6. LOW-TEMPERATURE PROPERTIES OF NON-CRYSTALLINE SOLIDS
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6. LOW-TEMPERATURE PROPERTIES OF NON-CRYSTALLINE SOLIDS T > 1 K : other low-frequency excitations, “soft modes”, and the Soft-Potential Model
description
6. LOW-TEMPERATURE PROPERTIES OF NON-CRYSTALLINE SOLIDS T > 1 K : other low-frequency excitations, “soft modes”, and the Soft-Potential Model. T > 1 K ?. U. Buchenau e t al ., Phys.Rev. B34 , 5665 (1986). Basic assumptions of the SOFT-POTENTIAL MODEL:. - PowerPoint PPT Presentation
Transcript of 6. LOW-TEMPERATURE PROPERTIES OF NON-CRYSTALLINE SOLIDS
6. LOW-TEMPERATURE PROPERTIESOF NON-CRYSTALLINE SOLIDS
T > 1 K: other low-frequency excitations, “soft modes”, and the Soft-Potential Model
T > 1 K ?
U. Buchenau et al., Phys.Rev. B34, 5665 (1986)
2
Basic assumptions of the SOFT-POTENTIAL MODEL:
The SOFT-POTENTIAL MODEL
V(x) = W (D1 x + D2 x2 + x4 )
(i) The soft modes can on average be characterized by a single energy W (ii) D1 and D2 are randomly distributed around the origin of the D1 - D2 plane:
P(D1 , D2) = P(0, 0) Ps (iii) The interaction between the soft modes and the sound waves isbilinear in the displacement of the soft mode and in the strain fieldof the sound wave:
Vl,t = l,t x l,t
D2 < 0
V(x) = W x4
D2 > 0D2=D1=0
V(x) = W (D1 x + D2 x2 + x4 )
TUNNELING MODEL SOFT POTENTIAL MODEL
2
02,DWPPP s 2/2Djj
WP
P jsj
22
0
2/3
20 32exp DW 21 2DWD
SPM universal constants:
SOFT-POTENTIAL MODEL: Specific heat
)(ln
91
6)( exp3/1
3/12
, Tt
WTkkPTC
min
BBsTLSp
56
, 212)(
WTkkPTC B
Bssmp
minTW 28.1
0.1 0.5 1 5 10
0.5
1
5
10
50
W /1.85
T5T
Phillips Zeller&Pohl
Debye
SiO2
(Cp-C
Deb)/
T3 (J/
g·K
)
T (K)
53)( TCTCTCTC smDTLSp
)(ln
91
6)( exp3/1
3/12
, Tt
WTkkPTC
min
BBsTLSp
56
, 212)(
WTkkPTC B
Bssmp ]·28.1[ minTW
SPM fit:
{
1 10
2
4
6
810
20
T3Debye
2020.5 5
(B2O3)84 (Na2O)16
C p/T3 (
J/g∙
K)
T (K)
,
M.A.R. , Philos. Mag. (2004)
CALCULATION OF THE THERMAL CONDUCTIVITY
(
( )
( )
( )
max·6.1 TW
CBTkW min85.1
Tc
max6.1 TkW B
2.1/1
QcBTkW
SPM
M.A.R. and U. Buchenau, Phys. Rev. B 55, 5749 (1997)
L. Gil et al., Phys. Rev. Lett. 70, 182 (1993)
2121 exp),( ADPDDP s 2/3)/(169.0 gBTkWA
4/14/3minmax 07.1 gTTT
Ioffe-Regel limit:
2/IR
SPM WCE IRIR3/122
phonon localization when itsmean free path decreasesdown to half the wavelength:
for Brillouin scattering:
