Session #17 - Homework Solutions
Problem #1
You are operating an x-ray tube with a chromium (Cr) target by applying an
acceleration potential (V) of 60 kV. Draw a schematic of the x-ray spectrum
emitted by this tube; label on it three characteristic λs and give the numerical
value of two of these.
Solution
A characteristic x-ray spectrum of Cr will show λSWL , Kβ , K α and the continuous
spectrum or Bremsstrahlung. We may quantify λK and λSWL .
α
24Cr:
ν
Kα
=
3 R(Z - 1)2
4
=
3
4
× 1.097 × 107 (23)2 = 4.35 × 109 m-1
λK = 2.3 × 10-10 m
α
λSWL =
hc
1.24 × 10-6 m
=
= 2.07 × 10−11 m
4
eV
6 × 10
Problem #2
(a) An X-ray tube with a silver (Ag) target at a plate voltage of 66 kV. Calculate the
value of λSWL , the shortest wavelength.
(b) Sketch the emission spectrum (intensity vs. wavelength) of the Ag target in part
(a). On your sketch, indicate the relative positions of the K α , Kβ , L α , and Lβ
lines and λSWL . It is not necessary to calculate the λ values of the Kβ , L α , and
Lβ lines.
(c) In one or two sentences explain the origin of the continuous spectrum.
Solution
(a) λSWL =
hc
12400
=
= 0.188 Å
eV
66 × 103
(b) See sketch above in answer to problem #1. The L α and Lβ lines will appear to the
right of the analogous K lines (at higher values of λ ), the L α to the right of the
Lβ .
(c) Incident electrons are deflected by the negative charge of electrons in the target.
Any change in velocity (speed or direction or both) is an acceleration.
Accelerating a charge emits radiation. The deflected electrons’ acceleration is
NOT QUANTIZED. Thus, the spectrum is continuous.
Problem #3
Determine the wavelength of λK for molybdenum (Mo).
α
Solution
Mo: Z = 42; Kα → ni=2; nf =1; σ=1
ν
ν
ν
⎡1
1⎤
= R(Z - 1)2 ⎢
⎥
2
Kα
ni2 ⎦⎥
⎣⎢ nf
Kα
Kα
λK
α
1⎤
⎡1
⎡1⎤
= 1.097 × 107 ⎢ ⎥ (42 − 1)2 ⎢
⎥
2
⎣m ⎦
⎣1
22 ⎦
= 1.38 × 1010 m−1
=
1
νK
= 7.25 × 10-11 m
α
Problem #4
Identify the element giving rise to K α with λ = 2.51 × 10-10 m .
Solution
1
λK
=ν
α
(Z − 1) =
Kα
⎡1
1⎤
3
= R(Z - 1)2 ⎢
⎥ = R(Z - 1)2 ×
2
2
4
ni ⎥⎦
⎣⎢ nf
4
= 22
3× λ ×R
Z = 23 (Vanadium)
MIT OpenCourseWare
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3.091SC Introduction to Solid State Chemistry
Fall 2009
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