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On the oscillations in the solidification rate during facet growth under rotation
Journal of Crystal Growth 11(1971)293—296
North-Holland Publishing Co.
ON THE OSCILLATIONS IN THE SOLIDIFICATION RATE DURING FACET GROWTH UNDER ROTATION J. BARTHEL and M. JURISCH Deutsche Akade,’nie der Wissenschaften zu Berlin, Zentralinstitut fur Fesfkorperphysik und Werkstofforschung, Dresden, DDR Received 5 July 1971, revised manuscript received 1 september 1971 Crystal growth from the melt with rotation of the crystal results in oscillations of the solidification rate in the regions of non-faceted growth. This phenomenon can be explained quantitatively by a simple geometrical evaluation. During facet growth the fluctuations have been found to be much smaller or zero by experiment. In the present work a calculation of these fluctuations using another geometrical model based on the dependence of the growth rate on the kinetic undercooling is attempted. This theoretical treatment verifies the smaller fluctuations in the regions of faceted growth; but it shows only a constant rate if the crystal axis coincides with the normal of the facet. The calculation shows a phase difference in the oscillations in the regions of faceted and non-faceted growth. By comparison of this theoretical work with experimental results it should be possible, to examine the usual assumptions on the mechanism of facet growth.
1. Introduction In case of crystal growth from the melt under rotation thermal asymmetries result in fluctuations of impurity concentration or so-called rotational striations’~5).The reason for these concentration fluctuations are oscillations of the instantaneous microscopic growth rate. In the region of non-faceted growth (offcore region) with the aid of a simple geometrical model the experimentally determined growth rate1’6) can be understood theoretically1’7). The model is based upon the concept of a growth process on a rough interface, En the region of faceted growth, the core-region, concentration fluctuations have been found, too”8’9). Experimental determination of the growth rate, however, showed no or only very small oscillations in the region of faceted growth6” 0). In the same crystal strong fluctuations ofthe rate occurredin the region of non-faceted growth. This result has been considered as the result of different growth mechanisms for rough and atomically smooth growth surfaces. An interpretation has not been attempted there. Development of the above mentioned model to the case of growth on a facet has led to a better understanding of the behaviour of growth-rate on the facet (facet displacement rate). The model uses the existing
knowledge of the solidification mechanism on a smooth interface. With this simply theory the influence of thermal asymmetry, rotation rate, pulling rate, angle between the facet normal and the rotation axis, shape of the solidification isotherm, temperature gradient and material constants can be studied. 2. Physical model The model for calculating the growth rate has been designed under the following assumptions: a) The crystal is grown from the melt under rotation with a rotation rate w and a pulling rate R. The thermal distribution and thus the solidification isotherms are constant in time. They are not symmetrical to the axis of rotation. The thermal distribution as well as the temperature gradient G near the interface are not disturbed by the facet growth. b) The non-faceted growth occurs with negligable kinetic undercooling. The growing interface coincides with the solidification isotherm. The resultant model concerning the growth rate v~has been described in ref. 7. c) The faceted growth takes place via a parallel displacement of a crystallographically defined plane. The angle between the rotation axis and the normal to the plane is ~(fig. I). The displacement rate v~is determined by the movement of the solidification isotherm relative
293
294
J. BARTHEL AN!) M. JIJRISCH
II
thermal ~isH~~rotat~ ~ I
~
r
)
First of all, according to c) it is necessary to cal-
off c~eregi~, ~ef
~_~~9~Wth
solid/f/cat ion isot~rm
suits in a differential equation for the displacement rate the facet:
i~(!) of
E
culate this tion purpose isotherni the point according in(with a stationary P~ onz by the to assumption solidification cylindrical a)coordinate isotherm. the solidificaFor crystal) teni =/(r X (r, isp). ~i, represented :) being the rotation axis ol systhe (I, The z-axis is identical withf~cetrotates the rotational axis pulling of the crystal. In this system the during with the rotation rate w and, conseqLiently can he described by a linear function with the rotation angle
Fig. I. Schematic representation of rotational crystal growth in the presence ofthernial asymmetry and facet grosvth. Pi~:point on the solidification isotherm with maximum distance a from the facet.
to the crystal that is by the maximum Lindercooling AT on the facet. The relation between t’~ and AT can be described as a function v~= ii(AT) or AT = g(v~). The maximum undercooling is obtained at that point of the facet showing the maximum distance a from the solidification isotherm (fig. 1). The corresponding point on the isotherm is ~ Between a and v~,therefore, there exists the correlation AT a
=
G
g(r~) =
‘Po
=
=
wi as a parameter: ii(r, p, p~,D) .
(2)
The other parameter D represented in this equation is the distance of the facet plane from any fixed point in the space; it has not yet been determined. By means of differential calculus these equations allow calculation of P~(p~ 1) and a(p0, D). Secondly, assumption c) gives with a(q0, D) = g(tj/G an equation which can be used to calculate D(p0, q(v~)/G).Consequently the position of the facet in the stationary system can be completely determined by =
u[r,
~,
~po,D(p~,q(v~)/G)].
(~)
--
G
The calculations have been done with linear as well as quadratic dependences of v~on ATi i), i.e. AT = v~/B~ +b1 andAT = ~v~/~+b. B1, B,b1,bareconstants characteristic of the material. d) The thermal asymmetry can cause a temporary remelting of a part of the crystal. During remelting the interface and the solidification isotherm coincide. Consequently on the faceted growth surface the velocity r~ calculated using the model does not have any physical meaning for v~< 0. e) For numerical calculations the solidification isotherm is considered as a spherical surface with the radius p. displaced relative to the rotation axis by a
Thirdly, the position of the facet in relation to the crystal during pulling may be described. This can he done by transition to a coordinate system ~ (P. ~. ~) fixed to the crystal. From the coordinate transformation = z + Rt, P = r, ~p = sumption a) results:
distance ci (fig. I). In the following way the above physical model re-
wi fulfilling the as-
= Ri+u[P, ~+wI, wi, D(wI. g(v~)/G)1. (4) Now the displacement rate t~ can he calculated. The
derivative of equation (4) by time t gives the ~-component of IC~ and by multiplication by cos ~ the rate ç itself:
[
p,,-~
cos c~’ = cos o~ R + I. ~5) Pt ~ The calculation results in the generally valid differeni’~ =
3. Derivation of the mathematical relations
—
tial equation:
295
ON THE OSCILLATIONS IN THE SOLIDIFICATION RATE
=
cos ~ R +wr~ sin c~ sin (co~—wt)— i~ Pg(v~) ‘
(6)
-
C
= h(AT), showed nearly the same rates v~(t)for the same values of the parameters. For this result it is necessary that the values B,, b1 and B, b are chosen in such a way, that suitable equations v~= h(AT) cor-
where r~and c~E are coordinates of ~ in the special case e) these coordinates can be cal-
respond at higher AT (table I). Therefore it is sufficient
culated and substituted in eq. (6). Thus, for a spherical
to discuss.
surface representing the solidification isotherm the differential equation for the determination of v~is given by:
The extrema of the rate v~ are obtained with sin (wtE—)L) = ±1. Their difference is used as a measure for the rate of oscillations and is given by
=
i~~g(v)
R ‘cos~—wd’s,nc’sInwl—---~ ——-fl. (I
(7)
In the case of the linear correlation between v~and integrable. As to the quadratic correlation, i.e. g(v~) = ~‘v~/B+b, this equation has been integrated
numerically. 4. Results The following is valid only for the case of a spherical solidification isotherm. The integration of the differential equation (7) with g(v~) = v~/B1+b1 results in the solution: R ‘cos
wd ~
~i
—
~ --~~
\I
(8)
sin (wt—A)
G)
1 + (w/B,
wcl sin Av~= 2—~-~-----~--.
(9)
OV~
AT, i.e. g(v~) = v~/B1+b~the differential equation is
=
to limit consideration to the linear case, which is easier
The following conclusions can be drawn from this equation: a) Av~is directly proportional to the thermal asymmetry d. b) Av~is approximately directly proportional to the angle c~between the rotation axis and the facet normal in the case of small values of c. c) Av~is independent of the curvature of the interface. d) For the dependence of Av~on the rotation rate w there exist two extreme cases: For w B1 G, Av~is independent of w; for w B1 G, Av~is approxi>~‘
‘~
mately directly proportional to w. e) For B1 G >~w the growth quantities B~and G, have no influence on Ave. With B, ~ w, Av~ becomes directly proportional to the product B~‘G.
with w
tan). = Equation (8) and the numerical solutions of the differential equation (7), using the quadratic dependence ~
WtE of the crystal where the extreme The rotation rates occur, angle can be determined directly from sin (wt~—2) = ±I. A detailed consideration shows that it is only in some defined places at the “core” = f)Liv~is independent 9E of b1.
TABLE
I
non-faceted growth O.5mmr=~lmm Au --
-=
R
~
(mm
-
min~)
2.08
4.20 70
~
z=6°
z~9°
0.42 (0.36)
0.86 (0.70)
1.28 (1.06)
7
14
21
(6)
(II)
(18) 0.075 mm
= 188 min ~, p = 6.5 mm, d= R =3mmmin~, G =5grdmm~, B 1=l2mmmin’~grd~, 1 grd2, bb1=0.4grd, = 0, ~ B = 7mm . min Absolute and relative change of the growth rate in the non-faceted region, calculated after ref. 7, and in the faceted region ofgrowth, calculated for the linear and the second order (in brackets) dependence v~=: h(AT).
parameters:
w
faceted growth a=3
296
J. BARTHEL AN!) M. JtJRISCFI
off core region
46
~‘mm
42 3,6
3.8
90
3,’ ~~60
2.2 2,6
U /4 I1~’
~
10
50
100
150
200
Fig. 2. Growth rate u in the non-faceted or off-core-region, calculated at the core boundary, and in the faceted or core region versus :. The growth parameters are the same as in table I.
II
lillilli
250 2/pm
10
50
100
150
200
250
Z
5/pm
Fig. 3. Growth rate on the facet for diffi.rcnt angles x hetssee.i the rotation axis and the direction perpendicular to the facet. The growth parameters are the same as in table I.
boundary that extreme growth rates on and off the designed on the existing concept of atomically smooth faceted region occur simultaneously. In general a growth reflects qualitatively the experimental findings change of the relative character of the growth rates of refs. 6, 10. For a quantitative comparison between takes place with the transition from rough to atoniic- theory and experiment the parameters necessary for the ally smooth growth. evaluation of the above equations as well as the rate In order to estimate the order of the rate of oscilla- field have to be measured. Such a coin parison together tion predicted by the model, equation (8) has been with results on the distribution of impurities shotild evaluated numerically for a special case. For compari- give detailed information on the mechanism of atomicson the oscillations in the non-faceted region have been ally smooth growth. calculated for the same parameters using equations given by ref. 7.The pulling parameters used approxiReferences mate to practical conditions, and are given in table I. For ~ the values 3°, 6 and 90 have been taken into I) K. Morizarie, A. F. Witt and H. C. Gatos, J. Flcctrochem. consideration. The growth parameters B~,B, b~and h (table I), were obtained from the dependence 1 )~v~= h(AT) determined experimentally for bismuth The results are given in table I and in figs. 2 and 3. The oscillations for rough growth, in fIg. 2 are calculated directly at the boundary of the region of faceted growth. The model shows widely different oscillations of the growth rate in the faceted and non-faceted regions of growth. A constant growth rate ill the core region. however, is only obtained with c~= 0 .
5. Conclusions The results of the calculations show that the model
Soc. 114 (1967) 738. 2) G. Baralis and M. C. Perosino, J. Crystal Growth 3/4(1968) 65!.Billig, Proc. Roy. Soc. (London) 229 ((955) 346. 3) E. 4) R. G. Rhodes, Imperfections and .4ctive Centers in Se,nicon— ductors (Pergamon Press, New-York, 1964). 5) K. F. Hulme and J. B. Mullin, Phil. Mag. 4(1959) 1286. 6) A. F. Witt and H. C. Gatos, J. Electrocheni. Soc. 115(1968) 70.
7) M. Juriseh and J. Barthel, Vortrag zurn 3. Internat. Symposium ‘Reinststoffe in Wissenschaft und Technik”, Dresden, 970 (to be published). 8) J. A. M. Dikhoff, Solid State Electronics 1 (1960) 202. 9) A. F. Witt and H. C. Gatos, J. Elcctrochcrn Soc 113(1966) 808. 10) K. Morizane, A. F. Wiit and H. C. Gatos, .1. E/lectrochcm.
Soc. 115 (1968) 747. II) G. A. Alfintzev and D. E. Ovsienko, in: (ristal Gras-tb Ed. H. S. Peiser (Pergamon Press, Oxford, 1967).
North-Holland Publishing Co.
ON THE OSCILLATIONS IN THE SOLIDIFICATION RATE DURING FACET GROWTH UNDER ROTATION J. BARTHEL and M. JURISCH Deutsche Akade,’nie der Wissenschaften zu Berlin, Zentralinstitut fur Fesfkorperphysik und Werkstofforschung, Dresden, DDR Received 5 July 1971, revised manuscript received 1 september 1971 Crystal growth from the melt with rotation of the crystal results in oscillations of the solidification rate in the regions of non-faceted growth. This phenomenon can be explained quantitatively by a simple geometrical evaluation. During facet growth the fluctuations have been found to be much smaller or zero by experiment. In the present work a calculation of these fluctuations using another geometrical model based on the dependence of the growth rate on the kinetic undercooling is attempted. This theoretical treatment verifies the smaller fluctuations in the regions of faceted growth; but it shows only a constant rate if the crystal axis coincides with the normal of the facet. The calculation shows a phase difference in the oscillations in the regions of faceted and non-faceted growth. By comparison of this theoretical work with experimental results it should be possible, to examine the usual assumptions on the mechanism of facet growth.
1. Introduction In case of crystal growth from the melt under rotation thermal asymmetries result in fluctuations of impurity concentration or so-called rotational striations’~5).The reason for these concentration fluctuations are oscillations of the instantaneous microscopic growth rate. In the region of non-faceted growth (offcore region) with the aid of a simple geometrical model the experimentally determined growth rate1’6) can be understood theoretically1’7). The model is based upon the concept of a growth process on a rough interface, En the region of faceted growth, the core-region, concentration fluctuations have been found, too”8’9). Experimental determination of the growth rate, however, showed no or only very small oscillations in the region of faceted growth6” 0). In the same crystal strong fluctuations ofthe rate occurredin the region of non-faceted growth. This result has been considered as the result of different growth mechanisms for rough and atomically smooth growth surfaces. An interpretation has not been attempted there. Development of the above mentioned model to the case of growth on a facet has led to a better understanding of the behaviour of growth-rate on the facet (facet displacement rate). The model uses the existing
knowledge of the solidification mechanism on a smooth interface. With this simply theory the influence of thermal asymmetry, rotation rate, pulling rate, angle between the facet normal and the rotation axis, shape of the solidification isotherm, temperature gradient and material constants can be studied. 2. Physical model The model for calculating the growth rate has been designed under the following assumptions: a) The crystal is grown from the melt under rotation with a rotation rate w and a pulling rate R. The thermal distribution and thus the solidification isotherms are constant in time. They are not symmetrical to the axis of rotation. The thermal distribution as well as the temperature gradient G near the interface are not disturbed by the facet growth. b) The non-faceted growth occurs with negligable kinetic undercooling. The growing interface coincides with the solidification isotherm. The resultant model concerning the growth rate v~has been described in ref. 7. c) The faceted growth takes place via a parallel displacement of a crystallographically defined plane. The angle between the rotation axis and the normal to the plane is ~(fig. I). The displacement rate v~is determined by the movement of the solidification isotherm relative
293
294
J. BARTHEL AN!) M. JIJRISCH
II
thermal ~isH~~rotat~ ~ I
~
r
)
First of all, according to c) it is necessary to cal-
off c~eregi~, ~ef
~_~~9~Wth
solid/f/cat ion isot~rm
suits in a differential equation for the displacement rate the facet:
i~(!) of
E
culate this tion purpose isotherni the point according in(with a stationary P~ onz by the to assumption solidification cylindrical a)coordinate isotherm. the solidificaFor crystal) teni =/(r X (r, isp). ~i, represented :) being the rotation axis ol systhe (I, The z-axis is identical withf~cetrotates the rotational axis pulling of the crystal. In this system the during with the rotation rate w and, conseqLiently can he described by a linear function with the rotation angle
Fig. I. Schematic representation of rotational crystal growth in the presence ofthernial asymmetry and facet grosvth. Pi~:point on the solidification isotherm with maximum distance a from the facet.
to the crystal that is by the maximum Lindercooling AT on the facet. The relation between t’~ and AT can be described as a function v~= ii(AT) or AT = g(v~). The maximum undercooling is obtained at that point of the facet showing the maximum distance a from the solidification isotherm (fig. 1). The corresponding point on the isotherm is ~ Between a and v~,therefore, there exists the correlation AT a
=
G
g(r~) =
‘Po
=
=
wi as a parameter: ii(r, p, p~,D) .
(2)
The other parameter D represented in this equation is the distance of the facet plane from any fixed point in the space; it has not yet been determined. By means of differential calculus these equations allow calculation of P~(p~ 1) and a(p0, D). Secondly, assumption c) gives with a(q0, D) = g(tj/G an equation which can be used to calculate D(p0, q(v~)/G).Consequently the position of the facet in the stationary system can be completely determined by =
u[r,
~,
~po,D(p~,q(v~)/G)].
(~)
--
G
The calculations have been done with linear as well as quadratic dependences of v~on ATi i), i.e. AT = v~/B~ +b1 andAT = ~v~/~+b. B1, B,b1,bareconstants characteristic of the material. d) The thermal asymmetry can cause a temporary remelting of a part of the crystal. During remelting the interface and the solidification isotherm coincide. Consequently on the faceted growth surface the velocity r~ calculated using the model does not have any physical meaning for v~< 0. e) For numerical calculations the solidification isotherm is considered as a spherical surface with the radius p. displaced relative to the rotation axis by a
Thirdly, the position of the facet in relation to the crystal during pulling may be described. This can he done by transition to a coordinate system ~ (P. ~. ~) fixed to the crystal. From the coordinate transformation = z + Rt, P = r, ~p = sumption a) results:
distance ci (fig. I). In the following way the above physical model re-
wi fulfilling the as-
= Ri+u[P, ~+wI, wi, D(wI. g(v~)/G)1. (4) Now the displacement rate t~ can he calculated. The
derivative of equation (4) by time t gives the ~-component of IC~ and by multiplication by cos ~ the rate ç itself:
[
p,,-~
cos c~’ = cos o~ R + I. ~5) Pt ~ The calculation results in the generally valid differeni’~ =
3. Derivation of the mathematical relations
—
tial equation:
295
ON THE OSCILLATIONS IN THE SOLIDIFICATION RATE
=
cos ~ R +wr~ sin c~ sin (co~—wt)— i~ Pg(v~) ‘
(6)
-
C
= h(AT), showed nearly the same rates v~(t)for the same values of the parameters. For this result it is necessary that the values B,, b1 and B, b are chosen in such a way, that suitable equations v~= h(AT) cor-
where r~and c~E are coordinates of ~ in the special case e) these coordinates can be cal-
respond at higher AT (table I). Therefore it is sufficient
culated and substituted in eq. (6). Thus, for a spherical
to discuss.
surface representing the solidification isotherm the differential equation for the determination of v~is given by:
The extrema of the rate v~ are obtained with sin (wtE—)L) = ±1. Their difference is used as a measure for the rate of oscillations and is given by
=
i~~g(v)
R ‘cos~—wd’s,nc’sInwl—---~ ——-fl. (I
(7)
In the case of the linear correlation between v~and integrable. As to the quadratic correlation, i.e. g(v~) = ~‘v~/B+b, this equation has been integrated
numerically. 4. Results The following is valid only for the case of a spherical solidification isotherm. The integration of the differential equation (7) with g(v~) = v~/B1+b1 results in the solution: R ‘cos
wd ~
~i
—
~ --~~
\I
(8)
sin (wt—A)
G)
1 + (w/B,
wcl sin Av~= 2—~-~-----~--.
(9)
OV~
AT, i.e. g(v~) = v~/B1+b~the differential equation is
=
to limit consideration to the linear case, which is easier
The following conclusions can be drawn from this equation: a) Av~is directly proportional to the thermal asymmetry d. b) Av~is approximately directly proportional to the angle c~between the rotation axis and the facet normal in the case of small values of c. c) Av~is independent of the curvature of the interface. d) For the dependence of Av~on the rotation rate w there exist two extreme cases: For w B1 G, Av~is independent of w; for w B1 G, Av~is approxi>~‘
‘~
mately directly proportional to w. e) For B1 G >~w the growth quantities B~and G, have no influence on Ave. With B, ~ w, Av~ becomes directly proportional to the product B~‘G.
with w
tan). = Equation (8) and the numerical solutions of the differential equation (7), using the quadratic dependence ~
WtE of the crystal where the extreme The rotation rates occur, angle can be determined directly from sin (wt~—2) = ±I. A detailed consideration shows that it is only in some defined places at the “core” = f)Liv~is independent 9E of b1.
TABLE
I
non-faceted growth O.5mmr=~lmm Au --
-=
R
~
(mm
-
min~)
2.08
4.20 70
~
z=6°
z~9°
0.42 (0.36)
0.86 (0.70)
1.28 (1.06)
7
14
21
(6)
(II)
(18) 0.075 mm
= 188 min ~, p = 6.5 mm, d= R =3mmmin~, G =5grdmm~, B 1=l2mmmin’~grd~, 1 grd2, bb1=0.4grd, = 0, ~ B = 7mm . min Absolute and relative change of the growth rate in the non-faceted region, calculated after ref. 7, and in the faceted region ofgrowth, calculated for the linear and the second order (in brackets) dependence v~=: h(AT).
parameters:
w
faceted growth a=3
296
J. BARTHEL AN!) M. JtJRISCFI
off core region
46
~‘mm
42 3,6
3.8
90
3,’ ~~60
2.2 2,6
U /4 I1~’
~
10
50
100
150
200
Fig. 2. Growth rate u in the non-faceted or off-core-region, calculated at the core boundary, and in the faceted or core region versus :. The growth parameters are the same as in table I.
II
lillilli
250 2/pm
10
50
100
150
200
250
Z
5/pm
Fig. 3. Growth rate on the facet for diffi.rcnt angles x hetssee.i the rotation axis and the direction perpendicular to the facet. The growth parameters are the same as in table I.
boundary that extreme growth rates on and off the designed on the existing concept of atomically smooth faceted region occur simultaneously. In general a growth reflects qualitatively the experimental findings change of the relative character of the growth rates of refs. 6, 10. For a quantitative comparison between takes place with the transition from rough to atoniic- theory and experiment the parameters necessary for the ally smooth growth. evaluation of the above equations as well as the rate In order to estimate the order of the rate of oscilla- field have to be measured. Such a coin parison together tion predicted by the model, equation (8) has been with results on the distribution of impurities shotild evaluated numerically for a special case. For compari- give detailed information on the mechanism of atomicson the oscillations in the non-faceted region have been ally smooth growth. calculated for the same parameters using equations given by ref. 7.The pulling parameters used approxiReferences mate to practical conditions, and are given in table I. For ~ the values 3°, 6 and 90 have been taken into I) K. Morizarie, A. F. Witt and H. C. Gatos, J. Flcctrochem. consideration. The growth parameters B~,B, b~and h (table I), were obtained from the dependence 1 )~v~= h(AT) determined experimentally for bismuth The results are given in table I and in figs. 2 and 3. The oscillations for rough growth, in fIg. 2 are calculated directly at the boundary of the region of faceted growth. The model shows widely different oscillations of the growth rate in the faceted and non-faceted regions of growth. A constant growth rate ill the core region. however, is only obtained with c~= 0 .
5. Conclusions The results of the calculations show that the model
Soc. 114 (1967) 738. 2) G. Baralis and M. C. Perosino, J. Crystal Growth 3/4(1968) 65!.Billig, Proc. Roy. Soc. (London) 229 ((955) 346. 3) E. 4) R. G. Rhodes, Imperfections and .4ctive Centers in Se,nicon— ductors (Pergamon Press, New-York, 1964). 5) K. F. Hulme and J. B. Mullin, Phil. Mag. 4(1959) 1286. 6) A. F. Witt and H. C. Gatos, J. Electrocheni. Soc. 115(1968) 70.
7) M. Juriseh and J. Barthel, Vortrag zurn 3. Internat. Symposium ‘Reinststoffe in Wissenschaft und Technik”, Dresden, 970 (to be published). 8) J. A. M. Dikhoff, Solid State Electronics 1 (1960) 202. 9) A. F. Witt and H. C. Gatos, J. Elcctrochcrn Soc 113(1966) 808. 10) K. Morizane, A. F. Wiit and H. C. Gatos, .1. E/lectrochcm.
Soc. 115 (1968) 747. II) G. A. Alfintzev and D. E. Ovsienko, in: (ristal Gras-tb Ed. H. S. Peiser (Pergamon Press, Oxford, 1967).