On the Formulation of the Exchange Field in the Landau-Lifshitz Equation for Spin-Wave Calculation in Magnonic Crystals

<table>The discrete lattice of spins. The angle between neighboring spins: <svg height="16.112499" id="M27" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 13.4 16.112499" width="13.4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.825)"><path d="M484 692v-218h-28q-25 83 -70 130q-54 56 -123 56q-51 0 -80.5 -27t-29.5 -72q0 -37 21 -59q23 -25 82 -55q14 -8 68 -36q67 -34 80 -41q50 -28 79.5 -74.5t29.5 -101.5q0 -96 -69 -154q-68 -59 -179 -59q-61 0 -130 24q-30 10 -41 10q-23 0 -30 -34h-29v248h29
q23 -97 60 -143q57 -72 144 -72q58 0 92 30.5t34 82.5q0 47 -57 88q-30 21 -122 65q-91 44 -131 91.5t-40 118.5q0 95 58.5 148t160.5 53q54 0 117 -23q27 -9 42 -9q14 0 20.5 6.5t11.5 26.5h30z" id="x1D412"></path></g>
<g transform="matrix(.012,0,0,-.012,9.513,15.9)"><path d="M238 681l-124 -585q-7 -31 4 -31q10 0 37.5 18.5t49.5 41.5l16 -22q-40 -48 -89.5 -81.5t-76.5 -33.5q-42 0 -16 122l105 488q7 32 0 41t-39 9h-35l5 26q35 3 71 13t58 17.5t26 7.5q14 0 8 -31z" id="x1D459"></path></g>
</svg> and <svg height="16.112499" id="M28" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 19.5875 16.112499" width="19.5875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,11.825)"><path d="M484 692v-218h-28q-25 83 -70 130q-54 56 -123 56q-51 0 -80.5 -27t-29.5 -72q0 -37 21 -59q23 -25 82 -55q14 -8 68 -36q67 -34 80 -41q50 -28 79.5 -74.5t29.5 -101.5q0 -96 -69 -154q-68 -59 -179 -59q-61 0 -130 24q-30 10 -41 10q-23 0 -30 -34h-29v248h29
q23 -97 60 -143q57 -72 144 -72q58 0 92 30.5t34 82.5q0 47 -57 88q-30 21 -122 65q-91 44 -131 91.5t-40 118.5q0 95 58.5 148t160.5 53q54 0 117 -23q27 -9 42 -9q14 0 20.5 6.5t11.5 26.5h30z" id="x1D412"></path></g>
<g transform="matrix(.012,0,0,-.012,9.513,15.9)"><path d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6
q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z" id="x1D45A"></path></g>
</svg> is <svg height="12.325" id="M29" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 10.0125 12.325" width="10.0125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,7.825)"><path d="M559 271q0 -109 -83 -187t-189 -89l-36 -213q-16 -30 -86 -43l-7 9q10 36 53 242q-91 11 -139.5 63.5t-48.5 132.5q0 186 242 271l11 -28q-76 -34 -119.5 -91.5t-43.5 -136.5q0 -66 29 -108.5t77 -55.5q39 205 56 260q21 71 65 111t93 40q50 0 88 -45.5t38 -131.5z
M484 273q0 65 -26.5 99t-52.5 34q-37 0 -53 -75l-59 -298q80 5 135.5 74.5t55.5 165.5z" id="x1D711"></path></g>
</svg>. We assume that <svg height="12.325" id="M30" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 10.0125 12.325" width="10.0125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,7.825)"><path d="M559 271q0 -109 -83 -187t-189 -89l-36 -213q-16 -30 -86 -43l-7 9q10 36 53 242q-91 11 -139.5 63.5t-48.5 132.5q0 186 242 271l11 -28q-76 -34 -119.5 -91.5t-43.5 -136.5q0 -66 29 -108.5t77 -55.5q39 205 56 260q21 71 65 111t93 40q50 0 88 -45.5t38 -131.5z
M484 273q0 65 -26.5 99t-52.5 34q-37 0 -53 -75l-59 -298q80 5 135.5 74.5t55.5 165.5z" id="x1D711"></path></g>
</svg> and the spin deviation from the <svg height="8.4250002" id="M31" style="vertical-align:-0.273pt" version="1.1" viewbox="0 0 8.5874996 8.4250002" width="8.5874996" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,0,0,-.017,.062,8.013)"><path d="M475 445l-64 -69q-119 -128 -289 -280q27 6 81 -15q66 -26 102 -26q48 0 111 93l23 -19q-41 -84 -75.5 -116.5t-73.5 -32.5q-37 0 -147 53q-43 21 -63 7q-22 -15 -38 -41q-4 -7 -11 -7q-8 10 -8 26q0 38 47 83l292 273q-32 -5 -67 -1q-63 7 -105 7q-32 0 -53.5 -22
t-47.5 -67l-25 13q15 39 38 78q40 66 83 66q57 0 120 -12q52 -10 80 -10q29 0 67 42z" id="x1D467"></path></g>
</svg> axis are small.</table>

Advances in Condensed Matter Physics

fig1

Figure 1

Figure 1: On the Formulation of the Exchange Field in the Landau-Lifshitz Equation for Spin-Wave Calculation in Magnonic Crystals 