Essentials of Materials Science and Engineering, SI Edition
Essentials of Materials Science and Engineering, SI Edition
4th Edition
ISBN: 9781337672078
Author: ASKELAND, Donald R., WRIGHT, Wendelin J.
Publisher: Cengage Learning
Question
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Chapter 5, Problem 5.58P
Interpretation Introduction

Interpretation:

The concentration of Ga at a depth of 2.0 micrometer if the surface concentration of Ga si 1023atoms/cm3 needs to be calculated.

Concept introduction:

The derived mathematical expression from Fick's second law is as follows:

  cscxcsc0=erf(x2 Dt)..........(1)

Here, cs is the concentration of diffusion at the surface, c0 is the concentration of diffusion at x, c0 is the uniform concentration of diffused atoms in the material, cx is the concentration of atoms diffusing at x and D is diffusion coefficient.

Expert Solution & Answer
Check Mark

Answer to Problem 5.58P

The value of cx si 3.55×1023 atoms/cm3.

Explanation of Solution

Gallium (Ga) content at any depth x beneath the surface can be calculated using the expression derived from Fick's Second Law.

Mathematical expression of Fick's Second Law is as given below:

  ct=x(Dcx)

The derived mathematical expression from Fick's Second Law is

  cscxcsc0=erf(x2 Dt)..........(1)

The diffusion coefficient D is 7×1013cm2/s.

When time t=3600s(1hr)

Substitute the above values in Fick's equation (1) as given below.

  cscxcsc0=erf(x 2 Dt ) 10 23cx 10 23=erf( 2× 10 4 2 ( 7× 10 13 )( 3600 ) )=erf(1.99)

The value of the error (x2 Dt) is 1.99, the corresponding value of the (erfx2 Dt) from table (1) is 0.9953.

Error function values for Fick's second law
Argument of the error

  x2Dt

Values of the error function

  erfx2Dt

  0

  

  0
  0.10  0.1125
  0.20  0.2227
  0.30  0.3286
  0.40  0.4284
  0.50  0.5205
  0.60  0.6039
  0.70  0.6778
  0.80  0.7421
  0.90  0.7969
  1.00  0.8427
  1.50  0.9961
  2.00  0.9953

  (erfx 2 Dt )=0.9953 10 23cx 10 23=0.9953cx=4.7×1020 atoms/cm3

Therefore, the value of cx is 4.7×1020 atoms/cm3.

When time t=7200s(2 hr)

Substitute the above values in Fick's equation (1) as given below.

  cscxcsc0=erf(x 2 Dt ) 10 23cx 10 23=erf( 2× 10 4 2 ( 7× 10 13 )( 7200 ) )=erf(0.995)

The value of the error (x2 Dt) is 0.995, the corresponding value of the (erfx2 Dt) from table (1) is 0.8427.

  (erfx 2 Dt )=0.8427 10 23cx 10 23=0.8427cx=1.57×1023 atoms/cm3

Therefore, the value of cx is. 1.57×1023 atoms/cm3

When time t=10800s(3 hr)

Substitute the above values in Fick's equation (1) as given below.

  cscxcsc0=erf(x 2 Dt ) 10 23cx 10 23=erf( 2× 10 4 2 ( 7× 10 13 )( 10800 ) )=erf(0.663)

The value of the error (x2 Dt) is 0.663 , the corresponding value of the (erfx2 Dt) from table (1) is 0.645 (extrapolation of the value since 0.663 lies between 0.6 and 0.7 ).

  (erfx 2 Dt )=0.645 10 23cx 10 23=0.645cx=3.55×1023 atoms/cm3

Conclusion

Therefore, the value of cx is. 3.55×1023 atoms/cm3

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Chapter 5 Solutions

Essentials of Materials Science and Engineering, SI Edition

Ch. 5 - Prob. 5.11PCh. 5 - Prob. 5.12PCh. 5 - Prob. 5.13PCh. 5 - Prob. 5.14PCh. 5 - Prob. 5.15PCh. 5 - Prob. 5.16PCh. 5 - Prob. 5.17PCh. 5 - Prob. 5.18PCh. 5 - Prob. 5.19PCh. 5 - Prob. 5.20PCh. 5 - Prob. 5.21PCh. 5 - Prob. 5.22PCh. 5 - Prob. 5.23PCh. 5 - Prob. 5.24PCh. 5 - Prob. 5.25PCh. 5 - Prob. 5.26PCh. 5 - Prob. 5.27PCh. 5 - Prob. 5.28PCh. 5 - Prob. 5.29PCh. 5 - Prob. 5.30PCh. 5 - Prob. 5.31PCh. 5 - Prob. 5.32PCh. 5 - Prob. 5.33PCh. 5 - Prob. 5.34PCh. 5 - Prob. 5.35PCh. 5 - Prob. 5.36PCh. 5 - Prob. 5.37PCh. 5 - Prob. 5.38PCh. 5 - Prob. 5.39PCh. 5 - Prob. 5.40PCh. 5 - Prob. 5.41PCh. 5 - Prob. 5.42PCh. 5 - Prob. 5.43PCh. 5 - Prob. 5.44PCh. 5 - Prob. 5.45PCh. 5 - Prob. 5.46PCh. 5 - Prob. 5.47PCh. 5 - Prob. 5.48PCh. 5 - Prob. 5.49PCh. 5 - Prob. 5.50PCh. 5 - Prob. 5.51PCh. 5 - Prob. 5.52PCh. 5 - Prob. 5.53PCh. 5 - Prob. 5.54PCh. 5 - Prob. 5.55PCh. 5 - Prob. 5.56PCh. 5 - Prob. 5.57PCh. 5 - Prob. 5.58PCh. 5 - Prob. 5.59PCh. 5 - Prob. 5.60PCh. 5 - Prob. 5.61PCh. 5 - Prob. 5.62PCh. 5 - Prob. 5.63PCh. 5 - Prob. 5.64PCh. 5 - Prob. 5.65PCh. 5 - Prob. 5.66PCh. 5 - Prob. 5.67PCh. 5 - Prob. 5.68PCh. 5 - Prob. 5.69PCh. 5 - Prob. 5.70PCh. 5 - Prob. 5.71PCh. 5 - Prob. 5.72PCh. 5 - Prob. 5.73PCh. 5 - Prob. 5.74PCh. 5 - Prob. 5.75PCh. 5 - Prob. 5.76PCh. 5 - Prob. 5.77PCh. 5 - Prob. 5.78PCh. 5 - Prob. 5.79PCh. 5 - Prob. 5.80PCh. 5 - Prob. 5.81DPCh. 5 - Prob. 5.82DPCh. 5 - Prob. 5.83DPCh. 5 - Prob. 5.84DPCh. 5 - Prob. 5.85DPCh. 5 - Prob. 5.86CPCh. 5 - Prob. 5.87CPCh. 5 - Prob. 5.88CPCh. 5 - Prob. K5.1KP
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