## 1. Introduction

[2] A graph of the ratio of saturation remanence to saturation magnetization, *M*_{rs}/*M*_{s}, against the ratio of remanent coercive force to ordinary coercive force, *H*_{cr}/*H*_{c}, was proposed by *Day et al.* [1977], and further developed by *Parry* [1982], as a method of discriminating domain state (single-domain, SD; pseudo-single-domain, PSD; multidomain, MD) and, by implication, grain size. For Day et al.'s sized grains of titanomagnetite (Fe_{3 – x}Ti_{x}O_{4}), values of the two ratios follow an approximately hyperbolic curve for all four compositions tested (*x* = 0, 0.2, 0.4, 0.6). There is a monotonic trend with grain size, the finer grains approaching SD endpoint values of *M*_{rs}/*M*_{s} and *H*_{cr}/*H*_{c} and the coarser grains approaching MD values. However, data for grains of the same size but different compositions lie on different parts of the curve. In addition, comparing the data of *Day et al.* [1977], *Parry* [1965, 1980, 1982], and *Dankers and Sugiura* [1981], different levels of internal stress introduced in sample preparation can be seen to shift points along the curve. In order to use the Day plot to determine grain size, one must therefore have independent information about mineral composition and state of internal stress.

[3] Day plots have been published as part of many paleomagnetic and environmental magnetic studies, for suites of oceanic rocks [e.g., *Dunlop*, 1981; *Tauxe et al.*, 1996; *Gee and Kent*, 1999], soils and lake and marine sediments [e.g., *Özdemir and Banerjee*, 1982; *King et al.*, 1982; *Smirnov and Tarduno*, 2000], and continental carbonate rocks [e.g., *Jackson*, 1990; *Channell and McCabe*, 1994; *Suk and Halgedahl*, 1996]. These and many other data sets show that there is no single curve that explains all the data. *Jackson* [1990] proposed that the strikingly different trends for remagnetized and unremagnetized carbonate rocks result from mixtures of SP and SD grains in the first case and SD + MD mixtures in the second case. *Gee and Kent* [1999] likewise propose SP + SD and SD + MD mixtures as the cause of different trends observed in interior to rim traverses of submarine basalt pillows. *Tauxe et al.* [1996] numerically modeled SP + SD mixtures and were able to explain data for submarine basaltic glasses, which do not fall on any simple trend.

[4] The purpose of the present study is to make a first-principles theoretical treatment of the parameters *M*_{rs}, *H*_{cr}, *H*_{c} and their correlation in the Day plot for SP, SD, PSD, and MD grains and mixtures of these domain states. The theoretical predictions will then be compared to published data for magnetite and other titanomagnetites (particularly Fe_{2.4}Ti_{0.6}O_{4} or TM60) with controlled grain sizes or mechanical mixtures of different sizes. In the companion paper by *Dunlop* [2002] the type curves will be compared to data for soils, sediments, carbonate rocks, and oceanic basalts and glasses.