```python
from sympy import *
```


```python
e1, p1, e2, p2, m = symbols("E_1 p_1 E_2 p_2 m",real=True, positive=True)
```

We work in 1-dimensional coordinates with $p = p_z$. We generally use $m = m_p = m_n$, in other words we neglect the mass difference between protons and neutrons. We consider a proton projectile and a nuclear target with $A$ nucleons.

We first compute the cms energy $\sqrt{s_{nn}}$ in the nucleon-nucleon system from the fixed target system.

In this system, we have these 4-vectors:
* projectile $(E, p)$
* target $(m, 0)$


```python
s2 = (e1 + m)**2 - (p1 + 0)**2; s2
```




$\displaystyle - p_{1}^{2} + \left(E_{1} + m\right)^{2}$




```python
s2 = s2.subs(e1, sqrt(p1 ** 2 + m ** 2)); s2
```




$\displaystyle - p_{1}^{2} + \left(m + \sqrt{m^{2} + p_{1}^{2}}\right)^{2}$




```python
s2.series(m, 0, 2)
```




$\displaystyle 2 m p_{1} + O\left(m^{2}\right)$




```python
s = sqrt((e1 + m)**2 - (p1 + 0)**2); s
```




$\displaystyle \sqrt{- p_{1}^{2} + \left(E_{1} + m\right)^{2}}$




```python
s = s.subs(e1, sqrt(p1 ** 2 + m ** 2)); s
```




$\displaystyle \sqrt{- p_{1}^{2} + \left(m + \sqrt{m^{2} + p_{1}^{2}}\right)^{2}}$




```python
s.series(m, 0, 2)
```




$\displaystyle \sqrt{2} \sqrt{m} \sqrt{p_{1}} + \frac{\sqrt{2} m^{\frac{3}{2}}}{2 \sqrt{p_{1}}} + O\left(m^{2}\right)$




```python
def sqrt_snn(energy):
    mass = (938.27 + 939.57) * 0.5e-9 # nucleon mass in PeV
    return sqrt(2 * mass * energy) * 1e3 # sqrt(sNN) in TeV
```


```python
0.5 * (sqrt_snn(320000+90000) - sqrt_snn(320000-90000))
```




$\displaystyle 110.127117815583$




```python
def beam_energy(sqrt_snn):
    mass = (938.27 + 939.57) * 0.5e-3 # nucleon mass in GeV
    return (sqrt_snn * 1e3) ** 2 / (2 * mass) / 1e6 # beam in PeV
```


```python
beam_energy(14)
```




    104.37523963702976



c45cca845436aadfd651c1105d52e93419c9eb25
ipynb
Jupyter Notebook
From fixed target to sqrt_s_nn and back.ipynb
HDembinski/essays
a3070c10c6ca2a9c4b24eb89cba5ec5518e085dd
MIT
2020-04-07T07:29:46.000Z
2022-03-04T18:09:31.000Z
From fixed target to sqrt_s_nn and back.ipynb
HDembinski/essays
a3070c10c6ca2a9c4b24eb89cba5ec5518e085dd
MIT
2020-04-08T11:31:52.000Z
2022-01-25T02:01:34.000Z
From fixed target to sqrt_s_nn and back.ipynb
HDembinski/essays
a3070c10c6ca2a9c4b24eb89cba5ec5518e085dd
MIT
2020-05-16T14:35:06.000Z
2022-03-01T05:53:21.000Z
Qwen/Qwen-72B
1. YES
2. YES