Propagation of Waves Along One-Dimensional Crystal with Two Kinds of Atoms

Regard the Amplitudes Ua & Ub as UNKNOWNS. And we have (7: Not again!) A SET of LINEAR HOMOGENEOUS EQUATIONS, for which A NON-TRIVIAL Solution EXISTS ONLY if the DETERMINANT of COEFFICIENTS of THE VARIABLES VANISHES... EXPANDING: (2β - m_Aω²) - (-2βcos ka) = 0 (-2βcos ka) - (2β - m_Bω²) = 0 (2β - m_Aω²) - (-2βcos ka) - (2β - m_Bω²) - (-2βcos ka) = 0 (2β - m_Aω²)(2β - m_Bω²) - (-2βcos ka)(-2βcos ka) = 0 m_A m_B ω^4 - 2β(m_A + m_B) ω² + 4β² - 4β²cos²ka = 0 m_A m_B ω^4 - 2β(m_A + m_B) ω² + 4β²(1-cos²ka) = 0 m_A m_B ω^4 - 2β(m_A + m_B) ω² + 4β²sin²ka = 0 ω^4 - 2β(1/m_A + 1/m_B) ω² + 4β²/m_A m_B sin²ka = 0 Solving, using quadratic formula: ω² = [2β(1/m_A + 1/m_B) ± √{4β²(1/m_A + 1/m_B)² - 4(4β²/m_A m_B)sin²ka}] / 2 ω² = β(1/m_A + 1/m_B) ± β√{(1/m_A + 1/m_B)² - 4/m_A m_B sin²ka} ω² = β(1/m_A + 1/m_B) ± β√{(1/m_A - 1/m_B)²} ω² = β(1/m_A + 1/m_B) ± β|1/m_A - 1/m_B| POSSIBLE SOLUTIONS ARE THE NORMAL (QUERY, WHICH IS NOT ALWAYS THE CASE) There are TWO EIGENVALUES. Each when substituted back into the ORIGINAL EQUATION would PROVIDE ratios from which WE COULD SOVE for Ua & Ub. THUS for A GIVEN K, THERE ARE NOW TWO DIFFERENT WAVES of ANGULAR FREQUENCY w which MAY BE PROPAGATED IN THE CRYSTAL INTERPRETATION OF THE SOLUTIONS NOTE that the GENERAL form of our RESULT THAT w is A CONSTANT (P) To which WE EITHER ADD-OR-SUBTRACT A TERM which is GIVEN BY THE SQUARE ROOT of A. CONSTANT LESS A TERM which is A FUNCTION of k, which we ADD-OR-SUBTRACT. THE MAXIMUM VALUE of the TERM which WE ADD-OR-SUBTRACT OCCURS when The Term which is A fn of k IS A MINIMUM - NAMELY sin²ka = 0 ka = nπ k = nπ/a THE MINIMUM VALUE of the TERM which WE ADD-OR-SUBTRACT OCCURS when The Term which is A fn of k IS A MAXIMUM - NAMELY sin²ka = 1 ka = (n+½)π k = (n+½)π/a MINIMUM VALUE ADDED AND MAXIMUM VALUE SUBTRACTED (A) Solutions for small k Solution with (+) Sign: At k = 0, ω² = β(1/m_A + 1/m_B) + β|1/m_A - 1/m_B| = 2β/m_A with increasing k, this maximum TERM ADDED BECOMES SMALLER. Solution with (-) Sign: At k = 0, ω² = β(1/m_A + 1/m_B) - β|1/m_A - 1/m_B| = 0 NEAR k=0: ω² = β(1/m_A + 1/m_B) - β|1/m_A - 1/m_B| - β(1/m_A - 1/m_B)² k² / 2 A TERM k² is ADDED TO THE EXPRESSION for ω². SO IT IS ALSO POSSIBLE to APPROXIMATE a VALUE for ω² NEAR k=0 ω² = β(1/m_A + 1/m_B) - β|1/m_A - 1/m_B| - β(1/m_A - 1/m_B)² k² / 2 TOP TERM is ABOUT HALF THE SIZE of THE BOTTOM TERM ω² = β(1/m_A + 1/m_B) - β(1/m_A - 1/m_B)² k² / 4 ω² = β(2/m_A) - β(1/m_A)² k² / 4 ω² = (2β/m_A) (1 - (k²/4m_A)) ω = (2β/m_A)^(1/2) k VALUES of AMPUTUDES SOLUTION With SIGN AT k=0: (2β - m_Aω²) U_A + (2βcos ka) U_B = 0 (2βcos ka) U_A + (2β - m_Bω²) U_B = 0 (2β - m_Aω²) U_A + (2βcos ka) U_B - (2β - m_Bω²) U_B - (2βcos ka) U_A = 0 (m_Bω² - m_Aω²) U_A + (2βcos ka) (U_B - U_A) = 0 (m_B - m_A) ω² U_A + 2βcos ka (U_B - U_A) = 0 (m_B - m_A) ω² U_A = -2βcos ka (U_B - U_A) (m_B - m_A) ω² = -2βcos ka (U_B/U_A - 1) ω² = -2βcos ka (U_B/U_A - 1) / (m_B - m_A) This is A MOTION in which the AMPLITUDES of A & B ARE INVERSELY PROPORTIONAL TO THE MASS of THE ATOMS AND OPPOSITE IN SIGN. THE CENTER OF MASS of THE UNIT CELL STAYS FIXED