3/2003 Rev 1 I.2.10 – slide 1 of 36 Part I Review of Fundamentals Module 2Basic Physics and...
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Transcript of 3/2003 Rev 1 I.2.10 – slide 1 of 36 Part I Review of Fundamentals Module 2Basic Physics and...
3/2003 Rev 1 I.2.10 – slide 1 of 36
Part I Review of Fundamentals
Module 2 Basic Physics and MathematicsUsed in Radiation Protection
Session 10 Basic Mathematics
Session I.2.10
IAEA Post Graduate Educational CourseRadiation Protection and Safety of Radiation Sources
3/2003 Rev 1 I.2.10 – slide 2 of 36
Introduction
Basic mathematics needed to perform health physics calculations will be reviewed
Students will learn about differentiation and integration; exponential and natural logarithmic functions; properties of logs and exponents; properties of differentials and integrals; and work example health-physics related problems
3/2003 Rev 1 I.2.10 – slide 3 of 36
Content
Concepts of differentiation and integration Exponential and natural logarithmic
functions Properties of logs and exponents Properties of differentials and integrals Solve sample health-physics related
problems
3/2003 Rev 1 I.2.10 – slide 4 of 36
Overview
Basic health-physics related mathematics will be discussed
Health physics-related sample problems will be worked to illustrate use of the mathematical principles discussed
3/2003 Rev 1 I.2.10 – slide 5 of 36
Definition of the Derivative
y = f(x)
f (x) is the derivative of f(x)
f (x) is also called the differential of y with respect to x
f (x) is defined as:
x 0 f (x) = = lim
dydx x
f(x + x) – f(x)
3/2003 Rev 1 I.2.10 – slide 6 of 36
Constant Rule for Differentiation
ddx (c) = 0 where c is a constant
ddx (cu) = c
dudx
3/2003 Rev 1 I.2.10 – slide 7 of 36
Sum and Difference Rule and Power Rule for Differentiation
ddx (u v) = du
dxdvdx
ddx (xn) = nxn-1
3/2003 Rev 1 I.2.10 – slide 8 of 36
Product and Quotient Rulesfor Differentiation
ddx (uv) = u + v
dvdx
dudx
ddx =
uv
dvdx
dudx
uv -
v2
3/2003 Rev 1 I.2.10 – slide 9 of 36
Chain and Power Rules for Differentiation
dydx =
dydu
dudx
dydx un = nun-1 du
dx
3/2003 Rev 1 I.2.10 – slide 10 of 36
Definition of the Anti-Derivative
A function F(x) is called an anti-derivative of a function f(x) if for every x in the domain of f:
F(x) = f(x)
3/2003 Rev 1 I.2.10 – slide 11 of 36
Definition of Integral Notationfor the Anti-Derivative
The notation for the anti-derivative (called the integral) of f(x):
f(x) dx = F(x) + C
where C is an arbitrary constant
F(x) is the anti-derivative of f(x)
That is, F (x) = f(x) for all x in the domain of f(x)
3/2003 Rev 1 I.2.10 – slide 12 of 36
Inverse Relationship Between Differentiation and Integration
f (x) dx = f(x) + C
ddx
f(x) dx = f(x)
3/2003 Rev 1 I.2.10 – slide 13 of 36
Basic Integration Rules
k dx = kx + C, where k is a constant
k f(x) dx = k f(x) dx
3/2003 Rev 1 I.2.10 – slide 14 of 36
[ f(x) g(x) ] dx = f(x) dx g(x) dx
Basic Integration Rules
3/2003 Rev 1 I.2.10 – slide 15 of 36
xn dx =
Basic Integration Rules
xn+1
n+1
3/2003 Rev 1 I.2.10 – slide 16 of 36
a
b
f(x) dx = F(b) - F(a)
Basic Integration Rules
3/2003 Rev 1 I.2.10 – slide 17 of 36
Definition of An Exponential Function
If a > 0 and a 1, then the exponential function with base “a” is given by
Y = ax
3/2003 Rev 1 I.2.10 – slide 18 of 36
Properties of Exponents
a0 = 1
(ab)x = axbx
axay = ax+y
(ax)y = axy
ax
ay = ax-y
1axa-x =
ab
x= ax
bx
3/2003 Rev 1 I.2.10 – slide 19 of 36
Natural Exponential Function
e = lim (1 + x)x 0
1x
Let y = ex
where “e” is the base of the natural logarithms (e = 2.71828. . .)
3/2003 Rev 1 I.2.10 – slide 20 of 36
Definition of the Natural Logarithmic Function
notation: ln(x) = loge(x)
ln(x) = b if and only if eb = x
ln(ex) = x and eln(x) = x
ex and ln(x) are inverse functions of each other
3/2003 Rev 1 I.2.10 – slide 21 of 36
Properties of Exponentialsand Natural Logarithms
ln(1) = 0
ln(e) = 1
ln(e-1) = -1
ln(2) 0.693
e0 = 1
e1 = e
eln(2) = e0.693 = 2
1ee-1 =
3/2003 Rev 1 I.2.10 – slide 22 of 36
Additional Propertiesof Natural Logarithms
ln(xy) = ln(x) + ln(y)
xyln( ) = ln(x) – ln(y)
ln(xy) = y ln(x)
3/2003 Rev 1 I.2.10 – slide 23 of 36
Derivative of theNatural Logarithmic Function
ddx
ln(x) = 1x
ddx
ln(u) = 1u
dudx
3/2003 Rev 1 I.2.10 – slide 24 of 36
Log Rule for Integration
( )( )dx = ln(u) + C
1x
dudx
1u
( )dx = ln(x) + C
where C is a constant
3/2003 Rev 1 I.2.10 – slide 25 of 36
Sample Problem No. 1
Solve (by integration) the basic differential equation for radioactive decay
= -N
where N is the number of radioactive atoms of a given radionuclide present at time t
dNdt
3/2003 Rev 1 I.2.10 – slide 26 of 36
is the radioactive decay constant, in units of sec-1
t is the elapsed decay time in seconds
Assume that the initial number of radioactive atoms at t = 0 is N0
Sample Problem No. 1
3/2003 Rev 1 I.2.10 – slide 27 of 36
( ) = - dt
Solution toSample Problem No. 1
dNdt = -N
dN = - Ndt
dN N = - dt
dN N
3/2003 Rev 1 I.2.10 – slide 28 of 36
ln(N) = -t + C
let C = ln(N0)
ln(N) = - t + ln(N0)
ln(N) - ln(N0) = - t
Solution toSample Problem No. 1
3/2003 Rev 1 I.2.10 – slide 29 of 36
ln ( ) = - t
Solution toSample Problem No. 1
NNo
NNo
NNo
N(t) = N0 e- t
= e- t
eln( ) = e(-t )
3/2003 Rev 1 I.2.10 – slide 30 of 36
Derive the rule of thumb:
Sample Problem No. 2
where A is the remaining activity of any radionuclide after an elapsed time of “n” half-lives and A0 is the initial activity at time t = 0
AAo
=12
n
3/2003 Rev 1 I.2.10 – slide 31 of 36
Recall from the previous problem that
N(t) = N0 e- t
Multiply both sides of the equation by
N(t) = N0 e- t
Solution toSample Problem No. 2
3/2003 Rev 1 I.2.10 – slide 32 of 36
Now recall that activity is simply A = N, so that the previous equation (which was in terms of radioactive atoms) can be written in terms of activity, as:
A = A0 e-t
Solution toSample Problem No. 2
3/2003 Rev 1 I.2.10 – slide 33 of 36
Solve the equation for an elapsed decay time “t” equal to “n” half-lives where T½ is the half-life
A = A0 e-t and recall =
Solution toSample Problem No. 2
ln(2)T½
AAo
= e-ln(2)
T½ nT½
3/2003 Rev 1 I.2.10 – slide 34 of 36
= e ln(2 )
= 2-n =
Solution toSample Problem No. 2
AAo
= e-nln(2)
-n
12
n
3/2003 Rev 1 I.2.10 – slide 35 of 36
Summary
Basic mathematics needed to perform health physics calculations was reviewed
Students learned about differentiation and integration; exponential and natural logarithmic functions; properties of logs and exponents; properties of differentials and integrals; and worked example health-physics related problems
3/2003 Rev 1 I.2.10 – slide 36 of 36
Where to Get More Information
Cember, H., Johnson, T. E., Introduction to Health Physics, 4th Edition, McGraw-Hill, New York (2008)
Martin, A., Harbison, S. A., Beach, K., Cole, P., An Introduction to Radiation Protection, 6th Edition, Hodder Arnold, London (2012)
Jelley, N. A., Fundamentals of Nuclear Physics, Cambridge University Press, Cambridge (1990)
Firestone, R.B., Baglin, C.M., Frank-Chu, S.Y., Eds., Table of Isotopes (8th Edition, 1999 update), Wiley, New York (1999)