ans. : () nd.5n, respectivel. For siplicit tke the dielectric constnt of the teril to be unit. When it is ecited b brodbnd rdition, n electron initill in the vlence bnd t k kes trnsition to the conduction bnd. The wvevector of the electron in the conduction bnd, in ters of the wvevector k t the edge of the Brillouin one, fter the trnsition is closest to () k / (b) k / (c) k / (d) Solution: The K vlue of electron in CB.. is K E 9 kg J / JS.. K t the Brillouin Zero is K 9 K K K H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co

25 Institute for NET/JRF, GATE, IIT JA,.Sc. Entrnce, JEST, TIFR nd GRE in Phsics Q. The electricl conductivit of copper is pproitel 95% of the electricl conductivit of silver, while the electron densit in silver is pproitel 7% of the electron densit in copper. In Drude s odel, the pproite rtio / of the en collision tie in copper Cu to the en collision tie in silver Ag Cu is (). (b).5 (c). (d).66 Ans. : (d) ne cu ncu cu cu n cu Ag Solution: n n Ag Ag Ag Ag Ag cu.95 cu Ag.7ncu.66 n Ag Ag cu Ag H.No. D, Ground Floor, Ji Sri, Ner IIT, Hu Khs, New Delhi 6 Phone: / Website: Eil: fiiks.phsics@gil.co 5

fiziks Institute for NET/JRF, GATE, IIT JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics

fiziks Institute for NET/JRF, GATE, IIT JAM, M.Sc. Entrance, JEST, TIFR and GRE in Physics Solid Stte Physics JEST-0 Q. bem of X-rys is incident on BCC crystl. If the difference between the incident nd scttered wvevectors is K nxˆkyˆlzˆ where xˆ, yˆ, zˆ re the unit vectors of the ssocited cubic