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2spinparticlesinanimpuresingletconsistingofasingletfractionxandarandomfraction1xforimpurityWxjsingletihsingletj1 41x16Where1istheidentitymatrixWithalltheindicesEq6becomesWmnxSm

2-spinparticlesinanimpuresinglet consistingofasingletfractionxandarandomfraction(1x)forimpurity:W=xjsingletihsingletj+1 4(1x)1(6)Where1istheidentitymatrix.Withalltheindices Eq.(6)becomesWm;n=xSm

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