| Center-of-mass energy in the symmetric collider | | Center-of-mass energy in the symmetric collider |
| | | |
| Each electron has total energy $ E = N m_e c^2$. | | Each electron has total energy $ E = N m_e c^2$. |
| When two identical particles collide head‑on with opposite momenta, the total energy in the center‑of‑mass (CM) system is simply the sum of their total energies: | | When two identical particles collide head‑on with opposite momenta, the total energy in the center‑of‑mass (CM) system is simply the sum of their total energies: |
| | | |
| $E_{\text{CM}} = 2E = 2N m_e c^2$ | | $E_{\text{CM}} = 2E = 2N m_e c^2$ |
| | | |
| This is the energy available for the interaction. | | This is the energy available for the interaction. |
| | | |
| Center-of-mass energy in the fixed target | | Center-of-mass energy in the fixed target |
| | | |
| Now we have an incident electron with total energy$ E_{\text{lab}}$ and a target electron at rest (energy $m_e c^2)$ | | Now we have an incident electron with total energy$ E_{\text{lab}}$ and a target electron at rest (energy $m_e c^2)$ |
| The square of the center‑of‑mass energy for this system is calculated using the relativistic invariant s | | The square of the center‑of‑mass energy for this system is calculated using the relativistic invariant s |
| | | |
| $s = (E_{\text{lab}} + m_e c^2)^2 - (p_{\text{lab}} c)^2$ | | $s = (E_{\text{lab}} + m_e c^2)^2 - (p_{\text{lab}} c)^2$ |
| | | |
| where$ p_{\text{lab}} $is the momentum of the incident electron. | | where$ p_{\text{lab}} $is the momentum of the incident electron. |
| | | |
| Using the relation | | Using the relation |
| $E_{\text{lab}}^2 - (p_{\text{lab}} c)^2 = (m_e c^2)^2$ | | $E_{\text{lab}}^2 - (p_{\text{lab}} c)^2 = (m_e c^2)^2$ |
| we expand: | | we expand: |
| | | |
| $s = E_{\text{lab}}^2 + 2E_{\text{lab}}m_e c^2 + (m_e c^2)^2 - (p_{\text{lab}} c)^2 | | $s = E_{\text{lab}}^2 + 2E_{\text{lab}}m_e c^2 + (m_e c^2)^2 - (p_{\text{lab}} c)^2 |
| = 2(m_e c^2)^2 + 2E_{\text{lab}}m_e c^2$ | | = 2(m_e c^2)^2 + 2E_{\text{lab}}m_e c^2$ |
| | | |
| The center‑of‑mass energy is E_{\text{CM}} = \sqrt{s}. | | The center‑of‑mass energy is E_{\text{CM}} = \sqrt{s}. |
| | | |
| For high energies $(E_{\text{lab}} \gg m_e c^2) $this approximates to: | | For high energies $(E_{\text{lab}} \gg m_e c^2) $this approximates to: |
| | | |
| $E_{\text{CM}} \approx \sqrt{2m_e c^2 E_{\text{lab}}}$ | | $E_{\text{CM}} \approx \sqrt{2m_e c^2 E_{\text{lab}}}$ |
| | | |
| Equating the center‑of‑mass energies | | Equating the center‑of‑mass energies |
| | | |
| We want the available energy in the fixed target to be the same as in the symmetric collider: | | We want the available energy in the fixed target to be the same as in the symmetric collider: |
| | | |
| $\sqrt{2m_e c^2 E_{\text{lab}}} = 2N m_e c^2$ | | $\sqrt{2m_e c^2 E_{\text{lab}}} = 2N m_e c^2$ |
| | | |
| Squaring both sides: | | Squaring both sides: |
| | | |
| $2m_e c^2 E_{\text{lab}} = 4 N^2 (m_e c^2)^2 | | $2m_e c^2 E_{\text{lab}} = 4 N^2 (m_e c^2)^2 |
| \quad\Rightarrow\quad | | \quad\Rightarrow\quad |
| E_{\text{lab}} = 2 N^2 m_e c^2$ | | E_{\text{lab}} = 2 N^2 m_e c^2$ |
| | | |
| The total energy that the incident electron must have is $E_{\text{lab}} = 2 N^2 m_e c^2$ | | The total energy that the incident electron must have is $E_{\text{lab}} = 2 N^2 m_e c^2$ |
| Since N = 1000, we get | | Since N = 1000, we get |
| | | |
| $E_{\text{lab}} = 2 \times (1000)^2\, m_e c^2 = 2 \times 10^6\, m_e c^2$ | | $E_{\text{lab}} = 2 \times (1000)^2\, m_e c^2 = 2 \times 10^6\, m_e c^2$ |
| | | |
| That is, the electron energy must be 2 million times its rest energy. | | That is, the electron energy must be 2 million times its rest energy. |
| | | |
| Comparing with the energy of each electron in the collider $(N m_e c^2 = 1000\, m_e c^2)$, the increase factor is: | | Comparing with the energy of each electron in the collider $(N m_e c^2 = 1000\, m_e c^2)$, the increase factor is: |
| | | |
| $\frac{E_{\text{lab}}}{N m_e c^2} = \frac{2 N^2}{N} = 2N = 2000$ | | $\frac{E_{\text{lab}}}{N m_e c^2} = \frac{2 N^2}{N} = 2N = 2000$ |
| | | |
| Therefore, an energy 2000 times greater than that of each electron in the colliding beams is needed. | | Therefore, an energy 2000 times greater than that of each electron in the colliding beams is needed. |
