Infinite1D lattice in atoms have the same mass but different spring constants

Jun 2nd, 2016
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Consider the 1-D lattice shown below in which the atoms have identical masses but alternating springs have different spring constants.

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Consider the 1-D lattice shown below in which the atoms have identical masses butalternating springs have different spring constants.Displacements Are Labelled:un for atoms at sites x = navn for atoms at sites x = na + dAssume Solutions Of Form:i qna wt un A ev Beiqdni qna wt e(a) Find equations of motion for atoms with displacementsiqdHint: Define B eB'unandvnFor convenience in calculationsThe equation of motion of uv and vn areM U n C1 (U n Vn 1 ) C 2 (U n Vn )M V n C 2 (Vn U n ) C1 (Vn U n 1 )Plug inuvi qna wt n Aei qna wt BeiqdeAnd let B e B , we then have,niqd' AM 2 e i ( qnat ) C1 ( Ae i ( qnat ) B' e iqa e i ( qnat ) ) C 2 ( Ae i ( qnat ) B' e i ( qnat ) ) B' M 2 e i ( qnat ) C1 ( Ae iqae i ( qnat ) B' e i ( qnat ) ) C2 ( Ae i ( qnat ) B' e i ( qnat ) )Divided by common factor ei ( qnat ) from both sides of the both above equations we have, AM 2

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