( )F.J.P. Schuurmans et al.rPhysics Letters A 264 2000 472–477 475arguments relating absorpti<strong>on</strong>, stimulated and sp<strong>on</strong>taneousemissi<strong>on</strong>. The corresp<strong>on</strong>ding dynamic polarw38xizability <strong>of</strong> the <strong>impurity</strong> in Õacuum is v 02I I 2 2 3 2v 0yv yi G0v rv 0a Ž v. sa Ž 0 . , Ž 10.Ž .Here, a Ž.I 0 is the static polarizability, v 0 the res<strong>on</strong>ancefrequency, and G0the res<strong>on</strong>ance width, whichis identical to the sp<strong>on</strong>taneous emissi<strong>on</strong> rate. The seta Ž. 0,v ,G 4I 0 0 has <strong>on</strong>ly two independent parameters,as the optical theorem relates them via w38xG0sa IŽ 0. v 0 4 r6p c03 , Ž 11.where c0is the speed <strong>of</strong> light in vacuum. Thisrelati<strong>on</strong> is c<strong>on</strong>sistent with Fermi’s Golden Rule resultfor G by taking the static polarizability a Ž.0 I 0s2< D< 2 r´ 0"v0 in terms <strong>of</strong> the dipole matrix elementD <strong>of</strong> two atomic levels. These relati<strong>on</strong>s holdfor a dipole in vacuum. When c<strong>on</strong>sidering a dipole ina background medium with dielectric c<strong>on</strong>stant ´ andneglecting local-<str<strong>on</strong>g>field</str<strong>on</strong>g> or cavity <str<strong>on</strong>g>effects</str<strong>on</strong>g>, Eq. Ž 11.ismodified to G Ž ´ . s ' ´G, 0 with G0determined bythe polarizability and the res<strong>on</strong>ance frequency as inEq. Ž 11 .. This is in agreement with the Nienhuis andAlkemade result, see Eq. Ž. 1 .Implementing the dynamic polarizability a Ž v.Ifor the static <strong>on</strong>e a in Eq. Ž.I 6 readily gives theeffective dynamic polarizability a Ž v.sub,I <strong>of</strong> a substi-tuti<strong>on</strong>al <strong>impurity</strong>asub,IŽ v.y12aIŽ v .Ž ´y1.L2empsaIŽ v.Lemp1y .9´VŽ 12.The polarizability <strong>of</strong> this ‘dressed’ <strong>impurity</strong> nearres<strong>on</strong>ance has a Lorentzian structure analogous toEq. Ž 10 .:asub,I Ž v. sasub ,I Ž 0.v 02 Ž ´ .=,2 2 3 2v 0Ž ´ . yv yiŽ G Ž ´ . v rv 0Ž ´ ..Ž 13.with parameters given by2v 0Ž ´ . 2a IŽ 0 .Ž ´y1.Lemps 1y2v 0 9´V, Ž 14.v 02 2asub,I Ž 0. sLemp a Ž 0 ., 2 Iv Ž ´ .Ž 15.2G ´ v 0 ´' 2emp 2G0 v 00Ž . Ž .s ´ L . Ž 16.Here, the sp<strong>on</strong>taneous emissi<strong>on</strong> rate G Ž ´ . <strong>of</strong> thesubstituti<strong>on</strong>al <strong>impurity</strong> follows from the polarizabilitya Ž. 0 and the res<strong>on</strong>ance frequency v Ž ´ .sub,I 0 byuse <strong>of</strong> the optical theorem, see Eq. Ž 11 .. It should benoted that a direct calculati<strong>on</strong> <strong>of</strong> G Ž ´ . from Eq. Ž 13.is not feasible, as the macroscopic derivati<strong>on</strong> presentedhere is in essence quasi-static, and hence <strong>on</strong>lyaccounts for a real polarizability. This drawback isnot encountered in the full, but more complicatedcalculati<strong>on</strong> <strong>of</strong> Ref. w37 x. In vacuum, ´s1, the originalresults are regained, that is v Ž. 1 sv , a Ž.0 0 sub,I 0sa Ž. 0 , and G Ž.I 1 sG 0. The res<strong>on</strong>ance frequency isred-shifted and the line width behaves in accordancewith the empty-cavity descripti<strong>on</strong>, identical to ourearlier results w37 x.The dynamical properties <strong>of</strong> an interstitial <strong>impurity</strong>are obtained in a similar fashi<strong>on</strong> as for thesubstituti<strong>on</strong>al <strong>on</strong>e. As a result, the polarizability <strong>of</strong>the ‘dressed’ interstitial <strong>impurity</strong> also has aLorentzian line shape given by Eq. Ž 13 .. The parametersfor the interstitial case are2v 0Ž ´ . 2a IŽ 0 .Ž ´y1.LLors 1y2v 0 9´V, Ž 17.v 02 2aint ,I Ž 0. sLLor a Ž 0 ., 2 Iv Ž ´ .Ž 18.2G ´ v 0 ´' 2Lor 2G0 v 00Ž . Ž .s ´ L . Ž 19.Again the res<strong>on</strong>ance frequency is red-shifted, althoughwith a different functi<strong>on</strong>al dependence <strong>on</strong> ´.The Lorentz local-<str<strong>on</strong>g>field</str<strong>on</strong>g> factor is found to determinethe modificati<strong>on</strong> <strong>of</strong> the sp<strong>on</strong>taneous emissi<strong>on</strong> rate, inagreement with Ref. w37 x.
476( )F.J.P. Schuurmans et al.rPhysics Letters A 264 2000 472–477The obtained results for both the substituti<strong>on</strong>aland interstitial <strong>impurity</strong> still depend <strong>on</strong> the volume<strong>of</strong> the cavity V. For the case <strong>of</strong> the interstitials they1volume is fixed: Vsr Ž Onsager’s assumpti<strong>on</strong> ..For substituti<strong>on</strong>als, V is generally taken to be thereal volume <strong>of</strong> the <strong>impurity</strong> atom or molecule. Forsmall <strong>impurity</strong> volumes, V-r y1 , <strong>on</strong>ly <strong>on</strong>e dipole<strong>of</strong> the dielectric is expelled from the cavity. If the<strong>impurity</strong> is large, V4r y1 , many dipoles <strong>of</strong> thedielectric are removed, and the <strong>impurity</strong> is a truemacroscopic cavity.The res<strong>on</strong>ance frequency <strong>of</strong> the <strong>impurity</strong> slightlyshifts to the red, see Fig. 1. This shift is a directc<strong>on</strong>sequence <strong>of</strong> the reacti<strong>on</strong> <str<strong>on</strong>g>field</str<strong>on</strong>g> produced by thepolarizati<strong>on</strong> in the medium that is induced by the<strong>impurity</strong> dipole. For small <strong>impurity</strong> polarizati<strong>on</strong> densitiesŽa Ž 0. rV
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