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tb@xmath0ga@xmath1o@xmath2 ( tgg ) is a dielectric material with a cubic garnet structure .
each tb@xmath3 ion has the same @xmath4 symmetry in its own local coordinate system which leads to pronounced crystal field ( cef ) effects in various physical properties .
tgg exhibits a transition to a antiferromagnetically ordered phase at 0.35k ( ref.@xcite ) .
the terbium ions are located on inter - penetrating triangular lattices , but they do not form a simple kagome lattice , where the triangles lie in a plane , but are arrayed in three dimensions , forming a so - called hyperkagome lattice @xcite , which like the kagome lattice is geometrically frustrated .
various interesting experiments have been performed recently with this substance .
one of them is the phonon hall effect . in analogy to the classical hall effect in conducting materials , the appearance of a thermal gradient in the direction perpendicular to both , the applied magnetic field and the thermal flux , was observed @xcite .
another experiment is the acoustic faraday effect where the rotation of a linearly polarized ultrasonic wave in a magnetic field applied along the propagation direction was observed and quantitatively analyzed @xcite . in another experiment the temperature dependence of the symmetry
elastic constants were measured and interpreted with a simple cef - model @xcite .
the aim of this paper is to show , that a cef - hamiltonian for the orthorhombic sorroundings of the tb ions is well fitted to describe the low energy spectrum and the thermodynamic properties of tgg .
we expect that this is also of intereset for a detailed model of transport phenomena like the phonon hall effect .
first we discuss the cef - hamiltonian , next we calculate the the magnetization in the three important crystal directions in high magnetic field and compare the result with existing experiments @xcite .
afterwards we show experimental results of the magnetic susceptibility and compare it with analogous calculations .
finally we discuss our esr results and interpret them using the same cef - model .
a tb@xmath3 ion has 8 4f - electrons leading with hund s rule to s=3 , l=3 and j=6 .
the cef for these ions with local orthorhombic symmetry can be described with a hamiltonian , introduced by guillot et al .
@xcite in 1985 , which in our notation reads :      here @xmath6 is the magnetic field in the local coordinate system of a tb@xmath3 ion .
the @xmath7 are the stevens operators as defined in @xcite , and the @xmath8 are the crystal field parameters .
the @xmath8 can be calculated from the @xmath9 given in ref .
we list the @xmath9 from @xcite in table [ table1 ] and give the explicit relation between the @xmath9 and the @xmath8 in the appendix . for a detailed discussion of the conventions used in defining the crystal field parameters see e.g.@xcite .
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we report on the measurement of the magnetic susceptibility and of esr transitions in the garnet substance tb@xmath0ga@xmath1o@xmath2 ( tgg ) . 
 the results are compared with a calculation in the framework of crystal field theory for the orthorhombic surroundings of the six inequivalent tb ions of tgg . 
 we also present a calculation of the magnetization for the three main crystal directions .