Project 1 : Introduction to MSE 510

Projects

Data Analysis Using Example Diffusion Data for Aluminum Ions

in Aluminum Oxide

Student Name

MSE 510

Due 01/24/2012

ü Project Objectives

1. Familiarize yourself with projects in this course

2. Familiarize yourself with Mathematica

3. Encourage you to review chapter 1 in the textbook

ü Summary

Diffusion data is selected from a peer-reviewed journal article and imported, plotted, and

analyzed using Mathematica. The source of the data is the diffusion studies of Paladino and

Kingery A1E and the following Mathematica functions are utilized:

Æ The raw data is imported from a .csv file using "Import" and plotted using "ListPlot"

Æ A non-linear mathematical fit is performed using "NonlinearModelFit"

Æ "Manipulate" is used to show the effects of changing two relevant parameters on the

fit of the data

Æ Relevant parameters for diffusion are extracted from the data fit and compare to the

published data

Æ Statistical error of the non-linear fit is analyzed

Æ Integration and differentiation are perfomed on the non-linear fit equation

Relevant discussion and comments are included where appropriate.

Import, Plot, and Fit Data

Import the data to a list designated as "data" :

In[1]:=

data =

Import@"D:\\Bill\\Bsu\\Courses\\MSE 510 ElecOptDielProps\\Projects Exams Quizzes

@MSE 510D\\Projects\\2013 Projects\\Project 1 @Ch 1D\\Student

Projects\\AaronWoodard\\Aaron Woodard diffdata.csv"D

Out[1]= 992178., 1.1 × 10−10=, 92103., 4. × 10

−11=, 92093., 3.5 × 10−11=, 92073., 2.8 × 10

−11=,

92043., 1.9 × 10−11=, 92003., 1.6 × 10

−11=, 91993., 1.1 × 10−11=, 91943, 1.9 × 10

−12==

Paladino and Kingery measured the amount of aluminum that diffused across an interface joining two cylinders of

aluminum oxide. A plot of the experimental results is generated showing the diffusion coefficient as a function of

anneal temperature. The diffusion coefficent describes the rate of atomic motion in the material.

In[2]:=

dataplot = ListPlotAdata, Frame → True,

GridLines → Automatic, PlotStyle → 8Red, AbsolutePointSize@5D<,

PlotRange → 881900, 2200<, 80, 1.4 ∗ 10^−10<<, FrameLabel → 9"T HKL", "D Hcm2êsL"=,

PlotLabel → "Diffusion Rate vs. T For Al in Al2O3"E

Out[2]=

1900 1950 2000 2050 2100 2150 2200

0

2. µ 10-11

4. µ 10-11

6. µ 10-11

8. µ 10-11

1. µ 10-10

1.2 µ 10-10

1.4 µ 10-10

T HKL

DHc

m2

êsL

Diffusion Rate vs. T For Al in Al2O3

From the plot, it is apparent that the diffusion rate increases with increasing temperature. It is well known that the

equation for the diffusion coefficient has the form of the Arrhenius rate equation, and is given as

D = Do expI -EA

kTM

where Do is the pre-exponential consant, EA is the activation energy for diffusion, k is the Boltzmann constant, and T is

temperature @2D. A model of this form is now fitted to the data.

In[3]:=

PrintA"D = ", fitexp = NonlinearModelFitAdata, a ∗ ãbêx

, 8a, b<, xEEH∗Calculates the fit model and outputs the resulting equation∗L

D = FittedModelB 47.0619 ‰-58 346.2êx F

2 MSE 510 - Project 1 - Excellent Example.nb

In[4]:=

plotdf =

PlotAfitexp@xD, 8x, 1900, 2200<, Frame → True, GridLines → Automatic, PlotStyle → Blue,

PlotRange → 881900, 2200<, 80, 1.4 ∗ 10^−10<<, FrameLabel → 9"T HKL", "D Hcm2êsL"=E;

Show@dataplot, plotdf, PlotLabel → "Diffusion Rate vs. T For Al in Al2O3

Blue Line = Fit, Red Dots = Data"D

Out[5]=

1900 1950 2000 2050 2100 2150 2200

0

2. µ 10-11

4. µ 10-11

6. µ 10-11

8. µ 10-11

1. µ 10-10

1.2 µ 10-10

1.4 µ 10-10

T HKL

DHc

m2

êsL

Diffusion Rate vs. T For Al in Al2O3

Blue Line = Fit, Red Dots = Data

The exponential model appears to fit the data well, as expected based on the known diffusion coefficient equation. In

general, the fit appears to be better at higher temperatures and diffusion rates. There is less agreement of the fit with

the observed data at lower temperatures. The accuracy of the fit will be analyzed during the error analysis performed in

a subsequent section.

The constants in the diffusion coefficent equation (excluding the Boltzmann constant) are dependent on the material

properties and conditions under which diffusion takes place. For example, crystal structure, grain structure, atomic

bonding, and defect characteristics affect the value of the pre-exponential constant and activation energy for a particular

situation. Using the built-in "Manipulate" function, the effects of changes in these values on the diffusion rate can be

investigated.

MSE 510 - Project 1 - Excellent Example.nb 3

In[6]:=

Clear@d, g, h, TD H∗initialize variables;

g = pre−exponential, h = activation energy∗Lk = 8.617 ∗ 10^−5;H∗Boltzmann constant, eVêK∗L

d@g__, h_, T_, kD := g ∗ ã

−h

k∗T

H∗define the function for the diffusion coefficient equation∗LManipulateAShowAPlotAd@g, h, T, kD, 8T, 1900, 2200<,

Frame → True, GridLines → Automatic, PlotStyle → Blue,

PlotRange → 881900, 2200<, 80, 1.4 ∗ 10^−10<<, FrameLabel → 9"T HKL", "D Hcm2êsL"=,

PlotLabel → "Diffusion Rate vs. T For Al in Al2O3"E, dataplotE,

99g, 47, "Pre−Exponential Hcm2êsL"=, 0, 2000, 1=,

88h, 5.03, "Activation Energy HeVL"<, 4.0, 8.0, .01<E

Out[9]=

Pre-Exponential Hcm2êsL

47

Activation Energy HeVL

5.03

1900 1950 2000 2050 2100 2150 2200

0

2. µ 10-11

4. µ 10-11

6. µ 10-11

8. µ 10-11

1. µ 10-10

1.2 µ 10-10

1.4 µ 10-10

T HKL

DHc

m2

êsL

Diffusion Rate vs. T For Al in Al2O3

Relevant diffusion parameters can be extracted from the fit of the data. The resulting equation for the non-linear fit

gives the estimated values of the pre - exponential and activation energy for the material and conditions studied by

Paladino and Kingery. These values and their corresponding units are listed in the table below.

In[10]:=

GridA98"Parameter", "Symbol", "Value", "Unit"<,

8"Activation Energy", "EA", "5.03", "eV"<,

9"Pre−Exponential", "DO", "47.06", "cm2ês"==, Alignment → Left,

Frame → AllEH∗generates a custom table of values∗L

Out[10]=

Parameter Symbol Value Unit

Activation Energy EA 5.03 eV

Pre−Exponential DO 47.06 cm2ês

By comparison, Paladino and Kingery found values for the pre-exponential and activation energy of 28 and 4.94,

respectively. By examination, using the manipulation chart presented above, these values appear to also produce a

good fit for the data. In fact, by further manipulation of the chart, there is an infinite number of combinations that will

produce a line that fits the data. In order to determine the correct parameters, the data must be plotted as Ln D vs. 1/T.

This is the general convention by which diffusion data is plotted.

4 MSE 510 - Project 1 - Excellent Example.nb

By comparison, Paladino and Kingery found values for the pre-exponential and activation energy of 28 and 4.94,

respectively. By examination, using the manipulation chart presented above, these values appear to also produce a

good fit for the data. In fact, by further manipulation of the chart, there is an infinite number of combinations that will

produce a line that fits the data. In order to determine the correct parameters, the data must be plotted as Ln D vs. 1/T.

This is the general convention by which diffusion data is plotted.

Import the new data set for Ln D and 1/T :

In[11]:=

data2 =

Import@"D:\\Bill\\Bsu\\Courses\\MSE 510 ElecOptDielProps\\Projects Exams Quizzes

@MSE 510D\\Projects\\2013 Projects\\Project 1 @Ch 1D\\Student

Projects\\AaronWoodard\\Aaron Woodard diffdata2.csv"D

Out[11]= 880.459137, −22.9<, 80.475511, −23.9<, 80.477783, −24.1<, 80.482393, −24.3<,

80.489476, −24.7<, 80.499251, −24.9<, 80.501756, −25.2<, 80.514668, −27.<<

By taking the natural logarithm of both sides of the diffusion coefficient equation, the equation becomes

Ln D = Ln Do - EA/kT

which is in the form of a straight line, y = mx + b @3D. Therefore, the pre-exponential and activation energy can be

obtained by a linear fit of the Ln D vs. 1/T data plot. The slope of the fit line is -EA/k and the intercept is Ln Do. Plot

the new data and perform a linear fit:

In[12]:=

dataplot2 = ListPlotAdata2, Frame → True,

GridLines → Automatic, PlotStyle → 8Red, AbsolutePointSize@5D<,

PlotRange → Automatic, FrameLabel → 9"1000êT HK−1L", "Ln D Hcm

2êsL"=,

PlotLabel → "Diffusion Rate vs. T For Al in Al2O3"E;

Print@"D = ", fitline = LinearModelFit@data2, x, xDDplotfitline =

PlotAfitline@xD, 8x, .45, .52<, Frame → True, GridLines → Automatic, PlotStyle → Blue,

PlotRange → Automatic, FrameLabel → 9"1000êT HK−1L", "Ln D Hcm

2êsL"=E;

Show@dataplot2, plotfitline, PlotLabel → "Fit = Blue Line; Data = Red Points"D

D = FittedModelB 7.19797 - 65.2783 x F

Out[15]=

0.46 0.47 0.48 0.49 0.50 0.51

-27

-26

-25

-24

-23

1000êT HK-1L

Ln

DHc

m2

êsL

Fit = Blue Line; Data = Red Points

MSE 510 - Project 1 - Excellent Example.nb 5

In[16]:=

Increasing Temperature

Increasing diffusion

coefficient

Slope = -E

A

k

0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52

-27

-26

-25

-24

-23

1000êT HK-1L

Ln

DHc

m2

êsL

Fit = Blue Line; Data = Red Points

Out[16]=

Increasing Temperature

Increasing diffusion

coefficient

Slope = -E

A

k

0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52

-27

-26

-25

-24

-23

1000êT HK-1L

Ln

DHc

m2

êsL

Fit = Blue Line; Data = Red Points

Again it is seen that the linear fit agrees well with the data at higher temperatures, but not as well at relatively lower

temperatures. The data point corresponding to the lowest temperature could be an outlier; it appears that by omitting

this point the accuracy of the fit could be improved. This point also corresponds to the lowest rate of diffusion mea-

sured by Paladino and Kingery. It is possible that the measurement device used was more accurate for higher diffusion

rates.

The resulting equation for the linear fit gives the estimated values of the pre-exponential and activation energy as shown

in the table below.

In[17]:=

GridA98"Parameter", "Symbol", "Value", "Unit"<,

8"Activation Energy", "EA", 65.2783 ∗ 1000 ∗ k, "eV"<,

9"Pre−Exponential", "DO", ã7.19797

, "cm2ês"==, Alignment → Left, Frame → AllE

Out[17]=

Parameter Symbol Value Unit

Activation Energy EA 5.62503 eV

Pre−Exponential DO 1336.71 cm2ês

The values reported here are not in agreement with those reported by Paladino and Kingery. The difference could be

due to the method of data extraction from the paper--values plotted here were extracted visually from the data chart in

the paper and the error could be significant. Raw data values for the paper were not available.

In summary, the data has been imported and plotted in its raw form. A non-linear equation was fit to the data, but the fit

was not adequate to obtain estimates of relevant diffusion coefficient equation parameters. The data was then plotted in

an alternate format, consistent with scientific convention for diffusion data. The resulting linear fit provided estimates

for the pre-exponential and activation energy parameters. An error analysis of the linear fit is performed in the next

section.

6 MSE 510 - Project 1 - Excellent Example.nb

The values reported here are not in agreement with those reported by Paladino and Kingery. The difference could be

due to the method of data extraction from the paper--values plotted here were extracted visually from the data chart in

the paper and the error could be significant. Raw data values for the paper were not available.

In summary, the data has been imported and plotted in its raw form. A non-linear equation was fit to the data, but the fit

was not adequate to obtain estimates of relevant diffusion coefficient equation parameters. The data was then plotted in

an alternate format, consistent with scientific convention for diffusion data. The resulting linear fit provided estimates

for the pre-exponential and activation energy parameters. An error analysis of the linear fit is performed in the next

section.

Error Analysis

The accuracy of the linear model is now analyzed by examining the statistical parameters associated with the fitting

equation. Recall the linear model used to fit the Ln D vs. 1/T data :

In[18]:=

Print@"D = ", fitline = LinearModelFit@data2, x, xDD

D = FittedModelB 7.19797 - 65.2783 x F

ü Fit Parameters

In[19]:=

Print@"Model Parameter Table = ", fitline@"ParameterTable"DD

Model Parameter Table =

Estimate Standard Error t-Statistic P-Value

1 7.19797 3.7537 1.91757 0.103615

x -65.2783 7.69559 -8.48256 0.000146805

The Estimates represent the calculated values for the intercept HLn DoL and the acivation energy H - EA êkL.The Standard Error represents how precisely the estimated value of the parameter was computed and estimates the

standard deviation of the sample mean based on the population mean @4D.

The t Statistic is the ratio of the Estimate to the Standard Error. It is used to calculate the P-Value @5D.

The P-Value represents the probability of getting a t Statistic greater than the calculated value by chance. A very low

value indicates that the parameter estimate is significantly different from zero; a rule of thumb is that a P-Value below

0.05 is significant @5D.

Based on the model parameters, the linear fit gives a good estimate of the activation energy for diffusion, since the P-

Value is very small. The estimate for Ln Do may not be accurate, as the P-Value indicates that there is at least a 10%

chance of obtaining the estimate by chance.

ü Coefficients of Determination

In[20]:=

PrintA"R2

= ", fitline@"RSquared"DEPrintA"Adjusted R

2= ", fitline@"AdjustedRSquared"DE

R2

= 0.923031

Adjusted R2

= 0.910203

The Coefficient of Determination represents how well the model fits the data. The value varies between 0 and 1, with 1

being a perfect fit, meaning that the model can accurately describe every data point. The Adjusted Coefficient of

Determination also represents how well the model fits the data, but also takes into account the degrees of freedom.

Here, the high values indicate that the linear model is a relatively good fit for the data. This agrees with the theoretical

Arrhenius behavior of the diffusion coefficient with temperature, which indicates that Ln D vs. 1/T should be a linear

relationship. As noted earlier, the data point corresponding to the lowest anneal temperature could be the reason that

R^2 is not higher @6D.

MSE 510 - Project 1 - Excellent Example.nb 7

The Coefficient of Determination represents how well the model fits the data. The value varies between 0 and 1, with 1

being a perfect fit, meaning that the model can accurately describe every data point. The Adjusted Coefficient of

Determination also represents how well the model fits the data, but also takes into account the degrees of freedom.

Here, the high values indicate that the linear model is a relatively good fit for the data. This agrees with the theoretical

Arrhenius behavior of the diffusion coefficient with temperature, which indicates that Ln D vs. 1/T should be a linear

relationship. As noted earlier, the data point corresponding to the lowest anneal temperature could be the reason that

R^2 is not higher @6D.

ü Residuals

The residuals represent the difference between the value predicted by the model and the value of the corresponding

obeserved data point.

In[22]:=

Print@"Residuals = ", fitline@"FitResiduals"DDListPlot@fitline@"FitResiduals"D, PlotRange → Automatic, Filling → Axis,

Frame → True, PlotLabel → "Ln D vs. 1êT Linear Fit Residuals"D

Residuals = 8−0.126285, −0.0574176, −0.109105,

−0.0081723, 0.0541939, 0.492289, 0.355811, −0.601315<

Out[23]=

0 2 4 6 8

-0.6

-0.4

-0.2

0.0

0.2

0.4

Ln D vs. 1êT Linear Fit Residuals

From the residual plot, the values least well described by the model are the three data points that correspond to measure-

ments taken at the lowest anneal temperatures. This supports the theory that the measurement device used may not be

sensitive enough to detect these lower amounts of diffused particles. It is also possible that, as temperature increases, it

becomes the dominant mechanism controlling the diffusion rate. At lower temperatures, there could be unknown

factors controlling the rate of diffusion.

ü Confidence Intervals

A confidence interval represents a range where the true value of the estimated parameter lies according to a given

probability. The confidence intervals for the parameters estimated by the linear fit are presented below.

8 MSE 510 - Project 1 - Excellent Example.nb

In[24]:=

Print@"80% Confidence Interval = ",

fitline@"ParameterConfidenceIntervals", ConfidenceLevel → .8DDPrint@"85% Confidence Interval = ",

fitline@"ParameterConfidenceIntervals", ConfidenceLevel → .85DDPrint@"90% Confidence Interval = ",

fitline@"ParameterConfidenceIntervals", ConfidenceLevel → .90DDPrint@"95% Confidence Interval = ",

fitline@"ParameterConfidenceIntervals", ConfidenceLevel → .95DDPrint@"99% Confidence Interval = ",

fitline@"ParameterConfidenceIntervals", ConfidenceLevel → .99DDListPlot@Table@fitline@"ParameterConfidenceIntervals", ConfidenceLevel → pD,

8p, 8.8, .85, .9, .95, .99<<D, Frame → True, GridLines → Automatic,

PlotLabel → "Confidence Intervals: 80%, 85%, 90%, 95%, 99%"D

80% Confidence Interval = 881.79356, 12.6024<, 8−76.3581, −54.1985<<

85% Confidence Interval = 881.00372, 13.3922<, 8−77.9774, −52.5793<<

90% Confidence Interval = 88−0.0961403, 14.4921<, 8−80.2322, −50.3244<<

95% Confidence Interval = 88−1.987, 16.3829<, 8−84.1087, −46.4479<<

99% Confidence Interval = 88−6.71859, 21.1145<, 8−93.8091, −36.7475<<

Out[29]=

-80 -60 -40 -20 0

-50

-40

-30

-20

-10

0

10

20

Confidence Intervals: 80%, 85%, 90%, 95%, 99%

The confidence intervals at all levels are relatively large. The range of values in which the true values may lie would

have a significant effect on the calculation of Do and EA. Using the 95% confidence interval as an example, the high

and low true values of Do and EA are shown in the following table:

MSE 510 - Project 1 - Excellent Example.nb 9

In[30]:=

GridA98"Parameter", "High Value", "Low Value", "Unit"<,

8"EA", 84.1087 ∗ 1000 ∗ k, 46.4479 ∗ 1000 ∗ k, "eV"<,

9"DO", ã16.3829

, ã−1.987

, "cm2ês"==, Alignment → Left, Frame → AllE

Out[30]=

Parameter High Value Low Value Unit

EA 7.24765 4.00242 eV

DO 1.30318 × 107

0.137106 cm2ês

The high range of values indicates that the estimates of the parameters are poor. To obtain more accurate estimates,

more data points could be measured or a more accurate measurement device could be used. It is also possible that

there is an unknown factor controlling the rate of diffusion that influences the behavior of the diffusion coefficent in

response to a temperature change. In that case, the diffusion coefficient might not completely obey the Arrhenius

relationship.

Integration and Differentiation

Since the data plotted as Ln D vs. 1/T fit a straight line, differentiation of the fit line will be a constant. However, as

noted above, the resulting constant is a useful parameter for diffusion--the activation energy. The value is actually

-EA ê H1000*kL, so EA is found by multiplying by Boltzmann's constant and by 1000 (since T was plotted as 1000/T).

The activation energy found for the analyzed diffusion data was presented earlier, however its calculation by differentia-

tion is demonstrated here. Since the second derivative would be zero, it is not considered.

Calculate the activation energy by differentiation of the linear fit :

In[31]:=

Print@"The activation energy for diffusion is = ", D@fitline@xD, xD ∗ −1000 ∗ k, " eV."D

The activation energy for diffusion is = 5.62503 eV.

The derivative can also be plotted. In this case, it is simply a straight line at 5.625 eV, as expected.

10 MSE 510 - Project 1 - Excellent Example.nb

In[32]:=

Needs@"PlotLegends`"Ddfitline@x_D = fitline'@xD;

Plot@dfitline@xD ∗ −1000 ∗ k, 8x, 1900, 2200<, PlotRange → 881900, 2200<, 80, 10<<,

Frame → True, GridLines → Automatic, PlotStyle → Blue,

FrameLabel → 8"T HKL", "dDêdT HeVL"<, PlotLegend → 8"Activation Energy"<,

PlotLabel → "Derivative of Diffusion Rate vs. T For Al in Al2O3"D

General::obspkg : PlotLegends` is now obsolete. The legacy version being loaded may

conflict with current Mathematica functionality. See the Compatibility Guide for updating information.

Out[34]=

1900 1950 2000 2050 2100 2150 2200

0

2

4

6

8

10

T HKL

dD

êdT

HeV

L

Derivative of Diffusion Rate vs. T For Al in Al2O3

Activation Energy

There is no value in performing integration on the diffusion data, since it will not yield useful information. However,

the built-in Mathematica integration function is used for demonstration purposes.

In[35]:=

intfitline@x_D = Integrate@dfitline@xD, xDH∗define a function that equals the integral of the derivative of the fitline∗L

Out[35]=

−65.2783 x

MSE 510 - Project 1 - Excellent Example.nb 11

In[36]:=

PlotAintfitline@xD, 8x, .45, .52<, Frame → True, GridLines → Automatic,

PlotStyle → Blue, PlotRange → Automatic, PlotLegend → 8"Original Linear

Fit Line

HIntercept = 0L"<, PlotLabel → "Integration Example",

FrameLabel → 9"1000êT HK−1L", "Ln D Hcm

2êsL"=E

Out[36]=

0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52

-34

-33

-32

-31

-30

1000êT HK-1L

Ln

DHc

m2

êsL

IntegrationExample

Original Linear

Fit Line

HIntercept = 0L

Since the integral of the derivative of the fitline was calculated, the fitline is also the result. As expected, the integration

does not give the intercept; however the slope is the same as that of the linear fit line. This indicates that the integration

function was used properly.

Applications

This type of analysis of diffusion data is not limited to the diffusion of aluminum in aluminum oxide. It can be applied

to the diffusion of any type of atom in any material and will show whether a particular diffusion process can be character-

ized by an Arrhenius relationship to temperature. Specifically for aluminum diffusion in aluminum oxide, one example

application is in characterization of the oxidation rate of aluminum alloys. By performing this type of analysis, the ion

diffusion rate can be determined over different operating temperatures, which assists in the determination of the rate of

oxide growth. Aluminum oxide is typically a protective oxide layer, so the data can be used to determine both the

temperature and length of time at which the alloy should be annealed in order to achieve the desired oxide thickness, or

the rate at which the alloy will oxidize in service.

In order to further characterize the process, the diffusion of oxygen in aluminum oxide could be investigated. Based on

a comparison of the respective diffusion rates of oxygen and aluminum in the oxide, the ion which controls the overall

oxide formation rate can be determined. If this is known, the rate of formation of oxide can be further controlled by

inhibiting or enhancing the rate of diffusion of the rate limiting ion. Even further experiments to determine the mecha-

nism by which diffusion occurs can allow for extensive control over the oxide formation rate. For example, diffusion

through a single crystal of aluminum can be compared to that through polycrystalline aluminum to determine the effects

of the presence of grain boundaries, which can be conduits by which diffusion occurs.

In summary, the analysis presented here can be repeated across different material types and structures to characterize

the rate of formation of any type of alloy, ceramic, polymer or phase for a given situation.

12 MSE 510 - Project 1 - Excellent Example.nb

References

[1] Paladino, A.E.; Kingery, W.D. Aluminum ion Diffusion in Aluminum Oxide. J.Chem.Phys. Vol 37, 5. 1962.

[2] Kasap, S.O. Principles of Electronic Materials and Devices. 3rd Ed. McGraw Hill, 2006.

[3] Atkins, P.; de Paula, J. Physical Chemistry. 7th Ed. Oxford University Press, 2002.

[4] Press,W.H.;Flannery,B.P.;Teukolsky,S.A.;and Vetterling,W.T. Numerical Recipes in FORTRAN:The Art of

Scientific Computing. 2nd Ed. Cambridge University Press, 1992.

[5] Bivariate Scatterplot and Fitting. JMP7: Statististical Discovery. Computer Software. SAS Institute, 2007.

[6] Interpreting Regression Results, OriginLabs - Origin Software - http://www.originlab.com/.

MSE 510 - Project 1 - Excellent Example.nb 13