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© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 Ch120a-
Goddard-
L01
1
Nature of the Chemical Bond
with applications to catalysis, materials
science, nanotechnology, surface science,
bioinorganic chemistry, and energy
William A. Goddard, III, [email protected]
316 Beckman Institute, x3093
Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics,
California Institute of Technology
Teaching Assistants: Caitlin Scott <[email protected]>
Hai Xiao [email protected]; Fan Liu <[email protected]>
Lecture 9 January 30, 2013
Ionic bonding and crystals
Course number: Ch120a
Hours: 2-3pm Monday, Wednesday, Friday
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 2
Ionic bonding (chapter 9)
Consider the covalent bond of Na to Cl. There Is very little
contragradience, leading to an extremely weak bond.
Alternatively, consider
transferring the charge
from Na to Cl to form
Na+ and Cl-
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 3
The ionic limit
At R=∞ the cost of forming Na+ and Cl-
is IP(Na) = 5.139 eV minus EA(Cl) = 3.615 eV = 1.524 eV
But as R is decreased the electrostatic energy drops as
DE(eV) = - 14.4/R(A) or DE (kcal/mol) = -332.06/R(A)
Thus this ionic curve crosses the covalent curve at
R=14.4/1.524=9.45 A
R(A)
E(eV)
Using the bond distance
of NaCl=2.42A leads to
a coulomb energy of
6.1eV leading to a bond
of 6.1-1.5=4.6 eV
The exper De = 4.23 eV
Showing that ionic
character dominates
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 4
GVB orbitals
of NaCl
Dipole moment
= 9.001 Debye
Pure ionic
11.34 Debye
Thus Dq=0.79 e
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 5
electronegativity
To provide a measure to estimate polarity in bonds, Linus
Pauling developed a scale of electronegativity () where
the atom that gains charge is more electronegative and
the one that loses is more electropositive
He arbitrarily assigned
=4 for F, 3.5 for O, 3.0 for N, 2.5 for C, 2.0 for B, 1.5 for
Be, and 1.0 for Li
and then used various experiments to estimate other
cases . Current values are on the next slide
Mulliken formulated an alternative scale such that
M= (IP+EA)/5.2
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 6
Electronegativity
Based on M++
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 7
Comparison of Mulliken and Pauling electronegativities
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 8
Ionic crystals
Starting with two NaCl monomer, it is downhill by 2.10
eV (at 0K) for form the dimer
Because of repulsion between
like charges the bond lengths,
increase by 0.26A.
A purely electrostatic calculation would
have led to a bond energy of 1.68 eV
Similarly, two dimers can combine to form
a strongly bonded tetramer with a nearly
cubic structure
Continuing, combining 4x1018 such
dimers leads to a grain of salt in which
each Na has 6 Cl neighbors and each Cl
has 6 Na neighbors
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 9
The NaCl or B1 crystal
All alkali halides
have this
structure except
CsCl, CsBr, CsI
(they have the B2
structure)
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 10
The CsCl or B2 crystal
There is not yet a good understanding of the fundamental
reasons why particular compound prefer particular
structures. But for ionic crystals the consideration of ionic
radii has proved useful
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 11
Ionic radii, main group
From R. D. Shannon, Acta Cryst. A32, 751 (1976)
Fitted to various crystals. Assumes O2- is 1.40A
NaCl R=1.02+1.81 = 2.84, exper is 2.84
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 12
Ionic radii, transition metals
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 13
Ionic radii Lanthanides and Actinide
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 14
Role of ionic sizes in determining crystal structures
Assume that the anions are large and packed so that they
contact, so that 2RA < L, where L is the distance between anions
Assume that the anion and cation are in contact.
Calculate the smallest cation consistent with 2RA < L.
RA+RC = L/√2 > √2 RA
Thus RC/RA > 0.414
RA+RC = (√3)L/2 > (√3) RA
Thus RC/RA > 0.732
Thus for 0.414 < (RC/RA ) < 0.732 we expect B1
For (RC/RA ) > 0.732 either is ok.
For (RC/RA ) < 0.414 must be some other structure
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 15
Radius Ratios of Alkali Halides and Noble metal halices
Based on R. W.
G. Wyckoff,
Crystal
Structures, 2nd
edition. Volume 1
(1963)
Rules work ok
B1: 0.35 to 1.26
B2: 0.76 to 0.92
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 16
Wurtzite or B4 structure
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 17
Sphalerite or Zincblende or B3 structure GaAs
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 18
Radius rations B3, B4
The height of the tetrahedron is (2/3)√3 a where a is the side of
the circumscribed cube
The midpoint of the tetrahedron (also the midpoint of the cube) is
(1/2)√3 a from the vertex.
Hence (RC + RA)/L = (½) √3 a / √2 a = √(3/8) = 0.612
Thus 2RA < L = √(8/3) (RC + RA) = 1.633 (RC + RA)
Thus 1.225 RA < (RC + RA) or RC/RA > 0.225
Thus B3,B4 should be the stable structures for
0.225 < (RC/RA) < 0. 414
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 19
Structures for II-VI compounds
B3 for 0.20 < (RC/RA) < 0.55
B1 for 0.36 < (RC/RA) < 0.96
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 20
CaF2 or fluorite structure
Like GaAs but
now have F at
all tetrahedral
sites
Or like CsCl
but with half
the Cs missing
Find for RC/RA > 0.71
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 21
Rutile (TiO2) or Cassiterite (SnO2) structure
Related to NaCl
with half the
cations missing
Find for RC/RA < 0.67
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 22
rutile
CaF2
rutile
CaF2
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25
Stopped L17, Feb 10
23
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 24
Electrostatic Balance Postulate
For an ionic crystal the charges transferred from all cations
must add up to the extra charges on all the anions.
We can do this bond by bond, but in many systems the
environments of the anions are all the same as are the
environments of the cations. In this case the bond polarity
(S) of each cation-anion pair is the same and we write
S = zC/nC where zC is the net charge on the cation and nC is
the coordination number
Then zA = Si SI = Si zCi /ni
Example1 : SiO2. in most phases each Si is in a tetrahedron
of O2- leading to S=4/4=1.
Thus each O2- must have just two Si neighbors
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 25
a-quartz structure of SiO2
Helical chains
single crystals optically active;
α-quartz converts to β-quartz
at 573 °C
rhombohedral
(trigonal)
hP9, P3121
No.152[10]
Each Si bonds to 4 O,
OSiO = 109.5°
each O bonds to 2 Si
Si-O-Si = 155.x °
From wikipedia
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 26
Example 2 of electrostatic balance: stishovite phase of SiO2
The stishovite phase of SiO2 has six coordinate Si, S=2/3.
Thus each O must have 3 Si neighbors
From wikipedia
Rutile-like structure, with 6-
coordinate Si;
high pressure form
densest of the SiO2
polymorphs
tetragonal
tP6, P42/mnm,
No.136[17]
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 27
TiO2, example 3 electrostatic balance
Example 3: the rutile, anatase, and brookite phases of TiO2
all have octahedral Ti.
Thus S= 2/3 and each O must be coordinated to 3 Ti.
top
front right
anatase phase TiO2
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 28
Corundum (a-Al2O3). Example 4 electrostatic balance
Each Al3+ is in a distorted octahedron,
leading to S=1/2.
Thus each O2- must be coordinated to 4 Al
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 29
Olivine. Mg2SiO4. example 5 electrostatic balance
Each Si has four O2- (S=1) and each
Mg has six O2- (S=1/3).
Thus each O2- must be coordinated to
1 Si and 3 Mg neighbors
O = Blue atoms (closest packed)
Si = magenta (4 coord) cap voids in
zigzag chains of Mg
Mg = yellow (6 coord)
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 30
Illustration, BaTiO3
A number of important oxides have the perovskite structure
(CaTiO3) including BaTiO3, KNbO3, PbTiO3.
Lets try to predict the structure without looking it up
Based on the TiO2 structures , we expect the Ti to be in an
octahedron of O2-, STiO = 2/3.
How many Ti neighbors will each O have?
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 31
Illustration, BaTiO3
A number of important oxides have the perovskite structure
(CaTiO3) including BaTiO3, KNbO3, PbTiO3.
Lets try to predict the structure without looking it up
Based on the TiO2 structures , we expect the Ti to be in an
octahedron of O2-, STiO = 2/3.
How many Ti neighbors will each O have?
It cannot be 3 since there would be no place for the Ba.
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 32
Illustration, BaTiO3
A number of important oxides have the perovskite structure
(CaTiO3) including BaTiO3, KNbO3, PbTiO3.
Lets try to predict the structure without looking it up
Based on the TiO2 structures , we expect the Ti to be in an
octahedron of O2-, STiO = 2/3.
How many Ti neighbors will each O have?
It cannot be 3 since there would be no place for the Ba.
It is likely not one since Ti does not make oxo bonds.
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 33
Illustration, BaTiO3
A number of important oxides have the perovskite structure
(CaTiO3) including BaTiO3, KNbO3, PbTiO3.
Lets try to predict the structure without looking it up
Based on the TiO2 structures , we expect the Ti to be in an
octahedron of O2-, STiO = 2/3.
How many Ti neighbors will each O have?
It cannot be 3 since there would be no place for the Ba.
It is likely not one since Ti does not make oxo bonds.
Thus we expect each O to have two Ti neighbors, probably at
180º. This accounts for 2*(2/3)= 4/3 charge.
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 34
Illustration, BaTiO3
A number of important oxides have the perovskite structure
(CaTiO3) including BaTiO3, KNbO3, PbTiO3.
Lets try to predict the structure without looking it up
Based on the TiO2 structures , we expect the Ti to be in an
octahedron of O2-, STiO = 2/3.
How many Ti neighbors will each O have?
It cannot be 3 since there would be no place for the Ba.
It is likely not one since Ti does not make oxo bonds.
Thus we expect each O to have two Ti neighbors, probably at
180º. This accounts for 2*(2/3)= 4/3 charge.
Now we must consider how many O are around each Ba, nBa,
leading to SBa = 2/nBa, and how many Ba around each O,
nOBa.
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 35
Illustration, BaTiO3
A number of important oxides have the perovskite structure
(CaTiO3) including BaTiO3, KNbO3, PbTiO3.
Lets try to predict the structure without looking it up
Based on the TiO2 structures , we expect the Ti to be in an
octahedron of O2-, STiO = 2/3.
How many Ti neighbors will each O have?
It cannot be 3 since there would be no place for the Ba.
It is likely not one since Ti does not make oxo bonds.
Thus we expect each O to have two Ti neighbors, probably at
180º. This accounts for 2*(2/3)= 4/3 charge.
Now we must consider how many O are around each Ba, nBa,
leading to SBa = 2/nBa, and how many Ba around each O,
nOBa.
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 36
Prediction of BaTiO3 structure : Ba coordination
Since nOBa* SBa = 2/3, the missing charge for the O, we have
only a few possibilities:
nBa= 3 leading to SBa = 2/nBa=2/3 leading to nOBa = 1
nBa= 6 leading to SBa = 2/nBa=1/3 leading to nOBa = 2
nBa= 9 leading to SBa = 2/nBa=2/9 leading to nOBa = 3
nBa= 12 leading to SBa = 2/nBa=1/6 leading to nOBa = 4
Each of these might lead to a possible structure.
The last case is the correct one for BaTiO3 as shown.
Each O has a Ti in the +z and –z directions plus four Ba
forming a square in the xy plane
The Each of these Ba sees 4 O in the xy plane, 4 in the xz
plane and 4 in the yz plane.
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 37
BaTiO3 structure (Perovskite)
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 38
How estimate charges?
We saw that even for a material as ionic as NaCl diatomic, the
dipole moment a net charge of +0.8 e on the Na and -0.8 e
on the Cl.
We need a method to estimate such charges in order to
calculate properties of materials.
First a bit more about units.
In QM calculations the unit of charge is the magnitude of the
charge on an electron and the unit of length is the bohr (a0)
Thus QM calculations of dipole moment are in units of ea0 which
we refer to as au. However the international standard for
quoting dipole moment is the Debye = 10-10 esu A
Where m(D) = 2.5418 m(au)
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 39
Fractional ionic character of diatomic molecules
Obtained from the experimental dipole moment in Debye, m(D), and bond
distance R(A) by dq = m(au)/R(a0) = C m(D)/R(A) where C=0.743470. Postive
that head of column is negative
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 40
Charge Equilibration
Charge Equilibration for Molecular
Dynamics Simulations;
A. K. Rappé and W. A. Goddard III;
J. Phys. Chem. 95, 3358 (1991)
First consider how the energy of an atom depends on
the net charge on the atom, E(Q)
Including terms through 2nd order leads to
(2) (3)
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 41
Charge dependence of the energy (eV) of an atom
E=0
E=-3.615
E=12.967
Cl Cl- Cl+
Q=0 Q=-1 Q=+1
Harmonic fit
= 8.291 = 9.352
Get minimum at Q=-0.887
Emin = -3.676
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 42
QEq parameters
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 43
Interpretation of J, the hardness
Define an atomic radius as
H 0.84 0.74
C 1.42 1.23
N 1.22 1.10
O 1.08 1.21
Si 2.20 2.35
S 1.60 1.63
Li 3.01 3.08
RA0 Re(A2)
Bond distance of
homonuclear
diatomic
Thus J is related to the coulomb energy of a charge the size of the
atom
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 44
The total energy of a molecular complex
Consider now a distribution of charges over the
atoms of a complex: QA, QB, etc
Letting JAB(R) = the Coulomb potential of unit
charges on the atoms, we can write
or
Taking the derivative with respect to charge leads to the
chemical potential, which is a function of the charges
The definition of equilibrium is for all chemical potentials to be
equal. This leads to
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 45
The QEq equations
Adding to the N-1 conditions
The condition that the total charged is fixed (say at 0)
leads to the condition
Leads to a set of N linear equations for the N variables QA.
AQ=X, where the NxN matrix A and the N dimensional vector A
are known. This is solved for the N unknowns, Q.
We place some conditions on this. The harmonic fit of charge to
the energy of an atom is assumed to be valid only for filling the
valence shell.
Thus we restrict Q(Cl) to lie between +7 and -1 and
Q(C) to be between +4 and -4
Similarly Q(H) is between +1 and -1
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 46
The QEq Coulomb potential law
We need now to choose a form for JAB(R)
A plausible form is JAB(R) = 14.4/R, which is valid when the
charge distributions for atom A and B do not overlap
Clearly this form as the problem that JAB(R) ∞ as R 0
In fact the overlap of the orbitals leads to shielding
The plot shows the
shielding for C atoms using
various Slater orbitals
And l = 0.5 Using RC=0.759a0
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 47
QEq results for alkali halides
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 48
QEq for Ala-His-Ala
Amber
charges in
parentheses
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 49
QEq for deoxy adenosine
Amber
charges in
parentheses
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 50
QEq for polymers
Nylon 66
PEEK
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 51
Perovskites
Perovskite (CaTiO3) first described in the 1830s
by the geologist Gustav Rose, who named it
after the famous Russian mineralogist Count Lev
Aleksevich von Perovski
crystal lattice appears cubic, but it is actually
orthorhombic in symmetry due to a slight
distortion of the structure.
Characteristic chemical formula of a perovskite
ceramic: ABO3,
A atom has +2 charge. 12 coordinate at the
corners of a cube.
B atom has +4 charge.
Octahedron of O ions on the faces of that cube
centered on a B ions at the center of the cube.
Together A and B form an FCC structure
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 52
Ferroelectrics The stability of the perovskite structure depends on
the relative ionic radii:
if the cations are too small for close packing with the
oxygens, they may displace slightly.
Since these ions carry electrical charges, such
displacements can result in a net electric dipole
moment (opposite charges separated by a small
distance).
The material is said to be a ferroelectric by analogy
with a ferromagnet which contains magnetic dipoles.
At high temperature, the small green B-cations can
"rattle around" in the larger holes between oxygen,
maintaining cubic symmetry.
A static displacement occurs when the structure is
cooled below the transition temperature.
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 53
c
a
Temperature 120oC 5oC -90oC
<111> polarized
rhombohedral
<110> polarized
orthorhombic
<100> polarized
tetragonal Non-polar
cubic
Different phases of BaTiO3
Six variants at room temperature
06.1~01.1a
c
Domains separated by domain walls
Non-polar cubic
above Tc <100> tetragonal
below Tc
O2-
Ba2+/Pb2+
Ti4+
Phases of BaTiO3
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 54
Nature of the phase transitions
1960 Cochran Soft Mode Theory(Displacive Model)
Displacive model
Assume that the atoms prefer to
distort toward a face or edge or
vertex of the octahedron
Increasing
Temperature
Temperature 120oC 5oC -90oC
<111> polarized
rhombohedral
<110> polarized
orthorhombic
<100> polarized
tetragonal Non-polar
cubic
Different phases of BaTiO3
face edge vertex center
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 55
Nature of the phase transitions
1960 Cochran Soft Mode Theory(Displacive Model)
Displacive model
Assume that the atoms prefer to
distort toward a face or edge or
vertex of the octahedron
Order-disorder 1966 Bersuker Eight Site Model
1968 Comes Order-Disorder Model (Diffuse X-ray Scattering)
Increasing
Temperature
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 56
Comparison to experiment
Displacive small latent heat
This agrees with experiment
R O: T= 183K, DS = 0.17±0.04 J/mol
O T: T= 278K, DS = 0.32±0.06 J/mol
T C: T= 393K, DS = 0.52±0.05 J/mol
Cubic Tetra.
Ortho. Rhomb.
Diffuse xray scattering
Expect some disorder,
agrees with experiment
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 57
Problem displacive model: EXAFS & Raman observations
57
(001)
(111)
d
α
EXAFS of Tetragonal Phase[1]
•Ti distorted from the center of oxygen octahedral in tetragonal
phase.
•The angle between the displacement vector and (111) is α= 11.7°.
Raman Spectroscopy of Cubic Phase[2]
A strong Raman spectrum in cubic phase is found in experiments.
But displacive model atoms at center of octahedron: no Raman
1. B. Ravel et al, Ferroelectrics, 206, 407 (1998)
2. A. M. Quittet et al, Solid State Comm., 12, 1053 (1973)
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 58
QM calculations
The ferroelectric and cubic phases in BaTiO3 ferroelectrics are also
antiferroelectric
Zhang QS, Cagin T, Goddard WA
Proc. Nat. Acad. Sci. USA, 103 (40): 14695-14700 (2006)
Even for the cubic phase, it is lower energy for the Ti
to distort toward the face of each octahedron.
How do we get cubic symmetry?
Combine 8 cells together into a 2x2x2 new unit cell,
each has displacement toward one of the 8 faces, but
they alternate in the x, y, and z directions to get an
overall cubic symmetry
Te
pe
ratu
re
x
CubicI-43m
TetragonalI4cm
RhombohedralR3m
OrthorhombicPmn21
y
z
o
FE AFE/
FE AFE/
FE AFE/
Px Py Pz
+ +
+ +
+ +
+ +
=
=
=
=
MacroscopicPolarization
Ti atom distortions
=
=
=
=
Microscopic Polarization
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 59
QM results explain EXAFS & Raman observations
59
(001)
(111)
d
α
EXAFS of Tetragonal Phase[1]
•Ti distorted from the center of oxygen octahedral in tetragonal
phase.
•The angle between the displacement vector and (111) is α= 11.7°.
PQEq with FE/AFE model gives α=5.63°
Raman Spectroscopy of Cubic Phase[2]
A strong Raman spectrum in cubic phase is found in experiments.
1. B. Ravel et al, Ferroelectrics, 206, 407 (1998)
2. A. M. Quittet et al, Solid State Comm., 12, 1053 (1973)
Model Inversion symmetry in
Cubic Phase
Raman Active
Displacive Yes No
FE/AFE No Yes
© copyright 2011 William A. Goddard III, all rights reserved Ch120a-Goddard-L25 60
stopped