ÿþ<HTML> <HEAD> <TITLE> Dorian Hatch's Publications </TITLE> <SCRIPT LANGUAGE="JavaScript"> <!-- this line hides JavaScript from old browsers /** * Put discription of JavaScript here * */ // this is the end of the script and comment structure --> </SCRIPT> </HEAD> <BODY BGCOLOR="#FFFFFF"> <BR> <HR> <CENTER> <TABLE> <TD><A HREF="Research.htm"><FONT SIZE=3>Research</FONT></A>&nbsp;&nbsp;&nbsp;&nbsp;</TD> <TD><A HREF="Biography.htm"><FONT SIZE=3>Biography</FONT></A>&nbsp;&nbsp;&nbsp;&nbsp;</TD> <TD><A HREF="http://www.byu.edu"><FONT SIZE=3>BYU Home</FONT></A>&nbsp;&nbsp;&nbsp;&nbsp;</TD> <TD><A HREF="http://physc1.byu.edu"><FONT SIZE=3>Physics & Astronomy</FONT></A>&nbsp;&nbsp;&nbsp;&nbsp;</TD> <TD><A HREF="hatch.htm"><FONT SIZE=3>Hatch Home</FONT></A>&nbsp;&nbsp;&nbsp;&nbsp;</TD> </TABLE> <HR> <BR> <CENTER><IMG SRC="images/colorbar.gif" width="600" height="1"></CENTER> <BR> </center> <CENTER> <H1>Publications</H1> </CENTER> <IMG height=10 src="hatch_files/reddot.gif" width=10> <FONT SIZE=4>Books</FONT> <BR><BR> 1. O.V. Kovolev edited by Harold T. Stokes and Dorian M. Hatch, "Representations of the Crystallographic Space Groups," (Gordon and Breach, Amsterdam, 1993). <BR><BR> 2. Harold T. Stokes and Dorian M. Hatch, "Isotropy Subgroups of the 230 Space Groups," (World Scientific, Singapore, 1988). <BR><BR> <IMG height=10 src="hatch_files/reddot.gif" width=10> <FONT SIZE=4>Papers</FONT> <BR> <BR> 92. H. T. Stokes, D. M. Hatch, J. Dong, and J. P. Lewis,  Mechanisms for the reconstructive phase transition between the B1 and B2 structure types in NaCl and PbS, Phys. Rev. B 69, 174111 (2004). <BR><BR> 91. D. M. Hatch, H. T. Stokes, and W. Cao,  Allowed mesoscopic point group symmetries in domain average engineering of perovskite ferroelectric crystals, Journal of Applied Physics 94, 5220-5227 (2003) <BR><BR> 90. D. M. Hatch, T. Lookman, A. Saxena, S. R. Shenoy,  Proper ferroelastic transitions in two dimensions: Anisotropic long-range kernels, domain wall orientations, and microstructure, Physical Review B 68, 104105 (2003). <BR><BR> 89. D. R. Trinkle, R. G. Hennig, S. G. Srinivasan, D. M. Hatch, M. D. Jones, H. T. Stokes, R. C. Albers, and J. W. Wilkins,  New mechanism for the alpha to omega Martensitic transformation in pure titatium, Physical Review Letters 91, 025701 (2003) <BR><BR> 88. D. B. Litvin, V. K. Wadhawan, and D. M. Hatch,  Latent symmetry and domain average engineered ferroics, Ferroelectrics 292, 65-70 (2003). <BR><BR> 87. H. T. Stokes and D. M. Hatch,  Symmetry of possible average multidomain structures at a phase transition, Ferroelectrics 292, 59-63 (2003) <BR><BR> 86. C. Capillas, E. Lroumova, M. I. Aroyo, J. M. Perez-Mato, H. T. Stokes, D. M. Hatch,  SYMMODES: a software package for group-theoretical analysis of structural phase transitions, J. Appl. Cryst. 36, 953 (2003). <BR><BR> 85. D. M. Hatch and H. T. Stokes,  INVARIANTS: Program for Obtaining a List of Invariant Polynomials of the Order Parameter Components Associated with Irreducible Representations of a Space Group. J. Appl. Cryst. 36, 951 (2003). <BR><BR> 84. H. T. Stokes, E. H. Kisi, D. M. Hatch, and C. J. Howard,  Group-Theoretical Analysis of Octahedral Tilting in Ferroelectric Perovskites, Acta Cryst. B58, 934-938 (2002). <BR><BR> 83. D. M. Hatch, H. T. Stokes, R. Ranjan, S. K. Mishra, D. Pandey, and B. J. Kennedy,  Antiferrodistortive Phase Transition in Pb(Ti0.48Zr0.52)O3: Space Group of the Lowest Temperature Monoclinic Phase, Phys. Rev. B 65, 212101 (2002). <BR><BR> 82. H. T. Stokes and D. M. Hatch,  COPL: Program for Obtaining a Complete List of Order Parameters for a Group-Subgroup Crystalline Phase Transition, J. Appl. Cryst. 35, 379 (2002). <BR><BR> 81. A. Saxena, S. G. Srinivasan, T. Lookman, R. C. Albers, and D. M. Hatch,  Modeling of Reconstructive Transformations in Titanium, Activities in Support of the Nuclear Weapons Program, LA-UR-01-6200, Winter 2001/2002. <BR><BR> 80. W. Klein, T. Lookman, A. Saxena, and D. M. Hatch,  Nucleation in Systems with Elastic Forces, Phys. Rev. Lett. 88, 085701 (2002). <BR><BR> 79. H. T. Stokes, C. Sadate, D. M. Hatch, L. L. Boyer, and M. J. Mehl, ``Analysis of the Ferroelectric Phase Transition in BaAl2O4 by Group Theoretical Methods and First Principles Calculations,'' Phys. Rev. B 65, 064105--1--4 (2002). <BR><BR> 78. G. M. Chechin, O.A. Lavrova, V. P. Sakhnenko, H. T. Stokes, and D. M. Hatch,  New Approach to Nonlinear Dynamics of Fullerenes and Fullerites, Physics of the Solid State Vol. 44, 581 (2002). <BR><BR> 77. H. T. Stokes and D. M. Hatch,  Procedure for Obtaining Microscopic Mechanisms of Reconstructive Phase Transitions in Crystalline Solids, Phys. Rev. B 65, 144111 (2002). <BR><BR> 76. D. M. Hatch, W. Cao, and A. Saxena,  Orientational Twins in an Improper Ferroelastic Phase Transition Driven by the M-5 Zone Boundary Phonon in Rag1-xInx, Phys. Rev. B 65, 094110 (2002). <BR><BR> 75. D. M. Hatch and H. T. Stokes,  Complete Listing of Order Parameters for a crystalline Phase Transition: A Solution to the Generalized Inverse Landau Problem , Phys. Rev. B 65, 014113 (2002). <BR><BR> 74. D. M. Hatch, T. Lookman, A. Saxena, and H. T. Stokes,  Systematics of Group - Nonsubgroup Transitions and Square to Triangle Transition, Phys. Rev. B: Rapid Communications 64, 060104-1 - 060104-4 (2001). <BR><BR> 73. W. Cao, A. Saxena and D. M. Hatch,  Antiphase Structures of an Improper Ferroelastic Phase Transition Driven by an M5- Zone Boundary Phonon in R Ag1-xInx, Phys. Rev. B 64, 024106-1 - 024106-12 (2001). <BR><BR> 72. R. A. Hatt and D. M. Hatch,  A Symmetry Comparison of Orientation - twin and Antiphase Domain Walls in Pb3(Vo4)2, J. Phys.; Condensed Matter 13, 3111-3121 (2001). <BR><BR> 71. G. M. Chechin, V. P. Sakhnenko, H. T. Stokes, A. D. Smith, and D. M. Hatch,  Non-linear Normal Modes for Systems with Discrete Symmetry, Intl. Jour. Of Non-Linear Mechanics 35, 497-513 (2000). <BR><BR> 70. W. R. Hurren and D. M. Hatch,  C60 and the Icosahedral Group Ih: Simplifications Methods and Group-Theoretical Applications, Phase Transitions 69 (3), 363-94 (1999). <BR><BR> 69. G. R. Barsch, T. Ohba, and D. M. Hatch,  Landau Theory of Structural Transformations in Titanium-Nickel and Gold-Cadmium Alloys, Mater. Sci. Eng. A 273-275, 161-165 (1999). <BR><BR> 68. Dorian M. Hatch and W. Cao,  Determination of Domain and Domain Wall Formation at Ferroic Transitions, Ferroelectrics, 222, 1-10, (1999). <BR><BR> 67. R. A. Hatt and D. M. Hatch,  Order Parameter Profiles in Ferroic Phase Transitions, Ferroelectrics, 226, pp. 61-78, (1998). <BR><BR> 66. Dorian M. Hatch, Richard A. Hatt, and Harold T. Stokes, "Systematic Approach to the Study of Domain Twins in Ferroic Phase Transitions," Ferroelectrics, 191, 29-35, (1997). <BR><BR> 65. Dorian M. Hatch, Ping Hu, Avadh Saxena, and Gerhard Barsch, "Systematic Technique for the Study of Interphase Boundaries in Structural Phase Transitions, Physical Review Letters, 76, 1288-1291 (1996). <BR><BR> 64. Dorian M. Hatch, Ping Hu, Avadh Saxena, and Gerhard Barsch, "Landau-Ginzgurg Model of Interphase Boundaries in CsCl-Type Ferroelastics due to M5 Mode Instability: La Ag1-x Inx," Journal de Physique III, suppl. 5, C8-131 (1995). <BR><BR> 63. John Flocken, Z. Mo, W.M. Mei, J.R. Hardy, and Dorian M. Hatch, "Molecular- dynamics Simulations of Some BaXF4 Compounds," Physical Review B 49, 5811 (1994). <BR><BR> 62. Dorian M. Hatch, Subrata Ghose, and John L. Bjorkstam, "The "ð-B Phase Transition in AlP04 Cristobalite: Symmetry Analysis, Domain Structure and Transition Dynamics," Physics and Chemistry of Minerals 21, 67 (1994). <BR><BR> 61. Avadh Saxena, Gerhard R. Barsch, and Dorian M. Hatch, "Lattice dynamics Representation Theory versus Isotropy subgroup Method with Application to M5- Modes Instability in CsCl Structure," Phase Transitions 46, 89 (1994). <BR><BR> 60. Hexiong Yang, Subrata Ghose, and Dorian H. Hatch, "Ferroelastic Phase Transition in Cryolite, Na3AlF6, a Mixed Fluoride Perovskite: High Temperature Single Crystal X-ray Diffraction Study and Symmetry Analysis of the Transition Mechanism," Physics and Chemistry of Minerals 19, 528 (1993). <BR><BR> 59. Harold T. Stokes, Dorian M. Hatch, and H. Mark Nelson, "Landau, Lifshitz, and Weak Lifshitz Conditions in the Landau Theory of Phase Transitions in Solids," Physical Review B 47, 9080 (1993). <BR><BR> 58. Dana T. Griffen and Dorian M. Hatch, "Crystal Chemistry of Birefringent Silicate Garnets: A Review," Trends in Mineralogy 1, 269, 1992. <BR><BR> 57. Ming Cai, Qi Wang, David D. Allred, L. V. Knight, Dorian M. Hatch, A. Reyes-Mena, Jesus González-Hernádez, and Guizhong Zhang, "Use of Raman Spectroscopy in Characterizing Soft X-Ray Multilayers: Tools in Understanding Structure and Interfaces," Optical Materials for High Power Lasers, Brian Newnam, Editor, 24th Boulder Damage Symposium, Boulder, CO, SPIE 1848, 521, 1992. <BR><BR> 56. David D. Allred, Ming Cai, Dorian M. Hatch, Qi Wang, and A. Reyes-Mena, "Raman Spectroscopic Analysis of Mo/Si. Multilayers," J. X-Ray Science and Technology 3, 222 (1992). <BR><BR> 55. Dana T. Griffen, Dorian M. Hatch, W. Revell Phillips, and Seyfi Kulaksiz, "Crystal Chemistry and Symmetry of a Birefringent Pyralspite75-Grandite25 Tetragonal Garnet," American Mineral 77, 399 (1992). <BR><BR> 54. Dorian M. Hatch and Harold T. Stokes, "Symmetry Analysis of a Ferroelastic Phase Transition in Solid Bromoform," in "Phase Transitions in Ferroelastic and Co-elastic Crystals," edited by E.K.H. Salje, (Cambridge University Press, Cambridge, 1990), pg. 326. <BR><BR> 53. Dorian M. Hatch and Subrata Ghose, "The "ð-B Phase Transition in Cristoballite, SiO2: Symmetry Analysis, Domain Structure, and the Dynamical Nature of the B-phase," Physics and Chemistry of Minerals 17, 554 (1991). <BR><BR> 52. Harold T. Stokes and Dorian M. Hatch, "Coupled Order Parameters in the Landau Theory of Phase Transitions in Solids," Phase Transitions 34, 53 (1991). <BR><BR> 51. Harold T. Stokes, Dorian M. Hatch and James D. Wells, "Group-Theoretical Methods for Obtaining Distortion in Crystals: Applications to Vibrational Model and Phase Transitions," Physical Review B 43, 11010 (1991). <BR><BR> 50. Subrata Ghose, Yoshiaki Ito, and Dorian M. Hatch, "A High-Temperature X-Ray Diffraction Study of the Paraelectric - Antiferroelectric Phase Transition in Titanite, CaTiSiO5: Order Parameter and Transition Mechanism," Physics and Chemistry of Minerals, 17, 591, 1991. <BR><BR> 49. Dorian M. Hatch, Johanne I. Artman, and Juliana Boerio-Goates, "Phase Transition in K2Cd2(SO4)3: Order Parameter and Microscopic Distortions," Phys. Chem. Minerals 17, 334, 1990. <BR><BR> 48. Dorian M. Hatch, Subrata Ghose, and Harold T. Stokes, "Phase Transitions in Leucite, KAlSi2O6. I. Symmetry Analysis with Order Parameter Treatment and the Resulting Microscopic Distortions," Physics and Chemistry of Minerals 17, 220, 1990. <BR><BR> 47. Dorian M. Hatch and Subrata Ghose "Symmetry Analysis of the Phase Transition and Twinning in MgSiO3 Garnet: Implications to Mantle Mineralogy," American Mineralogist 74, 1221, 1989. <BR><BR> 46. Dorian M. Hatch and Harold T. Stokes, "Group Theoretical Description of Phase Transitions in Solid Bromoform," Journal of Physics: Condensed Matter 2, 1121, 1990. <BR><BR> 45. Dorian M. Hatch and Harold T. Stokes, Comment on "Silent Soft Mode in Hexagonal Barium Titanate Observed by Hyper-Raman Scattering," Physical Review B 40, 4198, 1989. <BR><BR> 44. Dorian M. Hatch and Subrata Ghose, "A Dynamical Model for the I1- to P1- Phase Transition in Anorthite, CaAl2Si2O8 II. Order Parameter Treatment," Physics and Chemistry of Minerals, 16, 614 (1989). <BR><BR> 43. Subrata Ghose, Katsuhiro Tsukimura, and Dorian M. Hatch, "Phase Transitions in Ilvaite, a Mixed-Valence Iron Silicate," Physics and Chemistry of Minerals, 16, 483 (1989). <BR><BR> 42. Dorian M. Hatch and Dana T. Griffen, "Phases Transitions in the Grandite Garnets," American Mineralogist, 74, 151 (1989). <BR><BR> 41. Dorian M. Hatch, Harold T. Stokes, K.S. Aleksandrov, and S.V. Misyul, "Phase Transitions in the Perovskitelike A2BX4 Structure," Physical Review B 39, 9282 (1989). <BR><BR> 40. Dorian M. Hatch, Harold T. Stokes and Rand M. Putnam, "Space Group Representations Induced by Local Site Symmetries," Lecture notes in Physics, Vol. 313, Eds. H.D. Doebner, J.D. Hennig, T. D. Palev, (Springer, New York, 1988) P. 326. <BR><BR> 39. Dorian M. Hatch and Harold T. Stokes, "Classification of Octahedral Tilting Phases in the Perovskitelike A2BX4 Structure," Physical Review B 35, 8509 (1987). <BR><BR> 38. Dorian M. Hatch, Harold T. Stokes, and Rand M. Putnam, "Symmetry Analysis of the Microstructure and Phase Transitions of a Crystallographic Group: Application," Physical Review B 35, 4935 (1987). <BR><BR> 37. Dorian M. Hatch and Harold T. Stokes "Isotropy Subgroups for Reducible Representations of Space Groups," XV International Colloquium on Group Theoretical Methods in Physics (World Scientific, New Jersey, 1987), P. 710. <BR><BR> 36. Harold T. Stokes, Jai Sam Kim, and Dorian M. Hatch, "Continuous Solid-Solid Phase Transition Driven by an Eight-Component Order Parameter: Hamiltonian Densities and Renormalization-Group Theory," Physical Review B 35, 388 (1987). <BR><BR> 35. Serge Galam and Dorian M. Hatch, "Images, Landau Expansions, and Symmetry Changes," Physical Review B 34, 7813 (1986). <BR><BR> 34. Dorian M. Hatch and Harold T. Stokes, "Comment on `Irrelevant Variable, Landau Expansions, and Cubic Anisotropy, Physical Review B 33, 7811 (1986). <BR><BR> 33. Dorian M. Hatch and Harold T. Stokes, "Phase Transitions and Renormalization Group Hamiltonian Densities in the 80 Diperiodic Space Groups," Phase Transitions 7, 87 (1986). <BR><BR> 32. Dorian M. Hatch, Jai Sam Kim, and Harold T. Stokes, "Renormalization-Group Classification of Continuous Structural Phase Transitions Induced by Six-Component Order Parameters," Physical Review B 33, 6196 (1986). <BR><BR> 31. Jai Sam Kim, Harold T. Stokes, and Dorian M. Hatch, "Classification of Continuous Phase Transitions and Stable Phases II: Four Dimensional Order Parameters," Physical Review B 33, 6210 (1986). <BR><BR> 30. Harold T. Stokes, Dorian M. Hatch and Jai Sam Kim, "Images of Physically Irreducible Representations of the 230 Space Groups," Acta Crystallographia A 43, 81 (1987). <BR><BR> 29. Dorian M. Hatch, Harold T. Stokes and Jai Sam Kim, "Selection of Stable Fixed Points by the Toledano-Michel Symmetry Criterion; Six Component Example," Physical Review B 32, 7624 (1985). <BR><BR> 28. Dorian M. Hatch and Harold T. Stokes, "Systematic Classification of Symmetry Changes in Solid-Solid Phase Transitions," Computer Modeling of Phase Diagrams, (Metallurgical Society, Wrendale Penn., 1986). <BR><BR> 27. Dorian M. Hatch, Harold T. Stokes and Jai Sam Kim, "Isotropy Subgroups, Orbit Structure, and Minimization of Single Order Parameter Potentials," Proceedings of XIV International Colloquium on Group Theoretical Methods in Physics, Ed. Y. M. Cho, (World Scientific, Singapore, 1986), 390. <BR><BR> 26. Jai Sam Kim, Dorian M. Hatch, and Harold T. Stokes, "Analysis of Landau Potentials for Six-Dimensional Order Parameters and Realization of Non-Maximal Isotropy Subgroups," Japanese Journal of Applied Physics 24 (Supplement 24-2), 170 (1985). <BR><BR> 25. Dorian M. Hatch and Jeffrey W. Felix, "New Relevant Invariants in the Renormalization Group Study of Ferroelectric Transitions," Japanese Journal of Applied Physics 24 (Supplement 24-2), 176 (1985). <BR><BR> 24. Dorian M. Hatch and Harold T. Stokes, "Ferroelectric Transitions in Diperiodic Systems," Japanese Journal of Applied Physics 24 (Supplement 24-2) 173 (1985). <BR><BR> 23. Jai Sam Kim, Dorian M. Hatch, and Harold T. Stokes, "Classification of Continuous Phase Transitions and Stable Phases. I. Six-Dimensional Order Parameters," Physical Review B 33, 1774 (1986). <BR><BR> 22. Dorian M. Hatch and Harold T. Stokes, "Active Representations in Crystals of Cubic Symmetry," Physica Status Solidi B 130, 79 (1985). <BR><BR> 21. Harold T. Stokes and Dorian M. Hatch, "Equivalence Among Isotropy Subgroups of Space Groups," Physical Review B 31, 7462 (1985). <BR><BR> 20. Dorian M. Hatch and Harold T. Stokes, "Phase Transitions in Solids of Diperiodic Symmetry," Physical Review B 31, 4350 (1985). <BR><BR> 19. Jeffrey W. Felix and Dorian M. Hatch, "Example of a Group Action Determined Phase Transition," Journal of Chemical Physics 82, 1496 (1985). <BR><BR> 18. Dorian M. Hatch and Harold T. Stokes, "A Practical Algorithm for Identifying Sub-groups of Space Groups," Physical Review B 31, 2908 (1985). <BR><BR> 17. Jeffrey W. Felix and Dorian M. Hatch, "A Generalized Molien Function for Field Theoretical Hamiltonians," Journal of Mathematical Physics 26, 1442 (1985). <BR><BR> 16. Jeffrey W. Felix and Dorian M. Hatch, "New Fluctuation-Induced First-Order Anti-ferrodistortive Phase Transitions," Physical Review Letters 53, 2425 (1984). <BR><BR> 15. Dorian M. Hatch and Harold T. Stokes, "Symmetry-Restricted Phase Transitions in Two-Dimensional Solids," Physical Review B 30, 5156 (1984). <BR><BR> 14. Harold T. Stokes and Dorian M. Hatch, "A Table of Group-Subgroup Structural Phase Transitions: Comparison with Existing Tables," Physical Review B 30, 4962 (1984). <BR><BR> 13. Dorian M. Hatch and Harold T. Stokes, "Listings of Possible Structural Phase Transitions," Group Theoretical Methods in Physics, ed. W.W. Zachary (World Scientific, Singapore, 1984) pg. 427. <BR><BR> 12. Harold T. Stokes and Dorian M. Hatch, "Model for the Ordered Phases KCN and NaCN," Physical Review B 30 (7), 3845 (1984). <BR><BR> 11. W. Scott Stornetta and Dorian M. Hatch, "Isotropy Group Description of a Phase Transition in NaNO3," Journal of Chemical Physics 80 (11), 5745 (1984). <BR><BR> 10. Dorian M. Hatch, "Isotropy Groups of Space Groups--A Simple Method for Their Determination," Lectures Notes in Physics vol. 201, eds. G. Denardo, G. Ghiradi, and T. Weber (Springer, New York, 1984) pg. 390. <BR><BR> 9. S. T. Chui, G. Forgacs, and Dorian M. Hatch, "On the Ground State and the Nature of a Phase Transition in a Simple Cubic Fully Frustrated Ising Model," Physical Review B 25 6952 (1982). <BR><BR> 8. Dorian M. Hatch and Daniel L. Decker, "Landau Free Energy Form at the F-Point of the R3-c Structure," Journal of Chemical Physics 74, 2518 (1981). <BR><BR> 7. Dorian M. Hatch, "Order Parameter Symmetry for the B-(ð Transition in Pb3(VO4)2," Physica Status Solidi (b) 106, 473 (1981). <BR><BR> 6. Dorian M. Hatch, "Structural Phase Transitions in Crystals of R3-c Symmetry," Physical Review B 23, 2346 (1981). <BR><BR> 5. Dorian M. Hatch and Leo Merrill, "Landau Description of the Calcite-CaCO3(II) Phase Transition," Physical Review B 23, 368 (1981). <BR><BR> 4. Andrew L. Gardner, Dorian M. Hatch, A.I.Y. Chan, and Robert P. Evans, "Measurements on a Spherical Electrostatic Confinement System Employing Six Ion Guns," Annals of the New York Academy of Sciences 251, (1975) pg. 179. <BR><BR> 3. Dorian M. Hatch, "Selected Stable States in Large Quantum Systems," International Journal of Quantum Chemistry 9, 649-655 (1975). <BR><BR> 2. Alvin K. Benson and Dorian M. Hatch, "A Further Link with Inequivalent Representations in Quantum Field Theory," International Journal of Theoretical Physics 12, 32 (1975). <BR><BR> 1. Alvin K. Benson and Dorian M. Hatch, "Microscopic Mechanism for the Macroscopic Asymmetry of Ferromagnetism," Physical Review B., 8, 4410, (1973). <BR><BR> <CENTER><IMG SRC="images/colorbar.gif" width="600" height="1"></CENTER> <BR> <HR> <CENTER> <TABLE> <TD><A HREF="Research.htm"><FONT SIZE=3>Research</FONT></A>&nbsp;&nbsp;&nbsp;&nbsp;</TD> <TD><A HREF="Biography.htm"><FONT SIZE=3>Biography</FONT></A>&nbsp;&nbsp;&nbsp;&nbsp;</TD> <TD><A HREF="http://www.byu.edu"><FONT SIZE=3>BYU Home</FONT></A>&nbsp;&nbsp;&nbsp;&nbsp;</TD> <TD><A HREF="http://physc1.byu.edu"><FONT SIZE=3>Physics & Astronomy</FONT></A>&nbsp;&nbsp;&nbsp;&nbsp;</TD> <TD><A HREF="hatch.htm"><FONT SIZE=3>Hatch Home</FONT></A>&nbsp;&nbsp;&nbsp;&nbsp;</TD> </TABLE> <HR> </BODY> </HTML>