| The properties(dispersion relation and acoustic velocity) of dust lattice waves(DLW) in complex plasma crystals have been systematically studied in this thesis.In the first chapter,a brief review of the theories and experiments of complex plasmas and complex plasma crystals has been made.In the second chapter,we discuss the dispersion relations of DLW in complex plasma crystals systematically,propose the concept of dispersion relation matrix of DLW in complex plasma crystals,derive the dispersion relation matrices for DLW in body centred cubic(bcc) and face centred cubic(fcc) in three-dimensional complex plasma crystals,and obtain the simple forms of the dispersion relation matrices of DLW in the three characteristic directions ((1,0,0),(1,1,0) and(1,1,1)).We compute the dispersion relation matrices with the screened Coulomb interaction between a charged dust and all other particles being taken into account or with the screened Coulomb interaction between a charged dust and particles in the nearest eight cubic only being taken into account(screening parameterκ>>1).We then discuss the effects of screening parameters on the dispersion relations of DLW.In the third chapter,we discuss the acoustic velocity of DLW in the long wavelength region in plasma crystals.We obtain theκ-depending acoustic velocity of longitudinal DLW forκ>>1 andκ<<1,and transverse DLW forκ>>1 in two-dimensional plasma crystals, the acoustic velocity in the long wavelength region in the two characteristic directions(x-axis and y-axis) of plasma crystals is the same.The acoustic velocity we obtained is very close to the numerical result of Peeters and Wu,and also agrees with experimental result of Nunomura, "the acoustic velocity is the same for all wave propagation direction".We obtain theκ-depending acoustic velocity of longitudinal DLW forκ<<1 in the long wavelength region in the three characteristic directions in three-dimensional plasma crystals(bcc and fcc),the acoustic velocity in the long wavelength region in the three characteristic directions in three-dimensional plasma crystals is the same.Then we discuss the dispersion relations of DLW in the long wavelength region in two-dimensional and three-dimensional plasma crystalsIn the fourth chapter,we compute the "inversed" dispersion relations for the damped DLW in one-dimensional,two-dimensional and three-dimensional plasma crystals,with the screened Coulomb interaction between a charged dust and all other particles being taken into account.We compare the theoretical "inversed" dispersion relations quantitatively with experimental data from the reference,and they agree very well.We find that the "inversed" dispersion relations of DLW in the long wavelength region in the two characteristic directions of two-dimensional plasma crystals are the same,and the "inversed" dispersion relations of DLW in the long wavelength region in the three characteristic directions of three-dimensional plasma crystals are also the same,and obtain the explicit "inversed" dispersion relation formulae in the long wavelength region.In the fifth chapter,we consider the effects of damping of DLW in one- dimensional, two-dimensional and three-dimensional plasma crystals.For externally excited modes(with a real frequency and a complex wave number),the "inversed" dispersion of k_r=k_r(ω), k_i= k_i(ω) does not reach the "negative dispersion" region,due to the strong damping near the "standing wave" point where the group velocity goes to zero.For "naturally excited" modes(with a real wave number and a complex frequency),the dispersion curveω_r=ω_r(k) does extend all way to the "negative dispersion" region,while however a "cut off" is seen at the long wavelength end of the dispersion.If the damping is weak,the real-part of dispersion for both "externally" and "thermally" excited modes are very close to the no-damping dispersion in the "positive dispersion" region. It is in the "negative dispersion" region however where the small damping makes the dispersion curves depart far away from each other.
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