Pb-glass in E&M calorimeters has a radiation length of 1.2cm and a density of 5.

Question

Pb-glass in E&M calorimeters has a radiation length of 1.2cm and a density of 5.4g/cm^3. Assuming it is an equal mixture of 208/82 Pb, 28/14 Si, and 16/8 O, calculate its critical energy. What is the minimum size of a block of Pb glass that can be used to measure the energy of a 1 GeV electron? Explain how you would use a detector made out of blocks of the size calculated about to distinguish between electrons and pions. Pb-glass in E&M calorimeters has a radiation length of 1.2cm and a density of 5.4g/cm^3. Assuming it is an equal mixture of 208/82 Pb, 28/14 Si, and 16/8 O, calculate its critical energy. What is the minimum size of a block of Pb glass that can be used to measure the energy of a 1 GeV electron? Explain how you would use a detector made out of blocks of the size calculated about to distinguish between electrons and pions.

Explanation / Answer

PART 1:

Formulae to calculate critical energy for calorimeters is as follows:

q = qcal + qw

where qcal is the heat flow for the calorimeter and qw is the heat flow for the water.

Both of these individual heat flows can be related to the heat capacity and temperature change for the substance.

qcal = Ccal DT

qw = Cw DT

where Ccal is the heat capacity of the calorimeter and Cw is the heat capacity of the water. Because the water and calorimeter are in thermal equilibrium, they both have the same temperature and thus DT is the same for both. The consequence is that the heat capacity of the entire system (C) is the sum of the heat capacities for the individual components.

C = Ccal + Cw

Now using these formula,

1) Energy lost by the hot water:

q = m Cp T

q = (72.55 g) (4.184 J/g-1 °C-1) (24.3 °C)

q = 7376.24 J

2) Energy gained by the cold water:

q = m Cp T

q = (58.85 g) (4.184 J/g-1 °C-1) (24.9 °C)

q = 5818.54 J

3) The calorimeter got the rest:

7376.24 - 5818.54 = 1557.7 J

4) Heat capacity of the calorimeter:

1557.7 J / 24.9 °C = 62.558 J/°C (round off as you see fit)

PART 2:

Size of a block of Pb glass;

Minimum size of a block of Pb glass that can be used to measure the energy of a 1 GeV electron is

20 rad. length .

Also,

Effective area : 120x140 cm2

Dynamic range: 1.0 - 11.0 GeV/c

Energy resolution: ~6 % at 1 GeV/c

Pion rejection: ~100:1 at P > 1.5-2.0 GeV/c

Electron efficiency: > 98 % .

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