A simple model of harmonic vibrations in topologically disordered systems, such as glasses and supercooled liquids, is studied analytically by extending Euclidean random matrix theory to include vector vibrations. Rather generally, it is found that (i) the dynamic structure factor shows soundlike Brillouin peaks whose longitudinal/transverse character can only be distinguished for small transferred momentum, p; (ii) the model presents a mechanical instability transition at small densities, for which scaling laws are analytically predicted and confirmed numerically; (iii) the Brillouin peaks persist deep into the unstable phase, the phase transition being noticeable mostly in their linewidth; (iv) the Brillouin linewidth scales like in the stable phase, and like p in the unstable one. The analytical results are checked numerically for a simple potential. The main features of glassy vibrations previously deduced from scalar are not substantially altered by these new results.
REFERENCES
1.
F.
Sette
, M.
Krisch
, C.
Masciovecchio
, G.
Ruocco
, and G.
Monaco
, Science
280
, 1550
(1998
);E.
Courtens
, M.
Foret
, B.
Hehlen
, and R.
Vacher
, Solid State Commun.
117
, 187
(2001
).2.
T.
Scopigno
, S. N.
Yannopoulos
, D. Th.
Kastrissios
, G.
Monaco
, E.
Pontecorvo
, G.
Ruocco
, and F.
Sette
, J. Chem. Phys.
118
, 311
(2003
).3.
M.
Foret
, E.
Courtens
, R.
Vacher
, and J.-B.
Suck
, Phys. Rev. Lett.
77
, 3831
(1996
);P.
Benassi
, M.
Krisch
, C.
Masciovecchio
, V.
Mazzacurati
, G.
Monaco
, G.
Ruocco
, F.
Sette
, and R.
Verbeni
, Phys. Rev. Lett.
77
, 3835
(1996
);O.
Pilla
, A.
Cunsolo
, A.
Fontana
, C.
Masciovecchio
, G.
Monaco
, M.
Montagna
, G.
Ruocco
, T.
Scopigno
, and F.
Sette
, Phys. Rev. Lett.
85
, 2136
(2000
);B.
Rufflé
, M.
Foret
, E.
Courtens
, R.
Vacher
, and G.
Monaco
, Phys. Rev. Lett.
90
, 095502
(2003
).4.
5.
See, for example, J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic, London, 1986).
6.
M. C. C.
Ribeiro
, M.
Wilson
, and P. A.
Madden
, J. Chem. Phys.
108
, 9027
(1998
).7.
G.
Ruocco
, F.
Sette
, R.
Di Leonardo
, G.
Monaco
, M.
Sampoli
, T.
Scopigno
, and G.
Viliani
, Phys. Rev. Lett.
84
, 5788
(2000
).8.
J.
Horbach
, W.
Kob
, and K.
Binder
, J. Phys. Chem. B
103
, 4104
(1999
).9.
O.
Pilla
, S.
Caponi
, A.
Fontana
, M.
Montagna
, F.
Rossi
, G.
Viliani
, L.
Angelani
, G.
Ruocco
, G.
Monaco
, and F.
Sette
, cond-mat/0209519.10.
W.
Schirmacher
, G.
Diezemann
, and C.
Ganter
, Phys. Rev. Lett.
81
, 136
(1998
).11.
S. N.
Taraskin
, Y. L.
Loh
, G.
Natarajan
, and S. R.
Elliott
, Phys. Rev. Lett.
86
, 1255
(2001
);S. I.
Simdyankin
, S. N.
Taraskin
, M.
Elenius
, S. R.
Elliott
, and M.
Dzugutov
, Phys. Rev. B
65
, 104302
(2002
).12.
See, e.g., S. R. Elliott, Physics of Amorphous Materials (Longman, New York, 1990), and references therein.
13.
V.
Martı́n-Mayor
, G.
Parisi
, and P.
Verrocchio
, Phys. Rev. E
62
, 2373
(2000
).14.
15.
16.
V.
Martı́n-Mayor
, M.
Mézard
, G.
Parisi
, and P.
Verrocchio
, J. Chem. Phys.
114
, 8068
(2001
).17.
T. S.
Grigera
, V.
Martı́n-Mayor
, G.
Parisi
, and P.
Verrocchio
, Phys. Rev. Lett.
87
, 085502
(2001
).18.
M. L. Mehta, Random Matrices (Academic, London, 1991).
19.
A.
Cavagna
, I.
Giardina
, and G.
Parisi
, Phys. Rev. Lett.
83
, 108
(1999
).20.
A. P.
Sokolov
, U.
Buchenau
, W.
Steffen
, B.
Frick
, and A.
Wischnewski
, Phys. Rev. B
52
, 9815
(1995
);N. J.
Tao
, G.
Li
, X.
Chen
, W. M.
Du
, and H. Z.
Cummins
, Phys. Rev. A
44
, 6665
(1991
);D.
Engberg
, A.
Wischnewski
, U.
Buchenau
, L.
Börjesson
, A. J.
Dianoux
, A. P.
Sokolov
, and L. M.
Torell
, Phys. Rev. B
59
, 4053
(1999
).21.
A.
Wischnewski
, U.
Buchenau
, A. J.
Dianoux
, W. A.
Kamitakahara
, and J. L.
Zarestky
, Phys. Rev. B
57
, 2663
(1998
).22.
Y.
Inamura
, M.
Arai
, N.
Kitamura
, S. M.
Bennington
, and A. C.
Hannon
, Physica B
241–243
, 903
(1998
);Y.
Inamura
, M.
Arai
, O.
Yamamuro
, A.
Inaba
, N.
Kitamura
, T.
Otomo
, T.
Matsuo
, S. M.
Bennington
, and A. C.
Hannon
, Physica B
263–264
, 299
(1999
);23.
U.
Buchenau
, Yu. M.
Galperin
, and V. L.
Gurevich
, Phys. Rev. B
46
, 2798
(1992
).24.
25.
26.
See Foret et al. in Ref. 3.
27.
J. L.
Feldman
, P. B.
Allen
, and S. R.
Bickham
, Phys. Rev. B
59
, 3551
(1999
);P. B.
Allen
, J. L.
Feldman
, J.
Fabian
, and F.
Wooten
, Philos. Mag. B
79
, 1715
(1999
).28.
T. S.
Grigera
, V.
Martin-Mayor
, G.
Parisi
, and P.
Verrocchio
, J. Phys.: Condens. Matter
14
, 2167
(2002
).29.
T.
Scopigno
, E.
Pontecorvo
, R.
Di Leonardo
, M.
Krisch
, G.
Monaco
, G.
Ruocco
, B.
Ruzicka
, and F.
Sette
, cond-mat/0205615.30.
T. S.
Grigera
, V.
Martı́n-Mayor
, G.
Parisi
, and P.
Verrocchio
, Nature (London)
422
, 289
(2003
).31.
L.
Angelani
, R.
Di Leonardo
, G.
Ruocco
, A.
Scala
, and F.
Sciortino
, Phys. Rev. Lett.
85
, 5356
(2000
);K.
Broderix
, K. K.
Bhattacharya
, A.
Cavagna
, A.
Zippelius
, and I.
Giardina
, Phys. Rev. Lett.
85
, 5360
(2000
);T. S.
Grigera
, A.
Cavagna
, I.
Giardina
, and G.
Parisi
, Phys. Rev. Lett.
88
, 055502
(2002
).32.
P.
Turchi
, F.
Ducastelle
, and G.
Treglia
, J. Phys. C
15
, 2891
(1982
).33.
A.
Rahman
, M.
Mandell
, and J. P.
McTague
, J. Chem. Phys.
64
, 1564
(1976
);M.
Cho
, G. R.
Fleming
, S.
Saito
, I.
Ohmine
, and R. M.
Stratt
, J. Chem. Phys.
100
, 6672
(1994
);34.
Actually, a small number of terms in the perturbative expansion are not obtained correctly in this way, and the cactus equation is slightly modified (Ref. 40). However, the results of Sec. III are unchanged.
35.
T.
Nakayama
, K.
Yakubo
, and R. L.
Orbach
, Rev. Mod. Phys.
66
, 381
(1994
), and references therein.36.
37.
S.
Alexander
and R.
Orbach
, J. Phys. (France) Lett.
43
, L625
(1982
).38.
39.
C. A.
Angell
, Y.
Yue
, L.-M.
Wang
, J. R. D.
Copley
, S.
Borick
, and S.
Mossa
, J. Phys.: Condens. Matter
15
, S1051
(2003
).40.
S. Ciliberti, T. S. Grigera, V. Martin-Mayor, G. Parisi, and P. Verrocchio (unpublished).
This content is only available via PDF.
© 2003 American Institute of Physics.
2003
American Institute of Physics
You do not currently have access to this content.
