Chapter 4 Symmetry and Group Theory

Chapter 4 Symmetry and Group Theory
H
I
C1-..- / C, Cl c3:gc1 C1
C1 Top view
C3 rotations of CHCI,
Cross section of protein disk
of tobacco mosaic virus
C2, C, and C6 rotations
of a snowflake design
FIGURE 4-2 Rotations. The cross section of ilie tobacco mosaic virus is a cover diagram from
Nature, 1976,259. O 1976, Macmillan Journals Ltd. Reproduced with permission of Aaron Klug.
operation is designated c~~ and is also a symmetry operation of the molecule. Three successive
C3 operations are the same as the identity operation (c~~ = E). The identity operation
is included in all molecules. Many molecules and other objects have multiple rotation
axes. Snowflakes are a case in point, with complex shapes that are nearly always hexagonal
and nearly planar. The line through the center of the flake perpendicular to the plane of the
flake contains a twofold (C2) axis, a threefold (C3) axis, and a sixfold (C6) axis. Rotations
by 240" (~3~) and 300" (~6~) are also symmetry operations of the snowflake.
Rotation Angle Symmetry Operation
There are also two sets of three C2 axes in the plane of the snowflake, one set
through opposite points and one through the cut-in regions between the points. One of
each of these axes is shown in Figure 4-2. In molecules with more than one rotation axis,
the C, axis having the largest value of n is the highest order rotation axis or principal
axis. The highest order rotation axis for a snowflake is the C6 axis. (In assigning Cartesian
coordinates, the highest order C, axis is usually chosen as the z axis.) When necessary,
the C2 axes perpendicular to the principal axis are designated with primes; a single
prime (C2') indicates that the axis passes through several atoms of the molecule, whereas
a double prime (C2") indicates that it passes between the outer atoms.
Finding rotation axes for some three-dimensional figures is more difficult, but the
same in principle. Remember that nature is not always simple when it comes to
symmetry-the protein disk of the tobacco mosaic virus has a 17-fold rotation axis!
In the reflection operation (o) the molecule contains a mirror plane. If details
such as hair style and location of internal organs are ignored, the human body has a leftright
minor plane, as in Figure 4-3. Many molecules have mirror planes, although they
may not be immediately obvious. The reflection operation exchanges left and right, as if
each point had moved perpendicularly through the plane to a position cxactly as far from
the plane as when it started. Linear objects such as a round wood pencil or molecules

ผลลัพธ์ (อังกฤษ) 2:

Chapter 4 Symmetry and Group Theory
H
I
-..- C1 / C, Cl C3: gc1 C1
C1 Top View
C3 rotations of Chci,
Cross section of Disk protein
of tobacco Mosaic Virus
C2, C, and C6 rotations
of a Snowflake Design
FIGURE. 4-2 Rotations. The cross section of Ilie tobacco Mosaic Virus is a diagram Cover from
Nature, 1976.259. O one thousand nine hundred and seventy-six, Macmillan Journals Ltd. reproduced with permission of Aaron Klug.
Operation is designated C ~~ and is also a Symmetry Operation of the molecule. Three successive
C3 Operations are the Same as the Identity Operation (C ~~ = E). The Identity Operation
is included in all molecules. Many have multiple molecules and Other Objects Rotation
axes. Snowflakes are a Case in Point, with Complex Shapes that are nearly always Hexagonal
and nearly planar. The line Through the Center of the flake Perpendicular to the Plane of the
flake contains a Twofold (C2) axis, a threefold (C3) axis, and a Sixfold (C6) axis. Rotations
by 240 "(~ 3 ~) and 300" (~ 6 ~) are also Symmetry Operations of the Snowflake.
Rotation Angle Symmetry Operation
There are also Two sets of Three C2 axes in the Plane of the Snowflake, one SET
Through Opposite points. and one through the cut-in regions between the points. One of
each of these axes is shown in Figure 4-2. In molecules with more than one Rotation axis,
the C, axis having the largest Value of n is the highest Order or Principal axis Rotation
axis. The highest order rotation axis for a snowflake is the C6 axis. (In assigning Cartesian
Coordinates, the highest Order C, axis is usually Chosen as the Z axis.) When necessary,
the C2 axes Perpendicular to the Principal axis are designated with primes; a single
Prime (C2 ') Indicates that the axis passes Through several atoms of the molecule, whereas
a double Prime (C2 ") Indicates that it passes between the Outer atoms.
Finding Rotation axes for Some Three-dimensional figures is more difficult, but. the
Same in principle. Remember that nature is not always Simple when it comes to
Symmetry-the protein Disk of the tobacco Mosaic Virus has a 17-Fold Rotation axis!
In the Reflection Operation (o) the molecule contains a Mirror Plane. If Details.
such as Hair style and Location of internal organs are ignored, the Human Body has a Leftright
MINOR Plane, as in Figure 4-3. Many molecules have Mirror Planes, Although they
May not be immediately Obvious. The Reflection Operation Exchanges left and Right,. as if
each had moved Perpendicularly Through the Point Plane to a position as Cxactly Far from
the Plane as when it Started. Linear Objects such as a round or molecules Wood Pencil.

ผลลัพธ์ (อังกฤษ) 3:

Chapter 4 Symmetry and Group Theory


H I C1 - - - - C Cl, C3: GC1 C1
C1 Top view
C3 rotations, of CHCI
Cross section of protein. Disk

, of tobacco mosaic virus C2 C and C6, rotations
of a snowflake design
FIGURE 4-2 Rotations. The cross section of Ilie. Tobacco mosaic virus is a cover diagram from
Nature 1976 259. O 1976,,, Journals Macmillan Ltd. Reproduced with permission. Of Aaron Klug.
.Operation is designated C ~ ~ and is also a symmetry operation of the molecule. Three successive
C3 operations are the same. As the identity operation (C ~ = E). The identity operation
is included in all molecules. Many molecules and other objects. Have multiple rotation
axes. Snowflakes are a case in point with complex, shapes that are nearly always hexagonal
and nearly. Planar.The line through the center of the flake perpendicular to the plane of the
flake contains a twofold (C2), axis a threefold. (C3), axis and a sixfold (C6) axis. Rotations
by 240 "(~ 3 ~) and 300" (~ 6 ~) are also symmetry operations of the snowflake.
Rotation. Angle Symmetry Operation
There are also two sets of three C2 axes in the plane of, the snowflake one set
.Through opposite points and one through the cut - in regions between the points. One of
each of these axes is shown in Figure 4-2. In. Molecules with more than one rotation axis
the, C axis having, the largest value of n is the highest order rotation axis. Or principal
axis. The highest order rotation axis for a snowflake is the C6 axis. (In assigning Cartesian
coordinates,, The highest, order CAxis is usually chosen as the Z axis.) When necessary
the, C2 axes perpendicular to the principal axis are designated with. Primes; a single
prime (C2 ') indicates that the axis passes through several atoms of, the molecule whereas
a double prime. (C2. ") indicates that it passes between the outer atoms.
Finding rotation axes for some three-dimensional figures is more. Difficult but, the
.Same in principle. Remember that nature is not always simple when it comes to
symmetry-the protein disk of the tobacco. Mosaic virus has a 17-fold rotation axis!
In the reflection operation (o) the molecule contains a mirror plane. If details
such. As hair style and location of internal organs are ignored the human, body has a leftright
minor plane as in, Figure 4-3.Many molecules have, mirror planes although they
may not be immediately obvious. The reflection operation exchanges left. And, right as if
each point had moved perpendicularly through the plane to a position cxactly as far from
the plane as when. It started. Linear objects such as a round wood pencil or molecules.