**3. Statistical analysis**

Statistical analysis can generally be subdivided into three steps or sections. The first step involves obtaining the data and constructing the relevant variables. The second involves some basic statistical investigations, such as correlations between the variables or in the case of multiple sample comparisons, hypothesis testing. Often the first two steps may be sufficient, but in the event that the research requires an understanding of the exact relationships between the various variables, the quantitative effects one variable has on another, a further step in the form of a regression analysis becomes necessary. The discussion below outlines the steps encompassed in the statistical analysis.

#### **3.1 Data and the construction of variables**

The AFM data consist of individual 1 µm2 surface scans obtained from seven different RDX materials. Each scan produces one observation that contains the information on the actual surface variation, roughness of the area of the scan, R (RMS). The scans were taken from seven different materials, however, each material is comprised of particles, and differences between particles of the same material in terms of their surface characteristics are possible. Therefore, five particles of each material were selected at random and 5-6 scans of different regions of the surface of each particle were acquired.

This approach allows modeling of the surface heterogeneity between the seven materials in three possible ways: the average measure of observed surface roughness, the variability between the particles, and lastly the variation in the surface roughness across the surface of a particle. For further discussion see Bellitto & Melnik (2010) and Bellitto et al (2010).

Multiple scans per particle allow the construction of two measures of the surface characteristics of the particle, the particle average (Rpm),

$$R\_{pm} = \frac{1}{n} \sum\_{s=1}^{n} R\_{pms}$$

and the particle standard deviation (Spm),

$$S\_{p\text{nr}} = \left(\frac{1}{n-1} \sum\_{s=1}^{n} \left(R\_{p\text{ns}} - R\_{p\text{nr}}\right)^2\right)^{0.5} \text{.}$$

where the subscript p refers to the particle and m to the material. The subscript s refers to the individual scan and n is the number of scans per particle. Thus, Rpm represents the average value of all scans for that particle and Spm represents the standard deviation of roughness for that particle. These statistics are listed in Table 1.

Shown in Table 1 are the particle average (Rpm) and the particle standard deviation (Spm) values. These are limited to describing the individual particle characteristics but they can be used to construct variables describing the material characteristics. Rpm can simply be averaged to construct the average measure of observed surface roughness of the material (Rm),

Statistical analysis can generally be subdivided into three steps or sections. The first step involves obtaining the data and constructing the relevant variables. The second involves some basic statistical investigations, such as correlations between the variables or in the case of multiple sample comparisons, hypothesis testing. Often the first two steps may be sufficient, but in the event that the research requires an understanding of the exact relationships between the various variables, the quantitative effects one variable has on another, a further step in the form of a regression analysis becomes necessary. The

The AFM data consist of individual 1 µm2 surface scans obtained from seven different RDX materials. Each scan produces one observation that contains the information on the actual surface variation, roughness of the area of the scan, R (RMS). The scans were taken from seven different materials, however, each material is comprised of particles, and differences between particles of the same material in terms of their surface characteristics are possible. Therefore, five particles of each material were selected at random and 5-6 scans of different

This approach allows modeling of the surface heterogeneity between the seven materials in three possible ways: the average measure of observed surface roughness, the variability between the particles, and lastly the variation in the surface roughness across the surface of

Multiple scans per particle allow the construction of two measures of the surface

1 *<sup>n</sup> pm pms s R R n* 

1 <sup>1</sup> ( ) <sup>1</sup>

where the subscript p refers to the particle and m to the material. The subscript s refers to the individual scan and n is the number of scans per particle. Thus, Rpm represents the average value of all scans for that particle and Spm represents the standard deviation of

Shown in Table 1 are the particle average (Rpm) and the particle standard deviation (Spm) values. These are limited to describing the individual particle characteristics but they can be used to construct variables describing the material characteristics. Rpm can simply be averaged to construct the average measure of observed surface roughness of the material

*n pm pms pm s S RR n*

1

,

0.5 2

a particle. For further discussion see Bellitto & Melnik (2010) and Bellitto et al (2010).

discussion below outlines the steps encompassed in the statistical analysis.

**3.1 Data and the construction of variables** 

regions of the surface of each particle were acquired.

characteristics of the particle, the particle average (Rpm),

roughness for that particle. These statistics are listed in Table 1.

and the particle standard deviation (Spm),

(Rm),

**3. Statistical analysis** 



Predicting Macroscale Effects Through Nanoscale Features 183

Wilk test is employed to test for normality, to test if the underlying population can be assumed to be normally distributed. Table 3 presents the results of the test for the particle (Rpm) values for all seven samples (materials). The Shapiro-Wilk test fails to reject the null hypothesis of normality with p=0.10 for all of the samples. The null hypothesis assumes that the underlying population is normally distributed. If the test fails to reject the null hypothesis then that indicates that the assumption of the underlying population being normally distributed cannot be disproved. However, the Shapiro-Wilk test may at times be misleading and a visual examination of the data may be recommended. Interestingly enough, the F test used in ANOVA is relatively robust against the assumption of normality (Levine et al, 2010), but the

> Material W V z Prob>z I 0.921 0.929 -0.097 0.538 II 0.828 2.032 1.107 0.134 III 0.937 0.739 -0.379 0.648 IV 0.952 0.565 -0.682 0.752 V 0.946 0.634 -0.556 0.711 VI 0.948 0.617 -0.585 0.721 VII 0.897 1.218 0.273 0.392

The Levene test for homogeneity of variances is performed on the Rpm data and the groups are defined as the individual materials. The test compares multiple samples to determine if they are drawn from populations with equal variances. The value of the F statistic from the Levene test is 0.49367, while the critical value for the rejection of the null hypothesis of homogeneity of variances is 2.445259, the hypothesis of homogeneity of variances cannot be rejected by the test1. This enables us to perform the one way ANOVA on our Rpm data,

The ANOVA method enables us to check if there is enough statistical evidence to reject the hypothesis that the Rm values of the seven materials are statistically not different from each other. The ANOVA output is presented in Table 4 and it shows that the hypothesis of equal

The ANOVA results demonstrate that these materials differ substantially in terms of their

The focus of our research is to investigate a possible connection between the surface roughness and the shock sensitivity of the material. One simple way in which this connection can be examined is with the help of correlations, see Table 5. A clear negative correlation is observed between the measure of shock sensitivity (GPa) and the three

1 In the event the Levene test rejected the null hypothesis, we would not be able to proceed with ANOVA and would have to use weaker testing techniques such as the Kruskal-Wallis test. For further

assumption of homogeneity of variances is crucial to the validity of the test.

Table 3. Results of Shapiro-Wilk test

grouped by their corresponding materials.

discussion on the Levene test please see Levine et al (2010).

Rm values is rejected by the data.

particle average roughness.

$$R\_m = \frac{1}{N} \sum\_{p=1}^{N} R\_{pm} \quad \text{or} \quad \frac{1}{N}$$

where N represents the number of particles for the material. Similarly, the average measure of particle standard deviation can be constructed,

$$S\_m = \frac{1}{N} \sum\_{p=1}^{N} S\_{pm} \quad .$$

This simple average measure accounts for the average variability in the surface roughness across the particle surface.

At this point two measures of surface roughness and its variability have been constructed. One is the particle average level for the material (Rm) and the other is the average variation in surface roughness across the particle surface (Sm). To quantify the variation between particles the standard deviation of the distribution of Rpm, is introduced and expressed as

$$S\_{m\mu} = \left(\frac{1}{N-1} \sum\_{p=1}^{N} (R\_{pm} - R\_m)^2\right)^{0.5} \cdot \frac{1}{\cdot}$$

This measure enables us to account for the heterogeneity between the various particles of the material.

Cyclotetramethylene-tetranitramine (HMX), a major impurity within RDX, has been reported to be as high as 17% (Doherty & Watts, 2008). Table 2 provides a basic summary of the shock sensitivity of the seven materials and the mean % of HMX impurity. The HMX impurity is included since impurities can significantly alter the periodicity of a crystal and thus affect its surface roughness.


Table 2. Impurity (%HMX) and shock sensitivity (GPa) of the materials used in our study.

#### **3.2 Basic comparison of the materials**

The average surface roughnesses of the materials, shown in Table 1, are first compared using analysis of variance (ANOVA). ANOVA is a statistical method used to compare population characteristics for multiple populations. ANOVA relies on three important assumptions, randomness and independence, normality, and homogeneity of variances. As discussed previously, our process of particle selection met the assumption of randomness. The Shapiro-

1 *<sup>N</sup> m pm p R R N* ,

where N represents the number of particles for the material. Similarly, the average measure

1 *<sup>N</sup> m pm p S S N* .

This simple average measure accounts for the average variability in the surface roughness

At this point two measures of surface roughness and its variability have been constructed. One is the particle average level for the material (Rm) and the other is the average variation in surface roughness across the particle surface (Sm). To quantify the variation between particles the standard deviation of the distribution of Rpm, is introduced and expressed as

> 1 <sup>1</sup> ( ) <sup>1</sup> *N m pm m p S RR N*

This measure enables us to account for the heterogeneity between the various particles of

Cyclotetramethylene-tetranitramine (HMX), a major impurity within RDX, has been reported to be as high as 17% (Doherty & Watts, 2008). Table 2 provides a basic summary of the shock sensitivity of the seven materials and the mean % of HMX impurity. The HMX impurity is included since impurities can significantly alter the periodicity of a crystal and

**(Mean % HMX)** 

I 7.36 4.2 II 0.02 4.66 III 0.03 2.21 IV 8.55 3.86 V 0.82 5.24 VI 0.02 5.21 VII 0.19 5.06 Table 2. Impurity (%HMX) and shock sensitivity (GPa) of the materials used in our study.

The average surface roughnesses of the materials, shown in Table 1, are first compared using analysis of variance (ANOVA). ANOVA is a statistical method used to compare population characteristics for multiple populations. ANOVA relies on three important assumptions, randomness and independence, normality, and homogeneity of variances. As discussed previously, our process of particle selection met the assumption of randomness. The Shapiro-

 

.

**Material Impurity** 

of particle standard deviation can be constructed,

across the particle surface.

thus affect its surface roughness.

**3.2 Basic comparison of the materials** 

the material.

1

1

0.5 2

> **Sensitivity (GPa)**

Wilk test is employed to test for normality, to test if the underlying population can be assumed to be normally distributed. Table 3 presents the results of the test for the particle (Rpm) values for all seven samples (materials). The Shapiro-Wilk test fails to reject the null hypothesis of normality with p=0.10 for all of the samples. The null hypothesis assumes that the underlying population is normally distributed. If the test fails to reject the null hypothesis then that indicates that the assumption of the underlying population being normally distributed cannot be disproved. However, the Shapiro-Wilk test may at times be misleading and a visual examination of the data may be recommended. Interestingly enough, the F test used in ANOVA is relatively robust against the assumption of normality (Levine et al, 2010), but the assumption of homogeneity of variances is crucial to the validity of the test.


Table 3. Results of Shapiro-Wilk test

The Levene test for homogeneity of variances is performed on the Rpm data and the groups are defined as the individual materials. The test compares multiple samples to determine if they are drawn from populations with equal variances. The value of the F statistic from the Levene test is 0.49367, while the critical value for the rejection of the null hypothesis of homogeneity of variances is 2.445259, the hypothesis of homogeneity of variances cannot be rejected by the test1. This enables us to perform the one way ANOVA on our Rpm data, grouped by their corresponding materials.

The ANOVA method enables us to check if there is enough statistical evidence to reject the hypothesis that the Rm values of the seven materials are statistically not different from each other. The ANOVA output is presented in Table 4 and it shows that the hypothesis of equal Rm values is rejected by the data.

The ANOVA results demonstrate that these materials differ substantially in terms of their particle average roughness.

The focus of our research is to investigate a possible connection between the surface roughness and the shock sensitivity of the material. One simple way in which this connection can be examined is with the help of correlations, see Table 5. A clear negative correlation is observed between the measure of shock sensitivity (GPa) and the three

<sup>1</sup> In the event the Levene test rejected the null hypothesis, we would not be able to proceed with ANOVA and would have to use weaker testing techniques such as the Kruskal-Wallis test. For further discussion on the Levene test please see Levine et al (2010).


Table 4. ANOVA of Rpm


Table 5. Correlations and their statistical significance

measures of surface roughness of the material. This demonstrates that higher levels of surface heterogeneity are associated with lower levels of shock sensitivity.

Predicting Macroscale Effects Through Nanoscale Features 185

Furthermore, all of the correlation coefficients between the surface characteristics measures and the shock sensitivity are statistically significant at or above the 90% level of significance. The statistical significance of the coefficients is determined by their corresponding t values. The t statistics for the significance test of the correlation coefficients are reported in italics under their corresponding coefficient and those that are significant at or above the 90% level

> 2 2

*r* ,

1 *Nm t r*

where r is the correlation coefficient and Nm is the number of observations, which in this

Regression analysis is designed to establish numerical relationships between the regressors, the independent variables and the dependent variable. A multivariable regression enables one to interpret the regression coefficients as partial derivatives of the dependent variable with respect to the regressor. Various regression techniques exist, but given the simple setup of our problem the most basic model, the Ordinary Least Squares, can adequately serve the purpose. For further discussion of various regression techniques see Greene (2003). The greater concern is the fact that the data is limited to only seven materials and seven observations of shock sensitivity. Generally, regression techniques require satisfying the Central Limit Theorem requirements which demand a higher level of observations.

Unfortunately, the data is limited by the number of materials available in the study.

sensitivity of the material is correlated with the level of surface roughness.

Fig. 8. Plot of sensitivity of material versus average surface roughness (Rm)

However, the plot also demonstrates that there is potential for heteroscedasticity in the data. Although heteroscedasticity does not create a bias in determining the regression coefficients themselves, the statistical significance of those coefficients becomes essentially unknown as

0 5 10 15 20 **Average Surface Roughness (Rm)**

The relationship between the surface roughness characteristics of the RDX materials and their shock sensitivity is investigated. A simple plot (Figure 8) shows that the shock

are highlighted in bold font. The test statistic is obtained by the equation

case is limited to seven, the number of materials used in this study.

**3.3 Regression analysis** 

0 1

2 3 4

**Sensitivity (GPa)**

5 6 Furthermore, all of the correlation coefficients between the surface characteristics measures and the shock sensitivity are statistically significant at or above the 90% level of significance. The statistical significance of the coefficients is determined by their corresponding t values. The t statistics for the significance test of the correlation coefficients are reported in italics under their corresponding coefficient and those that are significant at or above the 90% level are highlighted in bold font. The test statistic is obtained by the equation

$$t = r \sqrt{\frac{N\_m - 2}{1 - r^2}} \quad \text{y}$$

where r is the correlation coefficient and Nm is the number of observations, which in this case is limited to seven, the number of materials used in this study.

#### **3.3 Regression analysis**

184 Atomic Force Microscopy – Imaging, Measuring and Manipulating Surfaces at the Atomic Scale

*Source of Variation SS df MS F P-value F crit*  Between Groups 705.2395 6 117.5399 3.964342 0.005398 2.445259

%HMX Sensitivity Rm Sm

Within Groups 830.1801 28 29.64929

Sensitivity -0.181 -0.411

Table 5. Correlations and their statistical significance

Rm 0.448 **-0.694**  1.120 *-2.155* 

Sm 0.284 **-0.896** 0.452 0.662 *-4.518* 1.133

surface heterogeneity are associated with lower levels of shock sensitivity.

Sm -0.170 **-0.706 0.656** 0.414 -0.386 *-2.230 1.945* 1.016

measures of surface roughness of the material. This demonstrates that higher levels of

Total 1535.42 34

Table 4. ANOVA of Rpm

*Groups Count Sum Average Variance*  I 5 91.352 18.2704 35.15459 II 5 42.12567 8.425133 37.45042 III 5 80.20038 16.04008 55.60776 IV 5 52.823 10.5646 18.57295 V 5 26.60883 5.321767 10.01659 VI 5 50.33267 10.06653 22.88255 VII 5 30.7508 6.15016 27.86016

SUMMARY

ANOVA

Regression analysis is designed to establish numerical relationships between the regressors, the independent variables and the dependent variable. A multivariable regression enables one to interpret the regression coefficients as partial derivatives of the dependent variable with respect to the regressor. Various regression techniques exist, but given the simple setup of our problem the most basic model, the Ordinary Least Squares, can adequately serve the purpose. For further discussion of various regression techniques see Greene (2003). The greater concern is the fact that the data is limited to only seven materials and seven observations of shock sensitivity. Generally, regression techniques require satisfying the Central Limit Theorem requirements which demand a higher level of observations. Unfortunately, the data is limited by the number of materials available in the study.

The relationship between the surface roughness characteristics of the RDX materials and their shock sensitivity is investigated. A simple plot (Figure 8) shows that the shock sensitivity of the material is correlated with the level of surface roughness.

Fig. 8. Plot of sensitivity of material versus average surface roughness (Rm)

However, the plot also demonstrates that there is potential for heteroscedasticity in the data. Although heteroscedasticity does not create a bias in determining the regression coefficients themselves, the statistical significance of those coefficients becomes essentially unknown as

Predicting Macroscale Effects Through Nanoscale Features 187

In Specification IV these three measures of surface characteristics are combined into one model. The test for heteroscedasticity failed to reject the hypothesis of homoscedasticity. Thus, the estimation is estimated without the HC3 technique. The overall explanatory power of the model improves substantially with the adjusted R-squared rising to 0.915. The coefficient on Rm is statistically significant at only 67% and the one on Sm is statistically significant at 84%, but the coefficient on Sm remains statistically significant at 98%. This specification includes multiple measures of surface characteristics that may in tern be correlated with each other. Although this has already been examined in the correlation table and no meaningful correlation was observed, this is verified with a computation of the variance inflation factor (VIF), see Table 7. The VIF computation confirms what the

correlation table suggested, no multicollinearity problems in specification IV.

Specification

 

Dependent Variable = Sensitivity

Rm -0.15 -0.04 1.07 1.14

Sm -0.29 -0.23 4.52 5.08

Sm -0.55 -0.23 2.23 1.87

Ind. Vars. I II III IV V

%HMX -0.05 0.41

Constant 6.00 6.02 7.26 7.31 4.47 5.31 14.34 5.41 14.38 8.39

R-sq 0.48 0.80 0.50 0.96 0.03

Adj. R-sq 0.08 0.76 0.40 0.92 -0.16

Variable VIF 1/VIF Rm 1.89 0.530049 Sm 1.81 0.552178 Sm 1.3 0.771688

Specification IV can be written as a simple equation where each coefficient can be

Table 6. OLS Regression with Heteroscedasticity correction

Table 7. Computation of Variance Inflation Factor.

interpreted as a partial derivative:

the standard errors become incorrectly computed by the basic OLS technique. To test for the presence of heteroscedasticity the Breusch-Pagan / Cook-Weisberg test is employed with the test statistics distributed as 2 with the degrees of freedom equal to the number of regressors. The Breusch-Pagan / Cook-Weisberg test for heteroscedasticity indicates 2(1)=3.71, which corresponds to a probability > 2 = 0.0542. Thus, with a p value of less than 0.05 the test fails to reject the hypothesis of no heteroscedasticty, however with a p value near 0.1, the hypothesis of no heteroscedasticty is rejected. In an effort to be conservative in the analysis a value of p=0.1 is selected. The problem is further amplified by the fact that the sample size is very small (only seven observations of shock sensitivity), which reduces the robustness of the White/Huber estimator, a commonly used method for heteroscedasticity correction. As a result, the HCCM estimator known as HC3 is employed, this estimation technique was discussed by MacKinnon and White (1985), and was later shown to perform better than its alternatives in small samples, see Long and Ervin (2000).

The test shows that the relationship between shock sensitivity and Sm does not exhibit any heteroscedasticity. The Breusch-Pagan / Cook-Weisberg test provides 2(1)=1.73, which corresponds to probability > 2 = 0.189. The test also finds no heteroscedasticity issues in the relationship between shock sensitivity and Sm

The regression analysis in essence plots a best fit line through the data plot, a line that minimizes the sum of squares in the differences between the predicted line Y values and the actual Y values. Table 6 presents the regression output for several specifications. In all of the reported specifications the dependent variable is the level of sensitivity (GPa). The coefficients are reported along with their corresponding t values, included below the coefficient.

Specification I simply examines the impact Rm has on the level of sensitivity of the material. This specification is equivalent to simply plotting the best fit line through the dataset in Figure 1 and is estimated using the HC3 method for the computation of errors. The model can be summarized by the following equation:

$$Sensitivity = 5.998 - 0.153 R\_m$$

However, the statistical validity of that equation is limited. First, the goodness of fit is low. As measured by the adjusted R-squared, the model explains only about 8% of volatility in the level of sensitivity. Secondly, the coefficient on Rm is statistically significant at only 67%. All the subsequent specifications are estimated without the HC3 technique, as no evidence of heteroscedasticity was found (see the discussion above).

Specification II models Sensitivity as a function of Sm. The level of statistical significance increases substantially. The coefficient on Sm is statistically significant at 99%, suggesting that the level of variation in surface roughness on the surface of the particle has a statistically significant impact on the level of shock sensitivity of the material. Furthermore, the overall goodness of fit of this specification has also improved. The adjusted R-squared suggests that model in Specification II explains over 76% of volatility in sensitivity.

Specification III examines the relationship between Sm and sensitivity. The coefficient on Sm is statistically significant at 92%. The overall fit of the model is also weaker with the adjusted R-square being at 0.398.

the standard errors become incorrectly computed by the basic OLS technique. To test for the presence of heteroscedasticity the Breusch-Pagan / Cook-Weisberg test is employed with the test statistics distributed as 2 with the degrees of freedom equal to the number of regressors. The Breusch-Pagan / Cook-Weisberg test for heteroscedasticity indicates 2(1)=3.71, which corresponds to a probability > 2 = 0.0542. Thus, with a p value of less than 0.05 the test fails to reject the hypothesis of no heteroscedasticty, however with a p value near 0.1, the hypothesis of no heteroscedasticty is rejected. In an effort to be conservative in the analysis a value of p=0.1 is selected. The problem is further amplified by the fact that the sample size is very small (only seven observations of shock sensitivity), which reduces the robustness of the White/Huber estimator, a commonly used method for heteroscedasticity correction. As a result, the HCCM estimator known as HC3 is employed, this estimation technique was discussed by MacKinnon and White (1985), and was later shown to perform better than its alternatives in small samples, see Long and Ervin (2000).

The test shows that the relationship between shock sensitivity and Sm does not exhibit any heteroscedasticity. The Breusch-Pagan / Cook-Weisberg test provides 2(1)=1.73, which corresponds to probability > 2 = 0.189. The test also finds no heteroscedasticity issues in the

The regression analysis in essence plots a best fit line through the data plot, a line that minimizes the sum of squares in the differences between the predicted line Y values and the actual Y values. Table 6 presents the regression output for several specifications. In all of the reported specifications the dependent variable is the level of sensitivity (GPa). The coefficients

Specification I simply examines the impact Rm has on the level of sensitivity of the material. This specification is equivalent to simply plotting the best fit line through the dataset in Figure 1 and is estimated using the HC3 method for the computation of errors. The model

*Sensitivity* 5.998 0.153*Rm*

However, the statistical validity of that equation is limited. First, the goodness of fit is low. As measured by the adjusted R-squared, the model explains only about 8% of volatility in the level of sensitivity. Secondly, the coefficient on Rm is statistically significant at only 67%. All the subsequent specifications are estimated without the HC3 technique, as no evidence

Specification II models Sensitivity as a function of Sm. The level of statistical significance increases substantially. The coefficient on Sm is statistically significant at 99%, suggesting that the level of variation in surface roughness on the surface of the particle has a statistically significant impact on the level of shock sensitivity of the material. Furthermore, the overall goodness of fit of this specification has also improved. The adjusted R-squared

Specification III examines the relationship between Sm and sensitivity. The coefficient on Sm is statistically significant at 92%. The overall fit of the model is also weaker with the

suggests that model in Specification II explains over 76% of volatility in sensitivity.

are reported along with their corresponding t values, included below the coefficient.

relationship between shock sensitivity and Sm

can be summarized by the following equation:

adjusted R-square being at 0.398.

of heteroscedasticity was found (see the discussion above).

In Specification IV these three measures of surface characteristics are combined into one model. The test for heteroscedasticity failed to reject the hypothesis of homoscedasticity. Thus, the estimation is estimated without the HC3 technique. The overall explanatory power of the model improves substantially with the adjusted R-squared rising to 0.915. The coefficient on Rm is statistically significant at only 67% and the one on Sm is statistically significant at 84%, but the coefficient on Sm remains statistically significant at 98%. This specification includes multiple measures of surface characteristics that may in tern be correlated with each other. Although this has already been examined in the correlation table and no meaningful correlation was observed, this is verified with a computation of the variance inflation factor (VIF), see Table 7. The VIF computation confirms what the correlation table suggested, no multicollinearity problems in specification IV.




Table 7. Computation of Variance Inflation Factor.

Specification IV can be written as a simple equation where each coefficient can be interpreted as a partial derivative:

*Sensitivity* 7.308 0.042 0.225 0.233 *RSS mmm*

**9** 

*1USA 2China* 

**AFM Application in III-Nitride** 

Z. Chen1, L.W. Su2, J.Y. Shi2, X.L. Wang2, C.L. Tang2 and P. Gao2

The nitride family is an exciting material system for optoelectronics industry. Indium nitride (InN), gallium nitride (GaN) and aluminium nitride are all direct bandgap materials, and their energy gaps cover a spectral range from infrared (IR) to deep ultraviolet (UV). This means that by using binary and ternary alloys of these compounds, emission at any visible

Atomic force microscopy (AFM) is a powerful tool to study the III-nitride surface morphology, crystal growth evolution and devices characteristics. In this chapter, we

Typical surface morphologies of GaN materials characterized by AFM are presented in §1. In additional, three types of threading dislocations, including edge, screw and mixed threading dislocation, are studied by AFM in this section. In §2, V-shape defects and other features in InN, InGaN film and InGaN/GaN multiple quantum wells are summarized. It is not easy to grow high quality AlN and AlGaN films, materials for UV light emitting diode and high electron mobility transistor, which usually have high dislocation density and three dimensional growth mode. Growth condition optimization of high quality, crack-free and smooth Al(Ga)N, with the assistance of AFM, is reviewed in §3. Applications of AFM in GaN based devices are discussed in §4, including the patterned sapphire substrates, as-

In recent years, the III-nitride-based alloy system has attracted special attention since highbrightness blue, green and white light-emitting diodes (LEDs), and blue laser diodes (LDs)

Good topography and crystal quality of GaN films are some of the key factors to improve devices performance. AFM helps researchers to observe the surface morphology of GaN, to study dislocations in GaN film, and to further understand and optimize the growth

provide an overview of AFM application in AlInGaN based materials and devices.

grown LED surface, backside polished surface, ITO surface investigation.

**1. Introduction** 

wavelength should be achievable.

**2. AFM study of GaN** 

condition of the GaN.

became commercially available.

*1Golden Sand River California Corporation, Palo Alto, California* 

**Materials and Devices** 

*2Lattice Power Corporation, Jiangxi* 

The last specification examines the relationship between the level of impurity (%HMX) and sensitivity. The regression analysis demonstrates that there is no statistically significant relationship between these two variables. For further analysis of shock sensitivity and its determinants in RDX materials see Bellitto and Melnik (2010) and Bellitto et al (2010).

#### **4. Conclusion**

Atomic force microscopy can be used to obtain a large number of data observations at the nanometric level. These data can statistically be used to investigate and establish quantitative relationships between various variables. This work demonstrates that surface characteristics data obtained from topographical scans can be used in investigating the relationship between the shock sensitivity of the materials at the macroscale and their surface roughness characteristics at the nanoscale. Statistical analysis can be used not only to show that there is a statistical relationship, but with the help of regression techniques, can precisely estimate any such relationships. As demonstrated in this chapter, the surface roughness variation on the surface of the particle has a substantial negative impact on the shock sensitivity in RDX materials. A one unit increase in the average standard deviation (Sm) reduces the shock sensitivity by 0.225 GPa. The statistical analysis can also be used to demonstrate absence of any meaningful relationship. For instance, the results demonstrate that the level of HMX impurity does not impact the shock sensitivity in the studied RDX materials.
