課件-plecture第三章量子理想氣體

第三 量子理想氣三節體熱容的統計經典理 .這種情況稱為定Thestructures graphiteanddiamondhavelongbeenknown.Howeveronlyrecentlyhasthescanningtunnelingmicroscope(STM)givenvisualconfirmation.ShownbelowisanSTMimageofagraphitesurface.Thisisanactualimage,notasimulation.Itclearlyshowstheinterconnected6-memberedringsofgraphiteandthetriangulargeometryabouteachcarbonatom.設原子數目為??,則系統自由度有3??63??（減去31 m2r31

1 h3ZN ,Zexpsh3S

3dPi

exp3 3NNElnZN, 3N,E3NkT E 低溫區域,熱容量隨溫度降低而變小.當溫度趨近絕對零度,熱容量趨于零.故公式在低溫下,不適用.頻率??

n1 2Z'

exp

1exphlnZ'h 3 3N NN

Z'

E3NE

3N CV CV 2 2令xh

CV

x22 在高溫極限下??→ CV在低溫極限下??→∞x 3Nkx2CVx當溫度趨于絕對零度,熱容趨于零,符合熱力學第三定律.實際上理論對能斯特建立熱力學第三定律有重要影響.但是能斯特發現溫度趨于零時,德拜理里.彈性波和熱輻射一樣,有很多簡正振動.不同??,??????Lgd=24 c L L ct cl 1 3 3Vdg cl1 1 2

c

c3

22 3

3

t l 2

B 3N

2exp2exp U00

kTexpkT

22U00

kT4 EU0

dU0

exp

3 kT 3kT U0B D

exp

D

y x

kT kT 31x30xexpy1dy=UkT9N0 3x0expy1 =E

=3NkDx

Dx3NkDx3NkT3Dx D2 D2 x

x12NkDx3Nk T

4Dx expxT 3

x V 3

430 x3 00T 3

V4

dy3xexpx4 3xexpx

T33

0

D低溫下實驗也符合得很好.四節.理想費米氣ln2S1gln1exp

dln1exp

xy

xx

p2p2

p2)

p2)lnV2S

2mV2S

2m

2m

2S

??????1??=???

x1/2dxexpnx

n5/

x1/2expx

12n5/1

n5/

f5/

nzexp,fmz n1ln2S12V2m

z2S1V

hTPhT

1ln

2S1

5/Eln32S1kT

3 35/NlnV2S1f5/2z3 3dfm

N2S1Vf5/2z2S1Vzf5/2z 2S1V z T33

2S1

y

用級數解

f3/2z2S法反解方程

zy

y214

y3 PkTf5/2zkTz25/2

f3/2

VVz23/2

kT11z...kT11yV

25/

V

25/ kT n ...V 25/

2S 狀態方程狀態方程:PVkTN1 ... 25/22S1PVPVkTN1nT25/22S...fmz

10m10

z1et,1 1n如果z1:fmz n1

z

mz1et010z

n1n,0,T對能級和其簡并度分別為????,????,粒子數在該能級的平

ak

expf

????溫度為零情f

1,<T

T

exp

0, 0其中2來源于自旋的簡并度2??+1=2, ???? dDV4p2dpV

dD2V2m3/

3/

NT

2

dD2d

FF

h2

/

2/F

2m8V

F 1/ FpF

F 3n;

TheFermisurfaceofbi-layerBSCCO,calculated(left)andmeasuredbyARPES(right).ThedashedrectanglerepresentsthefirstBrillouinzone.

3/22

355E02Dd 550E03

5

FETSECN5/3V2/3 壓強：PF

3低溫定義費米溫度????????/??米溫度,氣體處于強簡并狀態.這時??≈??,???

3/N2dDf0

dh0h

exp I

1 Id0

exp

exp

dd

exp

exp dkT

kyTd

kT0

0

expy1

0

expy1

expy1expy1expy1

dy

expkT

kTy

expkT

dy1y1T expy1expkTdy1y1kTexpy1expkT

kTexpkTdy1y1expy1kTexpkT(1)0( y

kT

exp

expy1

0

0

y1

y1

y y

expy1

dyexpy1 0000

dyexpy1

12;dyexpy

I

2

2kT2

;A

8 3N2/3

18

8 kT

F解可用

F

2 2

2

1kT

1

F 8F

F 8

kT2

2

8F

12

1 2

231/2

AV

5

2

25/2

2 2E

1

2F 2F

12

F

F FF

N

24

12 F

F

F不同于經典粒子,經典粒子的平均能量只是溫度的函數.我們看到,對理想費米氣體粒E 熱容:CV

T

N5

12 2T

FNNUTSuTsPvNPVPVkTlnnEln3ln3kTln33 PV23

FGPVN23 2

N

NF