# Intrabeam Scattering in the VLHC

#### 

#### Valeri Lebedev

### Fermilab

> VLHC mini-workshop
>
> SLAC, March 21-23, 2001.

##### Content

1.  Machine Parameters

2.  IBS theory

3.  IBS estimate

4.  Conclusions

VLHC Parameters

+----------------------------------+-----+-------------+-------------+
|                                  |     | **Stage 1** | **Stage 2** |
+----------------------------------+-----+-------------+-------------+
| Beam energy                      | *E~ | 20 TeV      | 87.5 TeV    |
|                                  | p~* |             |             |
+----------------------------------+-----+-------------+-------------+
| Luminosity                       | ##  | 10^34^      | 210^34^     |
|                                  | ### | cm^-2^s^-1^ | cm^-2^s^-1^ |
|                                  | # L |             |             |
+----------------------------------+-----+-------------+-------------+
| Magnetic field                   | *B* | 1.96 T      | 10 T        |
|                                  | ~0~ |             |             |
+----------------------------------+-----+-------------+-------------+
| Injection energy                 | *E  | 0.9 TeV     | 10 TeV      |
|                                  | ~in |             |             |
|                                  | j~* |             |             |
+----------------------------------+-----+-------------+-------------+
| Bunch spacing                    | *~  | 18.9 ns     |             |
|                                  | b~* |             |             |
+----------------------------------+-----+-------------+-------------+
| Circumference                    | *C  | 232 km      |             |
|                                  | *=2 |             |             |
|                                  | *R* |             |             |
+----------------------------------+-----+-------------+-------------+
| Revolution frequency             | *f* | 1294 Hz     |             |
|                                  | ~0~ |             |             |
+----------------------------------+-----+-------------+-------------+
| Number of bunches                | *N~ | 36943       |             |
|                                  | b~* |             |             |
+----------------------------------+-----+-------------+-------------+
| RF frequency                     | f~  | 477.9 MHz   |             |
|                                  | RF~ |             |             |
+----------------------------------+-----+-------------+-------------+
| Betatron tunes                   |     | \~214       |             |
+----------------------------------+-----+-------------+-------------+
| Momentum compaction              |     | 2.110^-5^   |             |
+----------------------------------+-----+-------------+-------------+
| Beta-function at IP              |     | 30 cm       | 50 cm       |
+----------------------------------+-----+-------------+-------------+
| Head-on beam-beam tune shift per |     | 1.810^-2^   | 1.810^-2^   |
| IP                               |     |             |             |
+----------------------------------+-----+-------------+-------------+
| Number of particles per bunch    | ##  | 2.510^10^   |             |
|                                  | ### |             |             |
|                                  | # N |             |             |
+----------------------------------+-----+-------------+-------------+
| Beam current                     | *I~ | 0.190 A     | 0.175 A     |
|                                  | b~* |             |             |
+----------------------------------+-----+-------------+-------------+
| RF voltage per turn,             | *V* | 50/50 MeV   | 50/50 MeV   |
| top/injection energy             | ~0~ |             |             |
+----------------------------------+-----+-------------+-------------+
| Synchrotron frequency,           | *f~ | 2.32/10.9   | 1.1/3.28 Hz |
| top/injection energy             | s~* | Hz          |             |
+----------------------------------+-----+-------------+-------------+
| Rms momentum spread,             | ~p~ | 1.5         | 0.          |
| top/injection energy             |     | /14.910^-4^ | 5/2.410^-4^ |
+----------------------------------+-----+-------------+-------------+
| Rms bunch length, top/injection  | ~s~ | 6.6/14.2 cm | 4.5/7.8 cm  |
| energy                           |     |             |             |
+----------------------------------+-----+-------------+-------------+

1.  **Intrabeam scattering theory**

If in the beam frame the longitudinal momentum spread is much less than
the transverse one

![](media/image1.wmf)

IBS calculations can be significantly simplified

#####  

##### The IBS growth rate for longitudinal degree of freedom

-   IBS transfers the energy from the transverse degrees of freedom to
    the longitudinal one and the growth rate can be approximated by the
    following formula

$\frac{d}{\text{dt}}\left( \theta_{{||}^{2}} \right) \equiv \frac{d}{\text{dt}}\left( \frac{\overline{p_{{||}^{2}}}}{p} \right) = \frac{e^{4}N_{i}}{4\sqrt{2}m_{p^{2}}c^{3}\gamma_{i^{3}}\beta_{i^{3}}\sigma_{\text{si}}}\left\langle \frac{\Xi_{||}\left( \theta_{x},\theta_{y} \right)}{\sqrt{\theta_{x^{2}} + \theta_{y^{2}}}}\frac{L_{C}}{\sigma_{x}\sigma_{y}} \right\rangle_{s}$
,

where averaging is performed along the beam orbit,

$\Xi_{||}(x,y) \approx 1 + \frac{\sqrt{2}}{\pi}\text{ln}\left( \frac{x^{2} + y^{2}}{2\text{xy}} \right) - 0\text{.}\text{055}\left( \frac{x^{2} - y^{2}}{x^{2} + y^{2}} \right)^{2}$,

$\sigma_{x} = \sqrt{\varepsilon_{x}\beta_{y} + D^{2}\theta_{{||}^{2}}}$,
$\sigma_{y} = \sqrt{\varepsilon_{y}\beta_{y}}$,
$\theta_{x} = \sqrt{\varepsilon_{x}/\beta_{x}}$and
$\theta_{y} = \sqrt{\varepsilon_{y}/\beta_{y}}$

> \- are the beam sizes and angular spreads along the ring, and
>
> \- *L~c~* - is the Coulomb logarithm.

-   In the smooth focusing approximation,

-   for equal horizontal and vertical emittances,
    $\varepsilon_{x} = \varepsilon_{y}$,

-   equal betatron tunes, $\nu_{x} = \nu_{y}$, and

-   small contribution of energy spread into the beam
    size,$D(\Delta p/p)\text{<<}\sigma_{x}$,

The momentum spread growth rate can be written in the following form

$\Lambda_{||} \equiv \frac{1}{\theta_{{||}^{2}}}\frac{d}{\text{dt}}\left( \theta_{{||}^{2}} \right) = \frac{e^{4}N_{i}L_{C}}{8m_{p^{2}}c^{3}\gamma_{i^{3}}\beta_{i^{3}}\sigma_{\text{si}}\varepsilon_{x^{\frac{3}{2}}}\theta_{{||}^{2}}}\sqrt{\frac{\nu_{x}}{R}}$
.

##### The IBS growth rate for transverse degree of freedom 

-   The heating of the longitudinal degree of freedom causes cooling for
    both transverse degrees of freedom;

-   Additional mechanism heats the horizontal degree of freedom

    -   At regions with non-zero dispersion, changes in longitudinal
        momentum change the particles reference orbits, which
        additionally excites the horizontal betatron motion,

$\frac{{d\varepsilon}_{x}}{\text{dt}} = \left\langle A_{x}\frac{{d\theta}_{{||}^{2}}}{\text{dt}} \right\rangle_{s}$
,

where
$A_{x} = \frac{D^{2} + \left( D^{'}\beta_{x} + \alpha_{x}D \right)^{2}}{\beta_{x}}$

Finally, one can write for the emittance growth rates

$$\begin{array}{r}
\begin{bmatrix}
\Lambda_{x} \\
\Lambda_{y}
\end{bmatrix} \equiv \frac{1}{\varepsilon_{x,y}}\frac{{d\varepsilon}_{x,y}}{\text{dt}} = \frac{e^{4}N_{i}}{8\sqrt{2}m_{p^{2}}c^{3}\gamma_{i^{3}}\beta_{i^{3}}\sigma_{\text{si}}\varepsilon_{x,y}} \\
\ \left\langle \frac{1}{\sqrt{\theta_{x^{2}} + \theta_{y^{2}}}}\frac{L_{C}}{\sigma_{x}\sigma_{y}}\begin{bmatrix}
2A_{x}\Xi_{||}\left( \theta_{x},\theta_{y} \right) - \frac{\beta_{x}}{\gamma_{i^{2}}}\Xi_{}\left( \theta_{x},\theta_{y} \right) \\
 - \frac{\beta_{y}}{\gamma_{i^{2}}}\Xi_{}\left( \theta_{y},\theta_{x} \right)
\end{bmatrix} \right\rangle_{s}\ \text{.}
\end{array}$$

> where

$\ \Xi_{}(x,y) \approx 1 + \frac{2\sqrt{2}}{\pi}\text{ln}\left( \frac{\sqrt{3x^{2} + y^{2}}}{2y^{2}}x \right) + \frac{0\text{.}\text{5429}\text{ln}(\frac{y}{x})}{\sqrt{1 + \text{ln}^{2}\left( \frac{y}{x} \right)}}$
.

-   In the smooth focusing approximation,

-   for equal horizontal and vertical emittances,
    $\varepsilon_{x} = \varepsilon_{y}$,

-   equal betatron tunes, $\nu_{x} = \nu_{y}$, and

-   small contribution of energy spread into the beam
    size,$D(\Delta p/p)\text{<<}\sigma_{x}$,

The emittance growth rate can be written in the following form

$$\Lambda_{x} = \frac{e^{4}N_{i}L_{C}}{\text{16}m_{p^{2}}c^{3}\gamma_{i^{3}}\beta_{i^{3}}\sigma_{\text{si}}\varepsilon_{x^{\frac{5}{2}}}}\sqrt{\frac{R}{\nu_{x}}}\left( \frac{2}{\nu_{x^{2}}} - \frac{1}{\gamma_{i^{2}}} \right)$$

2.  **Intrabeam Scattering Estimate for the First Stage**

![](media/image2.wmf)

3.  **Intrabeam Scattering Estimate for the Second Stage**

-   SR damping decreases beam sizes to the level where it will be
    stabilized by IBS or other heating mechanism

    -   SR damping time is \~ 2 hour

-   We need to prevent decrease of horizontal and longitudinal
    emittances to avoid stronger IBS and TMCI.

    -   Vertical beam size decrease is limited by increase in IBS for
        horizontal degree of freedom. That happens at the size aspect
        ratio,*~yx~* , of about 0.15.

        -   Vertical emittance needs to be kept at this level to prevent
            growth of horizontal emittance.

-   

4.  **Conclusions**

## IBS does not contribute into emittance growth for the first stage

## IBS is one of limiting factors for the second stage after the SR damping will cause significant cooling of beam emittances