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Microprocessors continue to grow in capabilities, complexity, and performance. Microprocessors typically integrate functional components such as logic and level two cache memory in their architecture. This functional integration of logic and memory results in improved performance of the microprocessor However, the integration also introduces a layer of complexity in the thermal design and management of microprocessors. As a direct result of functional integration, the power map on a microprocessor is typically highly nonuniform, and the assumption of a uniform heat flux across the die surface has been shown to be invalid post Pentium II architecture. The active side of the die is divided into several functional blocks with distinct power assigned to each functional block. Previous work (Kaisare et al., 2005, 'Thermal Based Optimization of Functional Block Distributions in a Non-Uniformly Powered Die,' InterPACK 2005, San Francisco, CA, Jul. 17-22) has been done, which includes numerical analysis and thermal based optimization of a typical package consisting of a nonuniformly powered die, heat spreader, thermal interface materials I and II, and the base of the heat sink. In this paper, an analytical approach to temperature distribution of a first level package with a nonuniformly powered die is carried out for the first time. The analytical model for two-layer bodies developed by Haji-Sheikh et al. (2003, 'Steady-State Heat Conduction in Multi-Layer Bodies,' Int. J. Heat Mass Transfer, 46(13), pp. 2363-2379) is extended to this typical package, which is a multilayer body. The solution is to begin by designating each surface heat flux as a volumetric heat source. An inverse methodology is applied to solve the equations for various surfaces to calculate the maximum junction temperature for a given multilayer body. Finally validation of the analytical solution is carried out using previously developed numerical model. [DOI: 10.1115/1.3068303]

The thermal behavior of synthetic richterite [sup.A][Na.sup.B] [(NaCa).sup.C][Mg.sub.5.sup.T][Si.sub.8][O.sub.22][(OH).sub.2], crystallized at 800 [degrees]C and 0.35 GPa under hydrothermal conditions, was studied by serial powder diffraction experiments using synchrotron radiation between 123 and 873 K. The a, b, and [beta] cell parameters show a non-linear behavior, whereas the c parameter shows a linear trend; values of the saturation temperature [[theta].sub.s] are 410(3), 215(2), and 300(2) K for a, b, and asin[beta], respectively. The axial expansion pattern below room temperature is [[alpha].sub.b] > [[alpha].sub.c] > [[alpha].sub.a], whereas above room T it is [[alpha].sub.b] > [[alpha].sub.a] > [[alpha].sub.c], the difference being related to the different saturation temperatures of the individual cell parameters. The thermal expansion was modeled following the Debye approximation for the density-of-state of phonons; the refined parameters are [V.sub.0] = 908.20 (0.04) [[Angstrom].sup.3], k = 1 (4), [Q.sub.0] = 32 (2) MJ, and the Debye temperature [theta]D = 586(31) K. The non-linear behavior at low T is well described without systematic differences between the data and the model. The volume thermal expansion coefficient [[alpha].sub.v] changes significantly with temperature, also at temperatures higher than room temperature. It is suggested that this may occur also in other amphiboles and pyroxenes, requiring critical re-examination of the available data. A comparison of the strain tensor for the thermal expansion between amphiboles and pyroxenes shows that in amphiboles the major deformation occurs onto the (010) plane, whereas in pyroxenes it occurs along the b axis. Moreover, the major deformation on (010) in Ca-bearing pyroxenes occurs along the bond of the M2 cation with the furthermost O3 atoms, whereas in compositionally related Ca-amphiboles (i.e., tremolite) it occurs in a direction rotated by 10-20[degrees] to [a.sup.*], i.e., in a direction not corresponding to that of the M4-O5 bonds. It is proposed that the M4 polyhedron contributes less to the thermal deformation of amphibole than the M2 polyhedron contributes to the thermal deformation of pyroxene. Keywords: Richterite, amphibole, thermal expansion, Debye model, comparison with pyroxenes