A bismuth-substituted rare earth iron garnet single crystal suitable for Faraday rotators and optical isolators with reduced insertion loss due to suppressed valence fluctuation of Fe ions is provided. The bismuth-substituted rare earth iron garnet single crystal of the present invention is characterized by the composition formula (Tb.sub.aLn.sub.bBi.sub.cMg.sub.3?(a+b+c))(Fe.sub.dGa.sub.eTi.sub.fPt.sub.5?(d+e+f))O.sub.12. In the composition formula above, 0.02?f?0.05, 0.02?{3?(a+b+c)}?0.08, and ?0.01?{3?(a+b+c)}?{f+5?(d+e+f)}?0.01. Ln is a rare earth element and may be selected from Eu, Gd, Ho, Tm, Yb, Lu, and Y.

The present invention provides polycrystalline silicon suitably used as a raw material for producing single crystal silicon. The polycrystalline silicon rod of the present invention is a polycrystalline silicon rod grown by chemical vapor deposition performed under a pressure of 0.3 MPaG or more, wherein when a plate-shaped sample piece collected from an arbitrary portion of the polycrystalline silicon rod is observed with a microscope with a temperature increased from a temperature lower than a melting point of silicon up to a temperature exceeding the melting point of silicon, a heterogeneous crystal region, which is a crystal region including a plurality of crystal grains heterogeneously assembled and including no needle-like crystal, having a diameter exceeding 10 ?m is not observed.

The present invention provides polycrystalline silicon suitably used as a raw material for producing single crystal silicon. The polycrystalline silicon rod of the present invention is a polycrystalline silicon rod grown by chemical vapor deposition performed under a pressure of 0.3 MPaG or more, wherein when a plate-shaped sample piece collected from an arbitrary portion of the polycrystalline silicon rod is observed with a microscope with a temperature increased from a temperature lower than a melting point of silicon up to a temperature exceeding the melting point of silicon, a heterogeneous crystal region, which is a crystal region including a plurality of crystal grains heterogeneously assembled and including no needle-like crystal, having a diameter exceeding 10 ?m is not observed.

A method for casting comprising: providing a seed, the seed characterized by: an arcuate form and a crystalline orientation progressively varying along an arc of the form; providing molten material; and cooling and solidifying the molten material so that a crystalline structure of the seed propagates into the solidifying material.

A method for casting comprising: providing a seed, the seed characterized by: an arcuate form and a crystalline orientation progressively varying along an arc of the form; providing molten material; and cooling and solidifying the molten material so that a crystalline structure of the seed propagates into the solidifying material.

A method for producing Si ingot single crystal by NOC growth method including a Si ingot single crystal growing step and a continuous growing step is provided. The growing step includes providing a low temperature region in the Si melt where the Si ingot single crystal is grown along the surface of the Si melt or toward the inside of the Si melt, and the Si ingot single crystal has distribution of a vacancy concentration and an interstitial concentration in which respectively a vacancy concentration and an interstitial concentration vary with a distance from the growth interface; and adjusting a temperature gradient and a growth rate in the Si melt, so that along with the increasing of the distance from the growth interface, the vacancy concentration and the interstitial concentration in the Si ingot single crystal respectively decrease come near to each other.

A method for producing Si ingot single crystal by NOC growth method including a Si ingot single crystal growing step and a continuous growing step is provided. The growing step includes providing a low temperature region in the Si melt where the Si ingot single crystal is grown along the surface of the Si melt or toward the inside of the Si melt, and the Si ingot single crystal has distribution of a vacancy concentration and an interstitial concentration in which respectively a vacancy concentration and an interstitial concentration vary with a distance from the growth interface; and adjusting a temperature gradient and a growth rate in the Si melt, so that along with the increasing of the distance from the growth interface, the vacancy concentration and the interstitial concentration in the Si ingot single crystal respectively decrease come near to each other.

A method for producing Si ingot single crystal by NOC growth method including a Si ingot single crystal growing step and a continuous growing step is provided. The growing step includes providing a low temperature region in the Si melt where the Si ingot single crystal is grown along the surface of the Si melt or toward the inside of the Si melt, and the Si ingot single crystal has distribution of a vacancy concentration and an interstitial concentration in which respectively a vacancy concentration and an interstitial concentration vary with a distance from the growth interface; and adjusting a temperature gradient and a growth rate in the Si melt, so that along with the increasing of the distance from the growth interface, the vacancy concentration and the interstitial concentration in the Si ingot single crystal respectively decrease come near to each other.

A method for producing Si ingot single crystal by NOC growth method including a Si ingot single crystal growing step and a continuous growing step is provided. The growing step includes providing a low temperature region in the Si melt where the Si ingot single crystal is grown along the surface of the Si melt or toward the inside of the Si melt, and the Si ingot single crystal has distribution of a vacancy concentration and an interstitial concentration in which respectively a vacancy concentration and an interstitial concentration vary with a distance from the growth interface; and adjusting a temperature gradient and a growth rate in the Si melt, so that along with the increasing of the distance from the growth interface, the vacancy concentration and the interstitial concentration in the Si ingot single crystal respectively decrease come near to each other.

A method for casting comprising: providing a seed, the seed characterized by: an arcuate form and a crystalline orientation progressively varying along an arc of the form; providing molten material; and cooling and solidifying the molten material so that a crystalline structure of the seed propagates into the solidifying material.