Useful formulas

• energy of a photon $E = hf = h \frac{c}{\lambda}$
• linear attenuation $I = I_0 \cdot e^{-\mu d}$
• mass attenuation coefficient $\mu_m = \frac{\mu}{\rho_m}$
• half-value layer $\frac{1}{2} = \textrm{e}^{-\mu d}$
• speed of light in vacuum $c \approx 3 \cdot 10^8 m \cdot s^{-1}$
• Planck's constant $h \approx 4.135 \cdot 10^{-15} eV \cdot Hz^{-1}$

A photon with wavelength $100~nm$ has energy of $12~eV$, what is the energy of a photon with wavelength $2~nm$? [600 eV]

The X-ray tube (rentgenka) generates X-rays from electrons with kinetic energy of $10~keV$. Compute the wavelength of the generated X-rays if we know, that only 1% of the energy is converted to radiation.

X-rays with intensity of $10~W/cm^2$ passes through a $10~cm$ segment of tissue with half-value layer of $2~cm$. What will be the intensity after the tissue passage? What is the tissue density in Hounsfield units (HU), consider the linear attenuation of water to be $\mu_w = 0.22~cm^{-1}$? What kind of tissue is it? For typical Hounsfield units ranges of tissues you can refer to wiki. [575, bone]

Consider a tissue block, that contains $30~cm$ width of tissue A followed by a block of $8~cm$ of tissue B. Let the half-value layers be A: $10~cm$, B: $3~cm$. What is the intensity on the tissue boundary A|B? And what is the residual intensity of the exiting radiation?

The linear attenuation coefficient for an unknown material shall be determined. The following data are obtained in a measurement made in “narrow beam geometry”. Material thickness: 2.5 cm. Measurement without material: 35 000 counts during 300 s (including background). Measurement with material: 25 700 counts during 300 s (including background). Measurement of the background (both measurements): 2350 counts during 600 s. Hint: The count rate r follows similar relationships like the activity; $r_d = r_0 e^{-\mu d}$ where $\mu$ is the linear attenuation coefficient, $r_0$ is the net count rate before attenuation and $r_d$ is the net count rate after the attenuation.

What fraction of 140 keV x‐rays incident upon a 0.5 mm thick lead apron will be transmitted? The mass absorption coefficient of lead for 140 keV x‐rays is: $μ_m = 2.0~cm_2 \cdot g^{‐1}$ and the density of lead is $\rho = 11.3~g \cdot cm^{-3}$.

Useful equations

• Exponential decay $ dN = -\lambda N dt$, where $N(0) = N_0$,
• the solution to which is $ N = N_0 \cdot \mathrm{e}^{-\lambda \cdot t}$, where $\lambda = \frac{\ln k}{t_k} $
• Decay constant and half-time $\lambda = \frac{\ln 2}{T_{1/2}}$
• Activity $ A = \frac{dN}{dt}$
• Avogadro number $N_A = 6.022 \cdot 10^{23} mol^{-1}$

By which factor does the mass of a radioactive isotope reduce in 3 years, if it reduces four times within a year? [$\frac{1}{64}=0.015625$]

The initial decay rate (the activity) of $1~g$ mass of isotope $_{88}^{226}Ra$ is $1~Ci \approx 3.7 \cdot 10^{10} Bq$. What is the half-life? The molar mass of this isotope is $226 \cdot 10^{-3} kg \cdot mol^{-1}$.

A sample of $_{18}F$ is measured at 10:40 and has an activity of 30 MBq.It is injected into a patient at 11:30. How much activity was injected? The half-life of $_{18}F$ is 109.8 min. [$21.9 \cdot 10^6 Bq$]

Let us consider the usual PET radiopharmaceutical with activity half-life of $130~[min]$ and half-life of elimination from the patient's body of $35~[min]$. The amount of $4\cdot 10^{-12}~[mol]$ of this pharmaceutical is produced $30~[min]$ before injection. What is the activity of the radiopharmaceutical at injection time? What is the activity after acquisition, which ends $15~[min]$ after injection?